Good morning, this is our last lecture on the topic of function approximation. In the previous lecture, we saw several ways of function approximation by the use of several families of Eigen functions of different stream level programs. And in particular, we also went through the Fourier series, which provides some additional facility other than what any family of Eigen functions of stream level problem would offer. Towards the end, we also discussed how to extend the idea of Fourier series to a function which is not periodic, which is defined over the entire real line and it is a non-periodic function. For that, we asked this question, how to apply the idea of Fourier series to a non-periodic function over an infinite domain. So, for that, we constructed the Fourier series over a finite interval of length 2 L and then we let the variable L increase and tend to infinity. And as a result, we arrived at this representation of the function which is the Fourier integral. Now in the Fourier integral, this is the coefficient available in the form of an integral. And similarly, this integral over an infinite domain of the continuous variable v. So, these two integrals are the coefficients. And the Fourier integral representation turns out to be a superposition of infinite such components with continuously changing value of e. Now in the case of the Fourier series, this integral was a summation. Now here, the summation becomes a sum of continuous components and therefore, you get an integral. And in the place of the two Fourier coefficients, we have got these in a similar manner in which, as we got it in the case of Fourier series. So, this Fourier integral can be now represented as this integral with the coefficients as coefficient functions. Earlier, these coefficients of these coefficients in the case of Fourier series were a n and b n that is they were indexed. Now these are continuous functions not for n equal to 1, 2, 3, 4 and so on, but p which varies continuously from 0 to infinity. So, in place of Fourier coefficients indexed with an integer variable n, we have got Fourier coefficients which turn out to be continuous functions of the frequency variable p. And these are in a way the corresponding amplitude functions a p and b p which are these two integrals. So, the only difference here is that this is an integral rather than a sum of terms for large number of n's and the coefficients are defined not for different values of n, but for continuous variable frequency variable which is p. Now using this itself, you can also represent this large double integral in a somewhat different manner in which you can push cos p x inside this integral and sin p x inside this integral and then you get this form of the Fourier integral expression which is called the phase angle form in which you find that cos p x cos p v plus sin p x sin p v is replaced with this and this turns out to become a straight forward double integral. Now from this double integral with a little more work, we can define an integral transform and how it is done is here. So, in this cosine, we can replace in this manner cos theta is e to the power i theta plus e to the power minus i theta by 2.  Now this term in the integrand can be omitted if we consider this limit also from minus infinity to infinity that gives us a more symmetric representation. To see that what we do is that we can substitute p equal to minus q. So, that means that for this integral if we try to separately evaluate this component, then in place of minus p we will have q here in place of d p we will have minus d q and therefore, the limit of this integral over p will become from 0 to infinity because of the sign change it will become minus infinity to 0. That is first of all it will become 0 to minus infinity and then because of the sign change we will have the limit reversed. So, now if we recognize this that the integral of this term f v into this turns out to be actually the integral of this same thing over minus infinity to 0 then the addition of these two terms is the addition of an integral from minus infinity to 0 and then from 0 to infinity. That means this integral and this integral together will give the complete integral from minus infinity to plus infinity and that gives us the complex form of the Fourier integral in which it is a single exponential function here with f v and we have this double integral. And then the in the complex form of the Fourier integral this term also rather than cosine and sine has become this exponential function with imaginary index and this c p the amplitude function has become the integral that you get when you take only f v into e to the power minus i p v and plus i p x remains here. So, from the original Fourier integral also we got similar a p and b p now in a complex form the two coefficients combined together to give us the complex coefficient function. Now this actually holds the root of the definition of the Fourier transform let us see that directly here in this chapter we will not go into great detail, but we will just see how the integral transform called Fourier transform gets defined from the complex form of the Fourier integral. So, with a different notation here in place of x we have got p here in place of p the frequency variable we have got w here and so on otherwise it is the same expression and the 1 by 2 pi that factor has been split into two parts one is here 1 by root over 2 pi and the other is here. Now here this is the same earlier complex coefficient function and, but with this factor here rather than the same function rather than the earlier factor that we had. Now this can be considered as the function representation in which we are talking about a composition that is addition the sum of an infinite number of functions which are these in the form e to the power minus w t by root over 2 pi that is e to the power minus w t divided by root over 2 pi and the coefficients of these superposition coefficients of this superposition turns out to be this that is the amplitude function and the amplitude function varies continuously with the frequency. Now this function representation is in a way the linear combination the composition of a large number of frequency components the frequency components are here and the corresponding coefficients are here corresponding amplitudes are here and this amplitude turns out to be a function of the frequency that you get from here. So this function of frequency which turns out to be the amplitude is the Fourier transform. So amplitude of a frequency component that is this bracketed term here and that is the Fourier transform and this is a function of the frequency variables. So that means a waveform has large number of components of continuous frequencies and now for every frequency this gives us the amplitude value and when with this amplitude value and this frequency we construct the term that that component completely we have got this and such terms infinite number distributed continuously over the frequency variable is the complete function. Now if we rather have this Fourier transform in our hand then that Fourier transform sitting here with e to the power i w t multiplied and d w put here and integrated like this that is in this entire place if we put this Fourier transform then we have got this and this integral will recover the original function that is the inverse Fourier transform. The way we defined earlier Laplace transform and inverse Laplace transform in a similar manner we have got this Fourier transform and inverse Fourier transform Fourier transform and inverse Fourier transform these are quite symmetric in their look here you have got e to the power minus i w t to be multiplied and then integrated here you have e to the power i w t multiplied and then integrated. So this gives you very similar expressions for evaluating the Fourier transform and the inverse Fourier transform. Now these transforms are extremely useful in the solution of differential equations partial differential equations and also in the field of signal processing we will not go into the detail of that because the main agenda of this lecture is another kind of function approximation in which the objective is different than all the function approximation techniques that we have discussed till now. Earlier when we were discussing the interpolatory approximation of functions in that context we studied we discussed three criteria of approximation of functions one was the interpolatory approximation which we studied in that particular lesson and the other was least square approximation which we have studied in several sections in this course. First the linear least square problem which we studied in the module of linear algebra and then we studied the non-linear least square problem which we studied in the context of non-linear optimization and finally the same least square approximation we have been discussing till recently which are least square approximation based on the error value developed through integrals rather than sums of finite number of components finite number of error values. So that least square approximation we have studied already. Now the third criteria of approximation of a function is a minimax approximation in which the objective is not the sum of square all over the sum of squares of errors all over the domain but the objective function to be minimized is the maximum error. So therefore this kind of an approximation is called minimax approximation and one particular family of Eigen functions that is the solutions of the Chebyshe equation and for the Chebyshe polynomials give us specific interesting properties which are directly useful in the case of minimax approximation and therefore this topic of minimax approximation we start with a discussion of Chebyshe polynomials. This differential equation resembles the Legendre equation to a great extent in the case of Legendre equation we had a 2 here minus 2 here we have got minus 1. Now in place of this minus 1 if we had minus 2 then we found that this minus 2 x turned out to be exact derivative of this and therefore these two terms together in the case of Legendre equation turned out to be the perfect differential coefficient of 1 minus x square into y prime. Now in this case that 2 is missing so it will this much right now is not in the self adjoint form right now not in the stream variable form. So if you can if you divide the entire equation with this square root of 1 minus x square then you will find that here then you will have square root of 1 minus x square and here you will have minus x by square root of 1 minus x square and in that case the terms together will be in the self adjoint form and together they will give us the derivative of this combined term and through the division here you will get this. Now there is one interesting point here that if we change the independent variable here and make this substitution x equal to cos theta then this differential equation quite easily boils down to this differential equation is the same differential equation which we saw in the case of Fourier series. Now from here we can very easily see that the solutions of this will be cosine n x and sin n x they will be the linearly independent solutions of this differential equation. Now in particular the solution sin n x will be very complicated functions but the other family of solutions cosine n x that is for different values of n cos 0 that is 1 then cos x sorry not x but theta. So the solutions of this will be cosine theta and sin theta. So the term sin theta when finally expressed in the suffix back by putting theta equal to cos inverse x will turn out to be quite complicated function. On the other hand the other solution that is y equal to cos theta will turn out to be polynomial sin x. So you can see that so that is why we are particularly interested in the polynomial solutions of this differential equation and note that this differential equation also will define a singular stream level problem just like the Legendre equation because here the function r x which appears here is 0 at x equal to minus 1 and at x equal to 1. That means you do not need any boundary conditions to define the stream level problem. So this differential equation will define a singular stream level problem over this interval Legendre equation also did the same thing. Now we are particularly interested in the polynomial solutions of this particular stream level problem and one family of solutions of that will turn out to be polynomial solutions. Now the way we discovered this fact in the case of Legendre polynomials it is possible to discover the same fact for Chebyshev polynomials also in the same manner by consideration of the series solution and noting that for integer values of n the series terminate and get you polynomial equations polynomial functions. The same conclusion will be reached through this arrangement in which we replace x with cos theta and solutions of this turn out to be cosine n theta sin n theta. We discard that sin n theta because that is not going to give us polynomial solutions but cosine n theta will give us polynomial solutions and for all values of n cosine n theta will have a value 1 when we take x as 1. So the closed form expression for that particular family of solutions which are going to turn out to be polynomial solutions is this that is t n x is cosine n theta theta is cos inverse x. Now if we put n equal to 0 then we get t 0 which is cos 0 that is 1. If we put n equal to 1 then we get cos cos inverse x which is x. If we put n equal to 2 then we have got cos 2 theta we know that cos 2 theta is twice cos square theta minus 1. Similarly, cos 3 theta will be 4 cos cube theta minus 3 cos theta and so on. So we can also very easily prove that the family this family of functions t 0 t 1 t 2 t 3 etcetera will satisfy this three term recurrence relation that is t k plus 1 will be twice x t k minus t k minus 1 which means that after evaluating the two of them we can evaluate others in terms of polynomials directly rather than writing multiple angle formulas of cosines from this recurrence relation. What other properties these polynomials have these solutions of the stream level problem have. Now that with a little exercise we can establish a lot of properties of these polynomials. In the exercises of chapter 39 and chapter 40 of the textbook we have the steps to establish many of these properties. So it is expected that the student will go through those exercises and execute those steps to learn this the learn the derivation of these properties first hand. Here we will just summarize the immediate observations immediate properties for proceeding further. Coefficient in a sebaceous polynomials are all integers which is different from the Legendre polynomials. In particular the leading coefficient of t n will be t 2 to the power n minus 1 for even n that is for t 2 t 4 t 6 t n x is an even number of n function because they will have only the even powers. While for odd n it is an odd function this is the same as the Legendre polynomials in the case of Legendre polynomials also we noticed this property. All the sebaceous polynomials will have the value 1 at x equal to 1 that is obvious because at x equal to 1 cos inverse x will be 0. So for all values of n n into 0 is 0 so cos of 0 will be 1. So value of this all these polynomials for x equal to 1 will turn out to be 1 that means all of them are bunched at 1. Then we find that at minus 1 the value will turn out to be this minus 1 or 1 depending upon whether n is odd or even this also is same as Legendre polynomials. Legendre polynomials that means also had this property and this is a very interesting property that is the value the absolute value of t n x over this interval is always less than equal to 1. This also is there in the case of Legendre polynomials. Again zeros also the distribution of zeros of a sebaceous polynomial also shares a similarity with Legendre polynomials. The zeros of the sebaceous polynomials are all real and lie within this interval. But the locations of the zeros of sebaceous polynomials polynomials turn out to be turn out to have very particular significance there at these locations. And these locations are called sebaceous accuracy points why that we will explore further. Further another interesting thing is that the zeros are interlaced by the zeros of t n the n th sebaceous polynomial are interlaced by those of t n plus 1. That means the n th degree sebaceous polynomial will have n zeros and every zero of the t n function will turn out to be n plus 1. And closed by two consecutive zeros of the next order sebaceous polynomials. This interlacing property is also interesting. Now we find that these three properties except this part the sebaceous polynomials share with the Legendre polynomials. And this next property extremar of sebaceous polynomials have a very unique property which no other familiar functions possesses. And that is all extremar of the sebaceous polynomial t n are of equal magnitude magnitude is unity. And they alternate in sign and occur at these values of x one maximum next minimum next maximum next minimum all these maximum and minimum of the same absolute value 1.  So, this is a very unique property and this property is called the minimax property. We will see the significance of this property. Orthogonality and norms. So, this is the integral here is equal to 0 for m not equal to n. That is the outcome of the result on the stream level problem here you see that the function p x is 1 by root over 1 minus x square. So, that is why two distinct solution that is two solutions of this stream level problem corresponding to different values of n that is different Eigen values distinct Eigen values will turn out to be orthogonal with respect to the weight function 1 by root over 1 minus x square. So, the statement of orthogonality of Shobishi polynomials turns out to be like this. So, this is the statement on orthogonality and the definition of norms will come when we put m equal to n. And for m equal to n which is not 0 we get it pi by 2 and for m equal to n which is 0 we get pi. This is similar to what we found in the case of the Fourier coefficients. Now with these properties we can have a close look at one such sample Shobishi polynomials between minus 1 and 1. So, between minus 1 and 1 the Shobishi polynomials varies like this. You see the zeros and extrema of this turn out to have a uniform distribution over the theta variable say for theta equal to 0 x is 1. So, the value of Shobishi polynomials is 1 then for theta equal to 30 degree pi by 6 x is here cos of cos 30 degree that is root 3 by 2. So, at that we have got a 0 of the Shobishi polynomials. So, then 0 30 degree minus 60 degree at theta equal to 60 degree we have got this negative value minus 1. So, here was 1 extrema value maximum next minimum. So, at 30 degree at 0 we had maximum value at 30 degree we had a 0 at 60 degree we have the minimum value minus 1 then at 90 degree x equal to 0 we have got t 3 x also 0. So, that is a second 0 from this side then again at 120 degree we have got the maximum value at 150 degree we have got a 0 at 180 degree we have got this extrema value. So, the zeros and extrema are distributed like this. The extrema values have the same absolute value 1 these maxima are at value 1 these minima are at value minus 1. This is a distinct property which is not shared by any other familiar functions and as you compare it with say Legendre polynomials you will find that this is the graph of the 8th order Legendre polynomials and 8th order Shobishi polynomials super imposed super imposed. So, this is the graph of p 8 Legendre polynomials and this is the graph of the Shobishi polynomials. You can see that the Shobishi polynomials has this equal ripple property that is equal amplitude oscillations. So, from here it goes all the way down in the case of Legendre polynomials as you proceed towards the center of the domain near 0 the amplitude tends to come down compared to here it comes down towards the center. In the case of Shobishi polynomials every extrema is at the extreme value of minus 1 or 1. So, 8th order polynomial it has got going down going up going down like 8 such trends you have and all the zeros you can see here 1 2 3 4 5 6 7 8 and they are distributed uniformly over the theta variable cos inverse x. So, being cosines and polynomials at the same time Shobishi polynomials possess a wide set of interesting properties which have this unique application in minimax approximation. The most striking property is this equal ripple oscillations leading to the minimax property. We will see this in two stages first what is this minimax property of Shobishi polynomials. The important result is that among all polynomials p and x of degree n with the leading coefficient equal to unity why this is needed because if we want to compare two polynomials then we can make no comparison if one of the polynomials can be multiplied by an arbitrary number. So, we need to fix something for both the polynomials to make a worthwhile comparison. So, we say that we will fix the coefficient of the largest degree term that is we are talking about monic polynomials. So, the leading coefficient will have unity that means x to the power n plus a 1 x to the power n minus 1 plus a 2 into x to the power n minus 2 and so on. So, the leading coefficient is 1. So, among all such polynomials of degree n with the leading coefficient unity that is among all monic polynomials of degree n this is the one which has the least deviation from 0. Now, why this factor because we already know that T n x has the leading coefficient which is 2 to the power n minus 1. So, to make it monic we must multiply it with 2 to the power 1 minus n. So, the odd divide with 2 to the power n minus 1. So, therefore, the monic version of the n th order is this and among all monic polynomials of n th degree the claim is that this one will have least deviation from 0 over this interval. So, mathematically we can say that maximum over this domain this interval of absolute value of p n x is always greater than equal to the corresponding value for this particular function. So, all other polynomials will deviate more from 0 and this will deviate least and that least deviation turns out to be this. Now, this equality part is easy to establish because we already know that T n x will deviate between minus 1 and 1. That means the absolute value of the deviation of T n x is 1. So, absolute value of the deviation of this factor into T n x will be this factor itself. This unit equality part is quite obvious for the inequality that is all other polynomials will deviate more we need to do something. So, for that we first assume that suppose there is some other polynomial other than this which has even less deviation. So, if there exists some monic polynomial p n x of degree n which deviate less which deviate less than this. If we assume that like that then what we can do is that we consider the values of this candidate polynomial at the extrema of the Chebyshev polynomials. Now, already know that the n th degree Chebyshev polynomial has n plus 1 alternate extrema positive negative positive negative and so on. So, each with this absolute value that we already know. So, at those extrema points of the Chebyshev polynomial we consider the values of this. Now, these points are giving us the maximum deviation of this which is 2 to the power 1 minus n and we know we have assumed that this candidate function deviates less. So, if it deviates less then it will never be able to cross the extrema value of this. So, that means that if we consider this difference then this part in the difference will always dictate the sign of the difference polynomial q n x. So, if this is positive then whatever be the value of this the difference will be positive. If this is negative then whatever be the value of p n the difference will be negative because this fellow's value has been assumed to be less than the extrema value of this and we are evaluating at the extrema points. Then this difference polynomial will have the same sign as the first part. Now, we already know the nature of the signs of this in the sequence that is n plus 1 locations n plus 1 alternating extrema this function has. So, its sign will be alternating at these n plus 1 locations positive negative positive negative and like that. So, with alternating signs at n plus 1 locations in sequence p n x and therefore q n x also will have n intervening zeros. So, because this q n x has the same alternation of sign as this because this one is going to dictate the sign of q n. So, q n with alternating signs at n plus 1 locations will have n intervening zeros positive here negative here. So, in between q n x must have a 0. So, between n plus 1 extrema of alternating sign it will not extrema of this difference function, but with alternating signs at n plus 1 locations in sequence it will have n intervening zeros. But then that is something very awkward because both of these were monic polynomials which means that in both of these cases the coefficient of x n was unity and in that case in the difference the x n term will get cancelled and this polynomial is actually an n minus 1 degree polynomial. But then we have just now found that it has n intervening zeros how can an n minus 1 degree polynomial have n zeros. So, this leads to a contradiction which means that this assumption of a monic polynomial of less variation less deviation is certainly wrong. So, through contradiction we get this result that among all polynomials of degree n it is the Chebyshev polynomial the scaled version of it to make it a monic polynomial deviates least and this turns out to be the deviation. Now how do we utilize this property? Let us explore with a linear combination of the Chebyshev polynomials will have this series which is called the Chebyshev series for which we can construct the coefficients a 0 a 1 a 2 a 3 etcetera as we have earlier studied that is how to construct the generalized Fourier coefficient for this Chebyshev series. There are the Chebyshev coefficients for which we have these expressions. Now typically we would not be able to construct this infinite series. So, we typically try to represent a function with a truncated series only up to a finite term and this approximation is called the Chebyshev economization. We are representing the function in an economical manner now the leading error term in that case will be the next term that is up to the term T n we are already incorporating in this truncated series. The next term which will be the leading error term will be this and we know from the foregoing discussion that among other among all the candidates for the polynomials this will deviate least from 0 over this interval and therefore it is qualitatively similar to the error function. Then the question arises that with this how to develop a Chebyshev series approximation? Do we try to find out so many Chebyshev polynomials and then evaluate the coefficients and then construct this series the truncated series while that could be an option. The good news is that even that much computation is not really needed to be done there is a little shortcut to do that. So, first of all this is a little bit of pre processing involved if the required domain of the function description function representation is not minus 1 to 1, but it is a to b then we always can scale it scale the variable T to x with this in which x belongs to this. Then onwards we work with this interval of x the economized series this one which we are constructing that will give the minimax deviation of the leading error term which will deviate least and that deviation will be the will have the equal ripple property which every Chebyshev polynomial has. Now, since we are discussing with reference to the leading error term now for a moment if we concentrate on this leading error term only and pretend that this happens to be the actual error and assuming this to be the actual error we know that there will be n plus 1 values of x where this actual error officially turns out to be 0 why because T n plus 1 has n 0s. So, there are n values of x from minus 1 to 1 in that interval of interest in which this function is 0. So, these are at the 0s of the n plus 1th Chebyshev polynomial. So, at these values of x we find that we if we take the exact actual function value at these points then there will be the correct values through which we can interpolate a function. So, that means that we can take these points these values of x and at these values of x which turn out to be the 0s of this we evaluate the function we have got the values of the function at n plus 1 values of x. And we know that if this happens to be the actual error then the polynomial that will be interpolated through these n plus 1 points will deviate least. So, that means that after we have got the 0s of this function this Chebyshev polynomial of n plus 1 is d z and got the Chebyshev accuracy points the so called Chebyshev accuracy points. Then the polynomial interpolated through these values at the Chebyshev accuracy points through any other means possibly with the Lagrange interpolation itself will give us the Chebyshev series approximation in terms of the actual function that we developed. Because here we were seeing that this series will have 0 error at those values which are those values of x which are roots of this 0s of this. And now we are telling that through these same points with we can construct a polynomial which is through Lagrange interpolation. Now n plus 1 values of an n s degree polynomial fixes the polynomial uniquely. So, therefore whether we construct the series by finding out the Chebyshev coefficients a 0 a 1 a 2 etcetera or whether we construct that by Lagrange interpolation over those points where we know the correct value of the function it will lead to the same polynomial. And therefore from the foregoing theory of Chebyshev polynomials we can just take at which points we give the correct values of the function and then from the values at those n plus 1 points we simply construct the polynomial through Lagrange interpolation. So, this is called Chebyshev Lagrange approximation. Now this much on approximation with Chebyshev polynomials themselves. Now the minimax property goes beyond because here this assumption has worked that the leading error is the important component of the error. But that will not be always the case because in the actual complete series there could be components there could be contributions from t n plus 2 t n plus 3 and so on. So, if those components those contributions are significant then even though the approximation resulting from Chebyshev Lagrange consideration gives us qualitatively similar error values. But still it will not be the perfect minimax approximation even the perfect minimax approximation perfect minimax approximation we can construct and that is based on this very important property very important theorem of by Chebyshev that is before that make note that the situations in which the minimax approximation is desirable are those in which we want to develop the approximation once and keep it for use in future. A priori we do not know at which point the function value will be needed. And therefore it will be a good idea if we can say something regarding the uniform quality control that the function evaluated by this method will never be too long too much mistaken that is there is a limit on the maximum deviation that it may have. So, minimax approximation gives us that guarantee that the maximum deviation is limited the maximum deviation is minimized. And the interesting thing is that that approximation is going to have this minimax property which has constant amplitude of the variation. And this is the Chebyshev minimax theorem that is how do you recognize that a particular represent has this minimax property. And criterion for recognizing that minimax approximation is the same as the maximum deviation going to be same in absolute value at all the extrema that is equal ripple oscillation is directly related to the minimax property. So, the Chebyshev minimax theorem says that of all polynomials of degree up to n a polynomial p x is the minimax polynomial approximation of a function that is it minimizes this maximum deviation. If and only if there are n plus two values of x where this difference assumes alternate maxima and minima. So, where that this difference takes its extreme values of the same magnitude and alternate sign that means the exact properties that we saw in Chebyshev polynomial that is it has extreme maximum and minima values of the same absolute value 1 1 minus 1 1 minus 1 they are they were the extrema. And that actually confers on the Chebyshev minimax property. Similarly, for any approximation the approximation being minimax is equivalent to its equal ripple characteristics. In the following we will not really have a formal proof of this theorem, but we will in a way in a schematic manner we will see the line of proof which also gives us the seed of an algorithm to construct that minimum minimax polynomial approximation. Suppose you have got a function f x and for which you have got a polynomial approximation of the fourth degree which is the polynomial p x and the error function is f x minus p x. Now, the theorem says that p x is a minimax polynomial approximation of f x if the error function f x minus p x has n plus 2 that is 6 extrema of equal magnitude and alternating sign. So, d minus d d minus d d minus d 6 extrema of same magnitude and opposite signs if f x minus p x has that characteristic then p x is the minimax polynomial approximation of f x this is the statement of the Chebyshev theorem. Now, suppose the polynomial that we have constructed does not have this property that is it lacks that property only by a little that is 5 extrema are of the same value d minus d d minus d this particular extrema is not at minus d, but at a little above that is this is not really the farthest extrema value that we were looking for that means here in this case there is a gap. So, in that case this is not the minimax polynomial approximation. Now, if there is any such gap now this idea of utilizing any gap to reduce the deviation at the other extrema gives us the conviction that if not a formal proof of the minimax theorem and also in a way a very fruitful algorithm. Say suppose for being the minimax polynomial approximation this extrema had to be here at value minus d now there is a gap it does not reach here. So, we say that we might increase this deviation that is rather than this value we can increase the deviation that is take this minimum even lower and at the cost of this deviation we try to reduce the deviation at other extrema then it will reduce the maximum value of the error. So, for that what we do if this gap is of magnitude epsilon then we construct a polynomial with largest possible value epsilon by 2. So, that means it is bounded within y plus minus epsilon by 2 we consider a polynomial of that type which is bounded in its value within plus minus epsilon by 2 that kind of a polynomial we can call as delta p x that is the polynomial by which p x has to be changed. And we impose the condition that this delta p x other than having no value beyond this plus minus epsilon by 2 band is positive here at this extrema negative here at this extrema then positive here at this extrema positive here at this extrema and again negative here at this extrema. So, at all the 5 extrema minimum maximum maximum at these maxima we are asking for positive positive positive and positive values of delta p and at the 2 minima here and here we are asking for negative and negative values for delta p. Now note that for satisfying these requirements what delta p has to do delta p has to cross how many times from positive to negative to positive and then finally negative here. Since the largest magnitude extrema here there is a pair of continuous maxima without there being a minimum in between between this and this this point and this point m and n the function the polynomial delta p did not have to change its sign. So, it had to change its sign once here second time here and third time here. So, thrice changing sign is perfectly fine for a 4 degree polynomial. So, we can certainly construct a 4 degree polynomial which is positive here negative here positive positive and negative here. So, that kind of a delta p we can construct and we can have a hold over its maximum value also because if its maximum value turns out to be larger than epsilon by 2 then you can multiply the polynomial throughout by a small number to bring it within epsilon by 2 and this limiting of its absolute value within epsilon by 2 will ensure that whatever damage the polynomial p x is cause through the addition of delta p will not damage this point too much beyond as other values come closer as other values come closer to 0 at at other extremum points. This one will not go beyond epsilon by 2. So, the best improvement at other point that is possible is epsilon by 2 and even after the watch damage at this point it will not go beyond epsilon by 2. So, that means through this exercise we can develop a suitable delta p which when added to the current p will give us a new polynomial in which all these extrema will be reduced in their magnitude all these extrema will be reduced in magnitude and perhaps this extremum will be increasing in its magnitude, but never beyond what will be the resulting value of the other extrema. That means that any gap like this can be utilized to reduce the deviation at the other critical extrema. Now, consider the same situation if this point were also here. In that case in order to reduce all the extrema we would need the values to come down at these three maxima and to come up at these three minima if there were no gap, but that would not be possible because in that case delta p would be required to be positive negative positive negative positive negative. A fourth degree polynomial can never show never have so much of so many of oscillations. So, therefore, if there is no gap like this then that will mean that we cannot improve the approximation further. On the other hand any gap that exists like this we can always utilize that gap to reduce the variation at the other point. So, this is the schematic demonstration of the truth of the minimax polynomial. In other standard books on approximation theory you can have a look at the formal proof. Incidentally this same reasoning turns out to be the basis of the construction of the minima minimax polynomial approximation and that algorithm is called Ramesh algorithm. The general Ramesh algorithm is actually for rational approximation, rational function approximation, but the same theme can be utilized for approximation with polynomials also and for this minimax polynomial approximation we will find that the Chebyshev polynomial turns out to be a very good starting point with the help of Chebyshev Lagrangian approximation as we have seen sometime back. So, you will find that in the light of this general minimax theorem you will find that T n plus 1 x the n plus 1 th degree Chebyshev polynomial which we considered earlier was actually qualitatively similar to this complete error function with its equal ripple properties. Now, this completes our module on approximation theory. Next couple of lectures we will devote on a quick recapitulation of the very important partial differential equation and after that couple of lectures on partial differential equations we will consider complex analysis. Thank you.


The new scramble for Africa. What do they really want from Africa? The first notable rise in foreign interest in Africa, known as the scramble, began in the 19th century when European invaders divided up the continent and grabbed Africa's land. The second was during the Cold War, when east and west fought for the support of newly independent African states. While America backed regimes that professed to promote capitalism, the Soviet Union supported tyrannical Marxists. Currently, there is a third less harmful surge. According to the UN, there will be more Africans than Chinese by 2025, which has attracted the attention of outsiders who understand the significance of and continuous growth of the continent. Around the world, corporations and governments are seeking to improve their diplomatic, strategic and commercial ties. The amount of involvement abroad is unprecedented. Between 2010 and 2016, more than 320 embassies were established in Africa, possibly the largest embassy construction boom in history. Additionally, there is a strengthening of military ties. The United States and France are providing technology and military support for the fight against jihadism in the Sahel. China presently supplies more weaponry to sub-Saharan Africa than any other country, and it also shares defense technology with 45 other countries. Russia has signed 19 weapons contracts with African countries since 2014. Arab countries with large oil reserves are building bases on the Horn of Africa and using mercenaries from Africa. This, in our opinion, is the new race for Africa. Major superpowers like America, China and Russia all desire to rule Africa at the present time. The race for Africa right now isn't about selling African labor to other countries. Instead, it's about how the European Union, the United States, Russia, China and other global powers are battling to make deals with Africa in order to expand their geopolitical influence on the continent. Stay with us to learn more, and if you're just joining us, we'd appreciate it if you'd like, share and most importantly comment so we can hear what you think about the latest race for Africa. Who are these powers that are scrambling for Africa and how do they go about it? The European Union. The colonization of Africa by European nations in the late 19th and early 20th centuries is known as the Scramble for Africa. During this time, European nations including Britain, France, Germany and Belgium fought to expand their imperial holdings by claiming colonies in Africa. As European powers forced their political and economic systems on African nations, this competition frequently resulted in war and exploitation. The Scramble for Africa had a huge impact on the continent, influencing current political boundaries as well as persistent economic and social problems. Native populations were frequently forcibly removed during European colonization and European political and economic institutions were imposed on African nations in addition to the extraction of natural resources. Along with significant upheaval and eviction, this resulted in the depletion of land resources and cultural traditions. In order to assist Morocco in making the transition to a green economy and the digital era, the European Union plans to invest 1.6 billion euros in the nation. When the head of the European Commission visited Morocco, this idea was unveiled. The European Union made an unusual choice in this case because it rarely invests heavily on infrastructure outside of its boundaries. But the EU is deliberately attempting to change the balance of power not only in Morocco but also throughout Africa. The European Union's planned Global Gateway Program, which will see the EU invest up to 150 billion euros to help the financing of green energy and digital infrastructure projects in Africa, kicks off with the 1.6 billion euros promised to Morocco. The project's entry into the game of global geopolitical power is a given. The topic of why the European Union is making such a large expenditure during this time of economic unpredictability and the primary drivers China remain after this massive investment. China has been able to increase its strategic influence on the continent through its belt and road plan, which involves significant infrastructural investments. European powers have continued to exercise political and economic influence over African countries in the years after the scramble for Africa. Through its economic policies and development aid programs, the European Union has significantly influenced African policy. Although some have argued that the European Union's policies do not effectively meet the needs and concerns of African countries, the European Union has come under fire for its attitude to Africa. Countries like France have been criticized for being exploitative, and the French government is using its colonial currency to fund a modern-day slavery program in its former colonies. In general, both in Africa and around the world, the legacy of the scramble for Africa is still being felt. For the continent and its people, the effects of European colonization, including ongoing economic and social problems, remain a significant challenge. It will be crucial to solve these issues and make sure the rewards of engagement are distributed fairly as the European Union and other global powers continue to engage with Africa. As we ride on, please like and share this video. In order to join the Africa Reloaded community, subscribe to our channel as well. Enable notifications to receive alerts whenever we upload new videos. Russia Russia and Africa have been involved since the Soviet era, when Russia provided military and economic assistance to many African countries. Recent Russian engagement in Africa has been primarily driven by strategic and economic factors. Russia has worked to deepen its economic relations with African countries through investments in natural resources like oil and minerals, as well as the shipment of weapons and other military gear. Russia's efforts in Africa have drawn criticism, and some detractors have voiced concerns about possible detrimental effects on the economy and communities of Africa. In the case of Russia's armaments trade with Africa, for instance, there have been claims of corruption and illegal activity, as well as worries about the viability of African economies that are highly reliant on the sale of natural resources. Some people have also condemned Russia's involvement in African crisis, claiming that it has fueled unrest and violence. Despite these reservations, Russia's presence in Africa has increased recently. To improve economic and political ties with African leaders, Russian President Vladimir Putin organized a meeting in Sochi, Russia in 2019. Additionally, Russia has attempted to increase its military footprint on the continent of Africa by taking part in peacekeeping missions there, such as those in the Central African Republic and Sudan. Russia's interactions with Africa are intricate and varied overall. It will be crucial to keep a careful eye on Russia's activities in Africa as they develop and make sure they serve the interests of both Russia and Africa. Before we go any further, we kindly ask that you like and subscribe to our channel because this is the only way it will develop, and we will be able to make more engaging videos for you. United States of America The United States was the first country to openly denounce Chinese engagement in Africa. Washington is starting to realize that Africa has a big economic and investment potential. The December 2022 U.S. Africa summit is a sign that the United States is starting to wake up. The United States engagement in Africa spans many years, and it has changed as a result of its presence there. The United States has prioritized advancing economic growth and development, bolstering democratic institutions, and resolving security issues in Africa in recent years. Through programs like the African Growth and Opportunity Act, the U.S. has helped African nations with their development while simultaneously attempting to boost commerce and investment. Along with providing aid for economic development, the United States has also worked to alleviate Africa's security issues. This has included constructing military outposts as in Djibouti, participating in peacekeeping missions and counterterrorism initiatives, and giving African nations military aid and training. The United States has also contributed to promoting stability and peace on the continent through mediating disputes. Although some have argued that the U.S. policies do not sufficiently meet the needs and concerns of African countries, the U.S. has come under fire for its approach to Africa. Instead, the U.S. is attempting to impose its social norms and policies on Africans. Attempting to persuade African officials to accept and legalize practices that Africans do not deem appropriate for their continent. Other global powers like China and Russia, which also wanted to expand their influence in Africa, presented competition for the U.S. Dr. Erichana Cheyenne Bariquil, a former ambassador of AU to the United States, thinks that the United States is losing the argument because it is disrespectfully attempting to direct Africans and their governments. She contends that there was no agenda when African leaders were invited to the U.S. Africa summit. They simply went to hear what the United States had to say. In general, the United States keeps a substantial presence in Africa, and over the coming years, its role in the continent is expected to evolve. In order to advance its interests on the continent, and ensure that any decisions it makes benefit both the U.S. and Africa, it will be necessary for the U.S. to take into account the viewpoints and needs of African nations. The People's Republic of China Some have referred to the recent expansion of China's influence and investment in Africa as a scramble for Africa, referring to the 19th and early 20th century colonization of the continent by Europeans. Although there may be tensions between China and other countries with interests in Africa, the term scramble implies rivalry or conflict, whereas China's presence in the continent is primarily motivated by its own economic and strategic interests. China invests in natural resources like oil and minerals, builds infrastructure like roads, railroads, and ports, and offers development aid in the form of loans and grants. These initiatives have drawn criticism, and some detractors have expressed worries about possible detrimental effects on African economies and societies. Interaction between China and Africa extends back to the Cold War era. Recently, China has taken a number of steps to deepen its political and economic ties with African countries, including trade agreements, monetary investments in infrastructure projects, and humanitarian aid. One of the most noteworthy examples of China's growing involvement in Africa is the Belt and Road Program, a huge infrastructure development project that aims to connect Asia, Europe, and Africa through a network of roads, railways, and ports. China has agreements with various African nations through the program to build roads, railroads, and other infrastructure projects, as well as to offer financial support. Kenya, Egypt, and Ethiopia are a few of the nations that have agreements with China under this initiative. One of the main critiques levied against China is the fact that its involvement in Africa is largely focused on boosting Chinese interests rather than the development of Africa. Critics claim that Chinese corporations frequently exploit resources from Africa and that Chinese infrastructure projects frequently use Chinese labor and materials without sufficiently boosting the local economy. In addition, some people have raised concern over the conditions of the loans and grants given by China, as well as the possibility of China becoming a debtor for African nations. Africa has faced both chances and challenges as a result of China's scramble for Africa. While Chinese investment has increased infrastructure and access to capital, it has also raised questions about the sustainability of African economies in the long run and the possibility of exploitation. As China's role in Africa grows, it will be vital to allay these concerns and ensure that the benefits of Chinese investment are dispersed fairly. Turkey Additionally, Turkey is expanding its influence across the continent. The increase in Turkish missions on the continent from 12 in 2009 to 44 today reflects the president's diplomatic strategies. Erdogan has traveled to 28 different African nations over his 38 trips since taking office, making him the world leader with the most trips to the continent. The trade ties between Ankara and the African nation have greatly improved as a result of this diplomatic endeavor. Over the previous 20 years, commerce between Turkey and Africa expanded from $5.4 billion to $25.3 billion in 2017. 19 African nations, including Morocco, Nigeria, Ethiopia, Ghana, Mali, Algeria, Djibouti, Senegal, Kenya and South Africa, received military aid from Turkey. India India is another country that has been covertly boosting its economic clout in Africa. The country is currently the third largest trading partner on the continent and has recently raised the number of its diplomatic missions there. Due to its common history as a colonial power, the usage of English and the presence of more than 3 million Indians in Africa, India believes it has a competitive advantage against China. Additionally, New Delhi claims that in contrast to China, which regularly imports laborers from its own country, Indian projects in Africa lay a great emphasis on hiring local people. There is little doubt that the U.S. perceives China's and Russia's expanding influence in Africa as a danger. Because it affords them more options, this is the time for African nations to embrace the competition for influence. By leveraging its resources and its 1.2 billion person market, the majority of whom are young people, Africa has the opportunity to accelerate its growth. To ensure a better future for the next generation, African youth must emerge from their slumber and embrace these chances right away. We need reliable contributions and partners. Africa again declines the opportunity to act as a venue for the confrontation of Chinese-American and European conflicts. If you liked the video, don't forget to subscribe to Africa Reloaded to see more intriguing videos on the continent. To receive alerts when we upload a new video, don't forget to switch on notifications. Don't forget to leave a comment on this video.


Thank you for staying with us. You're still watching the breakfast on plus TV Africa and right now it's time to look at our hot topic It says Zanfarra to spend three billion a Naira on consultancy services for government's projects and Joining me to just have a conversation about this is Shagrun Shakwitan. He's the chairman accountability candle and Transparency. Yeah. Good morning to thank you for joining us or transparency network. Good morning. Thank you for joining us. Yes Thanks for having me. I was a pleasure. Yes. Thank you for coming in Alright, so we're talking about this is a lot of money. We're talking about three billion Naira on consultancy fees for government projects and Some other when I did my research some other projects were 950 million to build the emmer's palaces 350 million for mosque and cemeteries and We were just seeing about 426 billion that was signed into law. It was the appropriation bill that was signed into law by the government governor in December 2023 But please I want just I just want you to walk us through all of this because there's a lot of numbers and it was looking at You know three billion Naira just for the consultancy fees That's a lot of money, but just walk us through it Well It's I think it's excellent that you guys have decided to bring this to take this as as A major story or a major issue to look into This morning and I think and I wish that more more media houses and Nigerians in general will do a bit more of this, you know We all tend to focus our attention on the federal government And they forget the sub-nationals, you know this we've got the federal government us as Some administrative Body Yeah, let me just say administrative body That runs their fears of the country at the national level and then you've got 36 others 36 others that that that also have a very direct in fact arguably a more direct impact on the life of the average Nigerian spread across the entire country and You know this this this guys fly below the radar all the time They they do whatever they want to do and they really get any kind of scrutiny from the public except, you know in exceptional cases So maybe legal state for example because it's you know, it is what it is The 50 to 60 percent of the commercial activity in Nigeria is based in legal So you tend to get a lot of focus in legal spotter caught You know river states again because a lot of our oil resources are you know tied to the activities in that state You know and a couple of other places maybe can occasionally again because of its commercial importance to Nigeria But look at the order that I just listed three out of 36 and the other 30 plus You know, these governors get away with blue murder and nobody nobody pays any attention to them So I think who knows to your team for bringing Zampara states in particular, you know to to the public focus Having said this So the The figures that we have For this conversation are basically figures from the budget of Zampara state for for this year for 2024 It's a total budget of four hundred and twenty six Billionaire That's roughly just shy of half a trillionaire so It might sound small if you compare that to the federal government budget, but How the trillionaire is not it's not you can change. It's a lot of money still And then what we're looking at is, you know some components of of that budget and how Important it is, you know for some of the items that have been placed on that budget Why are they there, you know what why for example, like you said Well, the will the government that is running a budget of just shy of a trillionaire be spending Almost half a billion era on consultancy alone, you know, I mean, this is this is it's In fact, what am I? It's almost a trillion, you know on consultancy. I mean, this is three billion Three billion on consultancy. Yeah, three billion on consultancy for government projects I am a consultant In my day job So when I see that figure I can relate to a certain extent that is definitely paid it for somebody It's absolutely paid it for somebody. So the question is what exactly will those people, you know be doing for the government when you say consultancy The the sad thing that some of us who play in that field Suffer is that people use that terminology very loosely and very widely and it used to cover pretty much anything and everything So so you'll be good to know exactly what these people are consulting what specific Expertise are they providing that justifies three billion naira, you know Of course, we have to understand that this is over a one-year period So even if you split that three billion you say for example to every on a monthly basis you're still talking of about Divided by ten is three hundred million. So you're still talking about 250 to 180 million every month Pay consultants. That's a lot of money. Yeah, you know, so I Think the people of Zanfar states should be asking questions of their governor to find out why What exactly are we paying for what are these consulting services? What what specific expertise do these guys provide that the staff of the civil service in that state cannot provide? Why are we spending a trillion a billionaire? Pardon me a billionaire to build Palaces for Amiens. Are we just appointing new Emirates? Are we creating new Emirates? You know, I mean is it that this Emirates did not have policies before it will be one thing to say you want to renovate You know the policies of Amiens to say you want to build even if it's a renovation, you know, nine hundred and fifty million Just to spend on policies of Traditional rulers who you know to be honest have very little role to play in the governor's Processing in the modern-day Nigeria, you know, that that's that's a lot of money And then three hundred and fifty million to build religious religious centers mosques Cemetery's These figures are just large And I think that's what we perhaps should be doing is to look into the details, you know Just be be a bit more thorough as they say the devil is in the details So how many how many cemeteries for example, how many mosques are we talking about, you know, will that 350? million Finance so if it's there, for example, then you'd be saying 35 million era for each mosque and cemetery I Think that again, it would calm down to you know, the people of Zanfar as they think this is this is appropriate spending of their resources Arring their more important Things that the government can spend money on that would have a direct impact on the on the on the on the well-being of the people Of Zanfar state, so there is there are so many questions to unpack, you know from from from looking at this budget Okay, so looking at the budget I mean it's it's interesting to know that the budget is has an estimate of hundred and 18 billion an era allocated for required expenditure Which is about 27% of the sum of 308 billion is about 35% which was allocated for capital expenditure So when I'm looking at the numbers, I'm wondering 118 billion, you know for recurrent expenditure then 950 million For the the construction or for the traditional rulers. It's it's a lot of numbers and Which brings me to my next question. We're talking about Zanfar estates This is a state whereby the residents are still being attacked kidnapped killed sometimes Shouldn't there be, you know, you know, they're moving the forms to talk about security of the state then actually building palaces for the emirs that is to the tune of of about 905 or 950 Yeah, 950 million Naira because shouldn't shouldn't they be thinking of of how to secure their lives and properties of the people of Zanfar mean look, it's it's It's it's all in the air and it's a matter of prioritization Like you say, you know, we all know the the issues that some Zanfar estates in particular Over the years in the last 10 years have suffered, you know with the issue of panditry It can even be argued that what the entire country is suffering today from banditry started from that we remember all the Controversies surrounding the then gobbels and for estates having meetings with these bandits Talking about them handing over their their arms and ammunition In exchange for some sort of clemency or amnesty programs, you know, and all of that, you know This this trended heavily at a point in this country So to to find that we come down to a point Two years later where well the governor doesn't seem to think that this is now a priority And in fact if you look at that budget, you actually find that There is a provision for purchase of Hilux vehicles 15 brand new cash gifts. Yeah, 15 brand new Hilux vehicles cash gifts The leaders of different repentant dance of pandits By the past governor who is the current minister of defense, you know You you really begin to wonder what exactly is going on in the state. So I think that there may be Serious misplacement of priorities as far as as far as this budget is concerned I think that's not focusing our energies on the thing that match the most to the life of the ordinary Zanfarah Citizen what are the things that we should be thinking about one? Security like you said, what are the things that can we can spend money on to better secure the lives of the people of Zanfarah state? Are there a single point solutions? For the state in terms of security that you know can be funded from the budget or perhaps should we be talking about some sort of collaboration? between Zanfarah state neighboring states and the federal government and then what what on top of funding would that require of Zanfarah state? You know, I think some some sort of coherent Thought in that regard would have been good to see In this document, let's not forget that budget documents Typical policy statements the the points to the mind the thinking of the people that are Presenting the budget. What are their plans? You know for that period that the budget covers and all of that so This budget doesn't seem to be focusing attention on this very obviously critical problem that the state still faces another points to look at is You know the the capital Expenditure to recurrent expenditure percentage is proportion of the budget and on the face of it. You see the sum of 308 billion being spent for capital expenditure, you know 75 percent of the budget and only 27 percent going to recurrent and One might be tempted to think wow. This is fantastic Because this is what we always call for But one of the things that I that I do realize is that if you look at the sub-national budgets compared to the national up to the federal Government's budget you find that the skew the recurrent capital Proposal skew is reversed for most states most states budget more for capital expenditure and less for recurrent and People have argued that this is really because the capital budgets are the easiest sources of How would I put it now to the that's where you can put the till for the money to become more readily accessible It's easier to to avoid the contract of 20 billion Knowing that 10 billion out of that 20 billion will be used to dispense you know shared Orderly amounts take orders So even that 75 to 75 to 25 percent skew in favor of capital expenditure is not something I would celebrate without Knowing the details of those projects. So I mean we already highlighted some of the items in the capital expenditure And we're talking about mosques, you know, we're talking about cemeteries Is it that cemeteries didn't exist before granted that the state might be growing and you could argue that you You know, you need more resources to manage the state better So, yeah, maybe maybe they need more cemeteries But I think scooting a scooting a scooting it must be the watchword in this case We need to be sure that all of these figures that are being thrown around in this budget And it's not just for Zampara state for the states Actually, I'm going to be you know deployed for the purposes that They stated on paper. Yeah Okay, so I mean I am big on transparency. I'm big on giving a report if you're asking for Funds, you know, you should use them for what what you asked them for like you should be deployed to the right Channels, right? So I'm big on transparency. Do we have like agencies? I know you work for You work for one, but do we have like agencies that actually, you know Just help to put these politicians on their toes that you know If you're asking for such money such amount of money, we need to see what exactly we're using it for and not just even writing that we needed to you know build Palaces and we need to build mosque and all of that. Is there a need For those palaces is there a need for those mosques there need for all the things that you've written on paper And you know, they're just people who are there to ensure that there's some level of accountability and transparency I Mean like you like you said, I I had a civil society organization That is specifically focused, you know, on this issue of accountability and transparency in public in a public life As as a country and we do our bit We've reached out to the federal government on several occasions asking for details of projects To be made available to us so we can scrutinize them and and raise alarm if there's any need to do so highlight those issues to the public if there's any need to do so and you know on occasion we do get feedback from from government But most times they don't cooperate or even when you get the feedback you find that what they do is they play around the Issue but they refer you to one document or refer you to some public Press release that they've done in the past and all of that. They never really quite subject themselves or make themselves open to Disputing it that is required. However, there are a lot of other civil society organizations as well who have put mechanisms in place to help citizens track Public expenditure public accountability issues and all of that and I think we as as a people must also Seek out, you know, these platforms and this this programs that organizations like ours are put out there To make sure that we are supportive of their actions and that the government does not See us see the civil society organizations as a problem But see us as representatives of the people because the people in themselves are showing interest in in the actions that that we're taking But absolutely we need more we need more Civil society presence. We need more civil society organizations to especially at the state level to Beam the satellite on the activities of the state government and then ensure that they just simply don't carry on as if this money is at the personal property What we have for the state level is that, you know, I like to say a lot that Our state governors are basically Empire emperors who is running the empires as they didn't fit They rarely Give account what's for the things that they do in this budget. For example, you will not see an item for security votes But it's there You know, this is a famous problem, you know that Nigeria is aware of now Every state governor is entitled to constitutionally entitled to something they call the security vote They can vote any amount that they want to that item and they are not required In fact, they are required by law not to account for it Because they claim that it's it's security votes and they cannot divulge what the money is spent for So what tends to happen is that this becomes a slush fund for government and for governance they basically take this money and Just use it for whatever they want to use it for and they call the security votes and that there that's the end of the matter So a lot of issues like this exists that we have a lot of work to do as a country to fix this issue now Let me let me say that We must remember that when we talk about the federal Republic of Nigeria and we look at the budget and we look at fiscal issues and you know expenditure issues the federal government to represent a smaller proportion of the total revenues that come into the country in terms of how That those monies are spent the federal government represents only about 35 or 40 percent 60 percent goes to the state government and the local government and the local governments We all know state governments money the resources on their behalf. So 60 percent is actually managed 60 percent of our expenditure as a country of our revenues as a country are in the hands of the governors So we all focus our attention on that 40 percent. Yes, it's in one single entity But if 60 percent spread across the 36 states is a lot of money as well And we we need to put in a lot more work to Ensure that this gentleman, you know in the 26th government houses account for what they do with this resources They are the resource of Nigerian people not their personal property. Yeah, I mean Like well, we don't really have a lot of time But I really hope that there's some level of accountability and I was even going to ask What's the conversion rate like so for the time that the CSOs Come out and talk about this. Is there anything being done? Is there? Do we do we have to you know, go to court? So is there any suit that needs to be filed? I wanted to know what the conversion rate has been so far So for instance with Zamfara if people are coming out and saying, you know, these numbers don't make sense There's the government, you know, go back and say, okay You know what we're going to cut down on this or not And I think maybe should just help me answer that quickly before we wrap it up Occasionally when there's enough of energy when there's enough of a groundswell of Attention to specific issues you have governments to reverse their position You have moments, you know look back and take remedial action But it's occasional and it really happens even when CSOs like ours go to court I get actual documents on specific issues, you know concerning governance, you know They just ignored we run a country that is run on a culture of impunity where there is no consequence for bad behavior Where people disobey court orders when they please and nothing happens thereafter, you know So invariably you find that we put in our efforts and we do the bit we can but the politicians ignore you most of the time Knowing fully well that nothing will happen. So yes, we do get some conversion But it is precious in little there's a whole lot that needs to be done But I think that's what could change that is for the people the citizens of the country themselves citizens of the states to be more Deliberate in paying attention to show interest in the issues that CSOs are pushing for them Syrup is one very famous. Yes. So that's news if the federal government sneezes. They will sue them And they do they do get a lot of judgments But if you just look at you know, the the history behind those judgements, you find a majority of those judgments are ignored majority of them are not obeyed and Nothing happens. So we as a people we as citizens of this country need to get to the point where we are in system That the government must be accountable when or court orders are given they should they must be obeyed You know, social media is very powerful, right? Yes, maybe I'll just round up by saying that social media is one place where I've seen that when there is enough of of Unit unit unity. Yes of a boss behind a specific issue We the people always prevail and I can give example of an example of an example, you know Where people have trended a particular hashtag and the government had to respond So I think Nigerians will need to you know, use, you know, the instrument of social media a bit more to get public officials To do the beating of the people rather than just do whatever pleases them at the time All right. Well, that's a very good way to leave it. I want to say thank you so much for coming in and sharing your valuable contributions Thank you so much That was a pleasure. Thanks for having me. Thank you All right, we've been speaking with Shagrin Shepitoa and we've just been talking about the fact that Sampara has three billion are in consultancy fees for certain projects that they're going to be having and some of these projects are about 950 billion narrow for 950 million are for more looking at palaces for the air mayors also looking at cemeteries and Yeah, church mosque as well. So anyways, we're going to short break when we return We're looking at our next hot topic. Please stay with us


Thank you for joining me as I sit down with Pastor David and Marie Rosales from Calvary Chapel, Chino Valley, as we discuss marriage, raising children, and managing difficulties that arise in the family. We're ready to begin, so let's talk marriage. You know, speaking on that pastor, would you be able to describe the different meanings of love that are represented in the Greek? Because I want to go somewhere with some of these different types of love that are described in Greek. What are they, pastor? You know, if anybody is really interested, I believe that C.S. Lewis wrote a book called The Four Loves, and he does a scholar's job on the Greek and all of that. You have eros, which was a physical attraction, physical love, erotic. You have stargay, which is more of a fleshly kind of love. It's not the deepest form of love, but it's a word that's used to describe it sometimes. You have agape, which is the highest form of love, and you have filial. And filial is the brotherhood or sisterhood, relational love. And so there are aspects of love that are found in scripture. The three loves that are most commonly used scripturally, though, are the eros and the filial and the agape. And so one of my professors said, whether this true or not, I don't want to pass bad information on to our viewers. But one of our professors said that agape was really a word that was coined to express the kind of love that God had that was almost, and I don't want to stumble any of my scholars who may very well stumble across this.  I'm not. But my professor said that agape was almost an invented word to try and describe the highest form, the highest form of love, which is a self-sacrificing, self-giving love. Filial was the more common friendship, kind of love that you would have. And eros was the physical aspect, and it didn't even have to be, quote unquote, a real love, is a physical drive that is consummated in physical activity. And so in marriage, the highest form of relational love should be agape, because Jesus loved us and we're to love our wives as a man, like Christ loved the church. And so the word agape is a word that would be used to describe that, which is the self-sacrificing, dying to self-kind of love. But at the same time, you know, Paul speaks concerning not defrauding my wife. And when you speak about not defrauding my wife, that he said, if you separate it all, make it for just a time for prayer and then come back. And what's he referring to, you know, not plain checkers, he's talking about physical love. He said, and the woman and the man actually have a debt to one another to fulfill that aspect of their love relationship. And so Paul made it very clear that when within the confines of marriage, sexual activity occurs, it has been created to be pleasurable. And it's intended by God to be enjoyed by those who are married, not simply to have children, because some can't have children, but because it is the highest form physically, physically, of showing the intimacy that occurs between a man and a woman who become literally one in their activity through intercourse. And so there's an actual unity that goes deeper. And so there's that erotic love. But the filial love, I would say that in my marriage with Marie over the years, you know, the filial has been the most common one. It's the friendship, it's that, you know, again, we hold hands, we don't even realize we're doing it.  You know, that's just us, we don't think about it. And that there is a friendship, that's friendship, what we're doing here, this we enjoy each other and we kind of, you know, that's the way it is. And so I would say if you asked, you know, what has been the the thing that kept you together? We're friends. We're friends, first and foremost, but I love her with a an agape love, you know, because she's my girl. And the other aspects of love that are natural for those relationships have been part of us. But if you said, what do you think is going to keep you going? And I'll give you an example of what I'm trying to say. What's going to keep you going when you're old and old and old? We're old now, but say God gives us 20 more years. You know, well, I saw a picture of an old man, probably 90 plus years old and an older woman, his wife, and they weren't well. And but they couldn't be apart from one another. And they were in a room together. And and she's in her little cot, because they're not well. And he's in his little cot. And but they had to be in the same room. And they're reaching out across that divide holding hands. And my daughter, daughter in law, but she's my daughter, my Kareena said, I don't know if you're going to love it. This is kind of corny. And and I my tears started for me. I said, That's us. See, as long as I have her, and it's not sexual. It's not if I got her hand. I mean, heaven. And that's that's love, you know, that's friendship. And that's us. That's us. People see that.  And when I have everything with her, I'm free to love other people. Because I don't have any agenda. I don't have an agenda. I'm, I'm, I'm, she's safe and I'm safe. We're safe with each other. So I can love other women deeply. But I don't have the same love. I have love only for this woman. That kind of love, right? And that's all built on our friendship. So which one of these meanings of love is difficult for a husband to practice? Well, I don't think he'll ever have problems with erotica. I don't even think of that. Aros, are you kidding? No, I don't think Aros, let's just throw that out right now. I don't think he'll ever have a problem with that. I think that the the love that requires the deepest sacrifice is the agape. Yeah, because that is dying to self. And you can be a selfish friend sometimes, but you can't really love with the agape and be selfish at the same time. So I would say that the more pure and the higher and the most holy is agape. And that's what you seek the Lord for. Agape love is the, uh, the mark of a believer. And agape love is evidence of God's Spirit's presence in your life. And so it's the fruit of the Spirit. And so I would say that's the one to be pursued because when, when you do have the agape for your wife, when you have that, I'll lay my life down for you the way Christ did for the church. That's the highest form of expression that you can have. And so friendship has kept us together. It will always keep us together. Agape is what has made it deep and it's made it multi-dimensional. Yeah, so I would say agape is to be pursued, but it's expressed through your friendship. Would you say that in order to have the other, the filial and the other forms of love in a marriage that agape is the one that will allow those other ones to take place? Agape is the root of all other love. Um, so yeah, if, if, if you walk in the Spirit, then you don't fulfill the desires of the flesh.  Well, who is, who is he if he's not a loving, caring, compassionate, sacrificial God? Right? And so that's been my prayer. I've said this to our church, John. Perhaps you've heard, I don't know if I've said this recently, but the oldest prayer that I've prayed over the years, and it's been almost 50 years, it'll be 50 years in December that I came to faith in Christ.  I have prayed one prayer consistently and that is God teach me to love. And so fill me with your spirit as I want to love.  And so that's the key.  She's held nothing back, nothing.  So I asked the Lord, help me to, to, to, to cherish and treasure that and to act the way that ought to provoke and be activity. Help me to act like I believe that. And I hope she knows that. I think she does, you know, all these years. I think she does. But that's what I do.   He already was out there tending the trees in the garden. He was already doing fellowship with God. And so he's commanded to love her. It's interesting. She was created to love him. She was taken out of him and became his woman. But he already had other things going before he had her. And I suspect that there's some general truth in that with guys and get caught up in the jobs and their, their hobbies and other things and neglect our wives. But to me, and I've said this in church more than once. I told my kids this too. I've said, you know, one day I won't be doing what I do. One day I'll be quote unquote retired. And I told him, I want to be ready. I want to know this woman so I don't have to know who she is after I retire. And because a lot of guys hiding their jobs and their hobbies. And now they have no more jobs. And they're, they're, they can't play softball anymore and they can't do things, their activities and their hobbies bowling or shooting their weapons or whatever they like to do. And now they're sitting on a chair with this woman they don't even know. Who are you? Yeah.   You know, well, I made my mind up. You know, Bible tells us, Peter said, do I with your wives according to knowledge? So I study her.  She knows this. She says it. No, you study me. And I do. And I study, I watch her. I, I, I could, I can tell you, I can tell you pretty much everything I could know of all these years of studying her. I know my wife. I know my wife. And she'll tell you that. There'll be times she'll start saying something. I'll say, oh, because, and she'll go, yeah. And that's, that's not bad. That's knowing your wife, you know. And the blessing is sometimes she'll say, uh, no, there was something else. Now I learned something new. So all of that's good. And same, same is true with her. And she, she knows me. And I'll look at her and I'll say, you think I'm going to say this, don't you? She'll go, yeah, I'm not. I'm saying something different today. Well, really, you know, new. This is new, you know, so. Well Marie, if I were to ask you, uh, from a woman's perspective, which meaning of love is it difficult for the wife to practice if there is a difficulty in, in, in the meanings that pastor had given the Eros, the phileo and the Sturgy? Well, I would say, I would say the sexual, I think women have a tendency. I think a lot of women would withhold from their husband, um, because they get upset with them. I think I would say that's probably the most, most, most of the women have problems in that area. I would think, because if they think that their husband's not loving them, then they're not going to want to love them. It's a, it's, it's, it's, it's a terrible thing to, what's the word? What's the word I wanted? Bargaining. Yeah. Yeah. Yeah. You shouldn't be bargaining. I mean, I mean that, that part is not to be bargained. But I think that, I think that many people do, um, you hear a lot of, uh, problems in that area. In that area. You know, what's interesting is, is, uh, at least from a male's perspective, I can't speak on a female's perspective. It's funny, uh, I've seen and I've heard also that there can be an argument between the husband and wife or a disagreement and then, you know, then the husband's, uh, ready to be intimate in some ways. And the wife's like, no way. There's that different connection where men have, can have the capability of just having intimacy, uh, even after an argument where a woman's like, there's much more involved for a woman. And so there can be that disparity sometimes, which I find interesting of having that, uh, as well. And it's very real. I believe it's very real with the women. They, they, uh, hold on to grudges. Can, and yet, that's. They become tunative and they bargain. You know, that does happen. A tick for tack or the, if you're going to do this, I'm going to do that. Well, if you didn't do this for me, then I'm. Or they use their sexuality to get something they want. That's true. That's very true.  He doesn't need to know who you are, where you're from, anything. And, and there was a time when women actually valued themselves enough to think that those things are important. And sometimes it's sad, but sometimes women today, uh, may not see the value, uh, of what intimacy actually is. And in marriage, like Maria's saying, of course, she's right about that. Um, yeah, women, women can, because we hear this, we're ministers. I mean, we've been doing this for a long time. These are, these are things that people say to us.  And there's no doubt about that, that they use, uh, sexuality as, as a bargaining chip, or they use the word, uh, tell me I'm wrong on this one, or they say, I don't feel like it. That's right. No, I don't feel, yeah. I don't feel like it. Now, man's always going to feel like it. You know, there's a difference between men and women. You know, women say, no, I don't feel like it. You know, and then she punishes them. Yeah, it's true. You know, or whatever. Now, Marie said more than one conversation like that. Yes. It's interesting how that, and then the man cannot love her and it, then she cannot kind of hold out in the same time. And so it starts this. Yeah. The man begins to feel like she's, um, she's holding out on him, withholding from him, punishing him. And then he'll go into whatever. Some guys will do this. So say, I don't really think he'd do, you know. I mean, there's someone on a street corner right now, or I've got somebody in, on the job. I, when Marie and I first got married, I, I had a job, um, working in an office and, and, um, I, I, as a younger man, it's, I, you know, people don't know me very well on a personal level. They only know me as a pastor, but somebody working on the job site, you know, it's kind of, I was a friendly person.  Uh, I am too though.  And when I was on the job site, I did that. And, and again, I'm a, a newly married man. And there's these two young women working in, in an office. And I had to, interact with them every day, every day, because they were part of my department, right? I'm not thinking anything of it. And I go walking in one day. And this one young lady says to me, um, my, my parents are gone. I was a young man then. My parents are gone for the weekend. And, and I live in Downey. And I at that time was living in Norwalk. I said, oh, she goes, yeah, I live in Downey. My parents are gone. Um, you ought to come and pick me up and bring me to work. And I looked at her and I didn't say anything. I just kind of like, I didn't, I, no, I didn't hear that.  And, and the girl, her friend said, did you hear what she just told you?  You know, she made it very clear. And I looked at her and I, I said, I said, something or other. And I went to my desk and I thought about it. And later on I told Marie, I said, I was hit on today, you know, and I was very uncomfortable with that. See, like I said, from the beginning I've told Marie these things. Well, John, that was when I was probably 25, 26, 26 years old. And so I planted our church. I was probably 31 or 32. And I used to stand in the back of the church at the end of services. And here comes this woman who had told me, did you hear what she said? And she had gotten saved. And she was going to Rawls Church in West Covina. And she says to me, do you remember me? Her name was Teresa. And I said, oh, yes, I do. You know, how are you, Teresa? I just, she said, I heard you got saved. I wanted to come to see, or I heard you pastor to church. I just wanted to come and see if it was you. It is you. And I wanted you to know that I gave my heart to Christ. And I'm serving the Lord at Rawls Church, my husband, this is my husband, so and so. And I'm looking back at a few years and I'm thinking, if I'd have been a jerk, if I'd have been flirtatious, if I'd have acted interested, what would that have done? Imagine, wow, what would that have done? You know, you never know. You never know. So it's for me, it's always, you know, just pursue the wife of your youth and always be ravished with her love like Proverbs says. If you can finish this sentence, if you both want to share, friendship with my husband, wife means. Chili beans. Speaking of chili beans, this is a shout out. Those cookies were really, really good. They weren't for you, John. I know. My wife had to hide them. They were so good. What was the question again? If friendship with my wife, was friendship with my husband slash wife means for to finish the question, finish that sentence. Just being with them. Just enjoying them.  You know, Marie and I, it doesn't matter where in the world we are. As long as she's next to me, I'm home. So I'm complete with her. That means the world to me friendship with him. And so, like he said, it's just, we come home and we're there. It's just us. Two old people. Two old people kind of rolling around a house banging on each other. No. We don't. Marie, you know, as we were speaking earlier about sometimes using the different types of love to hold out from the husband, are there any repercussions when a wife lays down her life for her husband only to receive approval from him? I would think so. I would think that there would be only to receive approval from him. I would think so, John. Some of those things can be, I mean, I'm only doing it for him. I'm only doing it for her, which then doesn't, we're not doing it for the Lord anymore, right? It's now become about that person and what they can give me. Yeah. It's a take, a mentality rather than a give, willfully, lovingly giving. That's dangerous. That's a dangerous place to be. A give and take situation. Because as Pastor mentioned, what happens is potentially it can open the door for, well, if you're not going to, you know, if you're not going to give me what the things I need or the appreciation I, then I'm going to find, I'm going to look elsewhere. Exactly. And it can be dangerous. Yeah. It will be dangerous if you live your life together like that. A give and take. No, it's not about me. It's about him. The most important thing that you have to have in your relationship, and I know that Marie would agree with this, is a personal walk with the Lord. Yes. You know, when a man's ways please the Lord, he makes even his enemies to be at peace with him. When a man is seeking to please the Lord, he can be at comfort in his home. And when a woman's, see, so Marie, Marie cannot, and she knows this, so I can, I'll answer for her, because I know this is what she's saying to you right now, and I'll say it my own way, but I think I'm saying what she's saying. Yes. Marie wants to please me. I mean, she's supposed to, and I want to please her too. But she knows that if she tried to always please me, she can't, because I can change my demands. I can change what pleases me right now. And so she'll always be trying to figure out what can I do to make him happy. But if she pleases the Lord first, if she pursues the Lord, if she's in the Word with the Lord in prayer, in fellowship with God, has friends, she's growing spiritually. And as she's growing, those are the things that make her fulfilled. And when she's fulfilled, she's able to give to me from that abundance, from that, not to try and make me, oh, do something in return, but because she loves doing this, because it blesses my life. And I am able to see that she's not doing this so that I'll buy her a purse or take her to dinner or, you know, whatever. I'll see that, you know. And the same is true with her. We learned this at the beginning. We were married, as knew that was John, we were a month, two months, whatever, married. I mean, we're just freshly married. I still remember this. I've shared this again with the church. It's just something I've been open about.  You know, you're beautiful. I love you. She looks at me and she goes, what do you want?   I said, we're married. I said, I'm not giving you some kind of a play. I'm not trying to seduce you. I said, I want anything except you. I said, that's what I want. So she and I together, again, we were young. We were young at that time. We already knew. I already knew. No, I've already married to you. I've got you available to me as a wife. And I also know that you love me. And so it's all okay before God. So no, I don't have to say you're beautiful. I don't. I don't have to. And I don't. I just love you because I love you. See, so we put away that bargaining thing a long time ago. We never really had that. That's not one of the things that she ever needed to worry about with me. And I don't think that way. See, that's not how I think. So that was clear when we first got married. And that's important because my question, the other question would be, when a husband lays down his life for his wife, and she doesn't appreciate how, doesn't appreciate that, how is he to respond to that? Well, why is he laying his life down for her, right? Exactly. I mean, isn't that part of laying your life down, even if she doesn't respond? I mean, again, that's manipulation.  Something in return, right? You're manipulating. And you know, pastors sometimes, and guys are going to probably get upset with this, but in that sense, sometimes men can be the biggest babies about it.  I guess it works both ways. But I've met with a lot of couples, and there's this mentality where, well, you know what? I'm laying down my life for her.  Shouldn't you be doing that anyways? I don't know what you're supposed to do. Any man, you know, again, you know, I don't want to make the men feel bad, and I don't want women hitting him with her elbow. I've seen that when I'm teaching. I've seen that when I'm teaching. I'll say something, and I'll see the woman turn and... I mean, it's so funny to see that, and it'll go all through the churches. And the guys, you know, moving, but everything like that. Okay, so part of being a man is simply growing up, you know, and if you have a conversation or relationship with your wife where you're open and you say, this is on my heart, you need to hear this, which is the way Marie and I are, I will tell her that. I will say, you need to sit down. We're going to have a talk because I want to share with you what's on my heart right now. And she knows that it's time to put down whatever it is she's doing. And well, I'm good. Okay, I'll be right here. And I have always done that with her. I am that guy that will say, we're going to talk. I'm not going to let this be something that gets between us. We're going to talk. And it isn't, I try so hard. You know, there have been times when I have felt that and there have been times when I have cried and I have said, you know, honey, be honest with you at this moment, I'm feeling kind of used. And I know that you're not using me. So I want to take responsibility for some foremost for even telling you that because I know you love me. But I need to tell you how I'm feeling right now, because feelings aren't always fact. And this is how I'm feeling. And I need to hear from you so you can help me to feel some security in this moment that I'm dealing with some stuff. And that's how I am with my wife to this day. If there's anything that I need to talk to her about, we will talk.  She already had four kids. She doesn't need to talk. And sometimes I think in both cases that we expect our wives or our husband to be mind readers and not having that conversation. You know, it's so important to be open like that. And so the importance of sacrificial love in a marriage.  You wake up the next morning, you do it again. And then wake up the next morning after that and do it again. Unless she kills you in the middle of the night. Oh, thank you guys both for spending time and speaking about the importance of friendship in a marriage and for spending time too, let's talk marriage. Love it. Thank you guys. Thanks again for tuning in. Let's Talk Marriage is a ministry of Calvary Chapel, Chino Valley. If you've enjoyed this video, then please like and share it. We will see you again next week on another episode of Let's Talk Marriage.