It can be understood by considering a 2n-dimensional functionspace with the set of the sines and cosines of the products of the integers between 0 and n and x as it basis; each of its namesake coefficients is then the dot product of one of the basis functions with a function f(x) and becomes an integral over the functionspace as n approaches infinity.
Suppose that f is a piecewise smooth function. Then, for f, this converges to the arithmetic mean of the left hand and right hand limits of f at points where f is discontinuous. It can be understood by considering a 2n-dimensional functionspace with the set of the sines and cosines of the products of the integers between 0 and n and x as it basis; each of its namesake coefficients is then the dot product of one of the basis functions with a function f(x) and becomes an integral over the functionspace as n approaches infinity. First introduced in its namesake French mathematician's 1822 treatise The Theory of Heat, FTP, name the type of infinite trigonometric series that is invaluable in the study of partial differential equations.
Suppose that f is a piecewise smooth function. Then, for f, this converges to the arithmetic mean of the left hand and right hand limits of f at points where f is discontinuous.
Fourier series