[**Error Bounds for Numerical Integration of Oscillatory Bessel Transforms with Algebraic or Logarithmic Singularities**]{}[^1].

[**Abstract**]{} In this paper, we present and analyze the Clenshaw-Curtis-Filon methods for computing two classes of oscillatory Bessel transforms with algebraic or logarithmic singularities. More importantly, for these quadrature rules we derive new computational sharp error bounds by rigorous proof. These new error bounds share the advantageous property that some error bounds are optimal on $\omega$ for fixed $N$, while other error bounds are optimal on $N$ for fixed $\omega$. Furthermore, we prove from the presented error bounds in inverse powers of $\omega$ that the accuracy improves greatly, for fixed $N$, as $\omega$ increases.

**Keywords:** oscillatory, Bessel transforms singularities, Clenshaw-Curtis-Filon methods, quadrature rules error bounds.\

Introduction
============

Highly oscillatory Bessel transforms arise widely in mathematical and numerical modeling of oscillatory phenomena in many areas of sciences and engineering such as astronomy, electromagnetics, acoustics, scattering problems, physical optics, electrodynamics, and applied mathematics [@Arfken; @Bao1; @Davies1; @Huybrechs1]. In this paper, we focus on new computational sharp error bounds of the quadrature rules for singular oscillatory Bessel transforms of the forms $$\begin{aligned}
\label{1.1}
I_1[f]&=&\int_0^bx^\alpha f(x)J_m(\omega x)dx,\\
\label{1.2} I_2[f]&=&\int_0^bx^\alpha \ln(x)f(x)J_m(\omega x)dx,\end{aligned}$$ where $f(x)$ is suitably smooth in $[0,b]$, $J_m(z)$ is the Bessel function [@Abram] of the first kind and of order $m$ with $Re(m)>-1$, $\omega$ is a large parameter, $b$ are real and finite, and $\alpha>-1$. In particular, it should be noticed that transforms and are integrals with algebraic and logarithmic singularities, respectively.

In most of the cases, such integrals cannot be calculated analytically and one has to resort to numerical methods [@Davis]. The numerical evaluation can be difficult when the parameter $\omega$ is large, because in that case the integrand is highly oscillatory. The singularities of algebraic or logarithmic type and possible high oscillations of the integrands in (1.1) and (1.2) make the above integrals very difficult to approximate accurately using standard methods, e.g., *Gaussian quadrature rules*. It is well known [@Davis] that a prohibitively large number of quadrature points is needed if one uses a classic rule such as Gaussian quadratures, or any quadrature method based on (piecewise) polynomial interpolation of the integrands.

In the last few decades, much progress has been made in developing numerical schemes for generalized Bessel transform $\int_a^bf(x)J_m(\omega g(x))dx$ without singularity. For example, the modified Clenshaw-Curtis method [@Piessens1] was introduced for efficiently computing $\int_0^1f(x)J_m(\omega x)dx$ for $m$ being an integer in 1983; the Levin method [@Levin], Levin-type method [@Olver], and generalized quadrature rules [@Evans; @Xiang] were also available for approximating $\int_a^bf(x)J_m(\omega x)dx$ for $Re(m)>-1$. However, *the Levin method*, *Levin-type methods* and *generalized quadrature rules* cannot be used if $0 \not\in[a, b]$. In addition, based on a diffeomorphism transformation, the reference [@Xiang1] extended the Filon-type method to the efficient computation of the integrals $\int_0^1f(x)J_m(\omega g(x))dx$ with the exotic oscillator $g(x)$ satisfying that for $r\geq0$, and $g(0)= g'(0)= \cdots = g^{(r)}(0) = 0, g^{(r+1)}(0)\neq0,
g'(x)\neq0$ for $x \in(0,1]$, where $Re(m) > {1}/({r+1})$. In many situations the accuracy of the Filon-type method proposed in [@Xiang1] is significantly higher than that of other methods. As a matter of fact, it requires the solution of a linear system that becomes more ill-conditioned as the number of interpolation nodes increases, and one has to adopt higher-order digit arithmetic to get the required accuracy. Furthermore, to avoid the Runge phenomenon, the *Clenshaw-Curtis-Filon-type method* [@Xiang2] based on Clenshaw-Curtis points is designed for computing Bessel transform $\int_a^bf(x)J_m(\omega x)dx$ without singularity. Here, it should be also mentioned that the homotopy perturbation method in [@Chen3; @Chen4] was presented to compute $\int_a^bf(x)J_m(\omega
x)dx$. Recently, Chen [@Chen1][@Chen2] also proposed two different complex integration methods for approximating $\int_a^bf(x)J_m(\omega x)dx$ if $0 \not\in[a, b]$. To the best of our knowledge, so far little research has been done on the numerical computation of the integrals (1.1) and (1.2) with an algebraic or logarithmic singularity.

Consequently, our aim is to demonstrate high efficiency of the proposed quadrature rules for such integrals (1.1) and (1.2) by constructing error bounds. In the next section, we propose the Clenshaw-Curtis-Filon methods for computing the integrals (1.1) and (1.2). Here, the required modified moments can be efficiently calculated by a recurrence relation. Section 3 sets up new and computational sharp error bounds of these quadrature rules by theory analysis. In Section 4, we design a higher order method and derive its error estimate in inverse power of $\omega$. From these new error bounds, it can be seen that for fixed $\omega$, the error bounds are optimal on $N$, while for fixed $N$ the error bounds are optimal on $\omega$. Moreover, for fixed $N$, the larger the values of $\omega$, the higher the accuracy.

**Clenshaw-Curtis-Filon methods for computing (1.1) and (1.2)**
===============================================================

Chebyshev interpolation has precisely the same effect as taking partial sum of an approximation Chebyshev series expansion [@Mason]. Suppose that $f(x)$ is absolutely continuous on $[0,b]$. Let $P_Nf(x)$ denote an interpolant of $f(x)$ of degree $N$ in the Clenshaw-Curtis points $$\label{2.2}
x_k=\frac{b}{2}+\frac{b}{2}\cos\left(\frac{k\pi}{N}\right),\quad
k=0,1,...,N.$$ Then, the polynomial $P_Nf(x)$ can be expressed by (see \[24, Eq. 6.27, 6.28\]) $$\label{2.3}
P_Nf(x)=\sum_{j=0}^N {''}a_jT^{**}_j(x),\ \text{where} \,\,
a_j=\frac{2}{N}\sum_{k=0}^N {''}f(x_k)T^{**}_j(x_k),$$ where the double primes indicate that the first and last terms of the sum are to be halved, $T_j^{**}(x)$ denotes the shifted Chebyshev polynomial of the first kind of degree $j$ on $[0, b]$. The coefficients ${a_j}$ can be computed efficiently by FFT [@Dahlquist; @Trefethen].

The *Clenshaw-Curtis-Filon* (CCF) methods for (1.1) and (1.2) are defined, respectively, as follows, $$\begin{aligned}
\nonumber I_1^{CCF}[f]&=&\int_0^bx^\alpha P_Nf(x)J_m(\omega
x)dx\\
\label{2.4}&=&b^{\alpha+1}\sum_{j=0}^N{''}a_jM_j,\end{aligned}$$ and $$\begin{aligned}
\nonumber I_2^{CCF}[f]&=&\int_0^bx^\alpha \ln(x)P_Nf(x)J_m(\omega x)dx\\
\label{2.5}&=&b^{\alpha+1}\sum_{j=0}^N{''}a_j[\ln(b)M_j+\widetilde{M}_j],\end{aligned}$$ where, for $r=b\omega,$ $$\begin{aligned}
\label{2.6} M_j&=&\int_0^1x^\alpha T^*_j(x)J_m(r x)dx,\\
\label{2.7}\widetilde{M}_j&=&\int_0^1x^\alpha \ln(x)T^*_j(x)J_m(r
x)dx,\end{aligned}$$ are called the modified moments, where $T_j^{*}(x)$ denotes the shifted Chebyshev polynomial on $[0, 1]$, and which can be computed efficiently, as described below.

**Fast computations of the modified moments:**

The homogeneous recurrence relation of the modified moments $M_j$, was provided by Piessens [@Kythe] [@Piessens], as follows: $$\begin{aligned}
\label{2.8}
&&\nonumber \frac{r^2}{16}M_{j+4}+[(j+3)(j+3+2\alpha)+\alpha^2-m^2-\frac{r^2}{4}]M_{j+2}\\
&&\nonumber+[4(m^2-\alpha^2)-2(j+2)(2\alpha-1)]M_{j+1}\\
&&\nonumber-[2(j^2-4)+6(m^2-\alpha^2)-2(2\alpha-1)-\frac{3r^2}{8}]M_{j}\\
&&\nonumber+[4(m^2-\alpha^2)-2(j-2)(2\alpha-1)]M_{j-1}\\
&&+[(j-3)(j-3-2\alpha)+\alpha^2-m^2-\frac{r^2}{4}]M_{j-2}+\frac{r^2}{16}M_{j-4}=0,\end{aligned}$$ It is worth to notice that $$\frac{\partial}{\partial\alpha}M_j=\widetilde{M}_j.$$ Therefore, by differentiating the above recurrence relation with respect to $\alpha$, we find $\widetilde{M}_j$ satisfying the following recurrence relation: $$\begin{aligned}
\label{2.9}
&&\nonumber \frac{r^2}{16}\widetilde{M}_{j+4}+[(j+3)(j+3+2\alpha)+\alpha^2-m^2-\frac{r^2}{4}]\widetilde{M}_{j+2}\\
&&\nonumber+[4(m^2-\alpha^2)-2(j+2)(2\alpha-1)]\widetilde{M}_{j+1}\\
&&\nonumber-[2(j^2-4)+6(m^2-\alpha^2)-2(2\alpha-1)-\frac{3r^2}{8}]\widetilde{M}_{j}\\
&&\nonumber+[4(m^2-\alpha^2)-2(j-2)(2\alpha-1)]\widetilde{M}_{j-1}\\
&&\nonumber+[(j-3)(j-3-2\alpha)+\alpha^2-m^2-\frac{r^2}{4}]\widetilde{M}_{j-2}+\frac{r^2}{16}\widetilde{M}_{j-4}\\
&&\nonumber=-2(\alpha+j+3)M_{j+2}+4(2\alpha+j+2)M_{j+1}+4(3\alpha+1)M_{j}\\
&&\ \ \ +4(2\alpha-j+2)M_{j-1}+2(j-\alpha-3)M_{j-2}.\end{aligned}$$

Because of the symmetry of the recurrence relation of the Chebyshev polynomials $T_j(x)$, it is convenient to get $T_{-j}(x)=T_j(x),
j=1,2,...,$ and, consequently $T^*_{-j}(x)=T^*_j(x),$ $M_{-j}=M_j$ and $\widetilde{M}_{-j}=\widetilde{M}_j.$ It can be verified easily that both and are valid, not only for $j\geq5$, but for all integers of $j$. Unfortunately, for and both the forward recursion and the backward recursion are asymptotically unstable [@Kythe; @Piessens]. Nevertheless, in practical applications the instability is less pronounced if $\omega\geq 2j$. Practical experiments demonstrate that $M_j$ and $\widetilde{M}_{j}$ can be computed accurately using the forward recursion as long as $\omega\geq 2j$. But for $\omega<2j$ the loss of significant figures increases and recursion in the forward direction is no longer applicable. In this case Lozier’s algorithm [@Lozier] or Oliver’s algorithm [@Oliver] has to be used. This means that both and have to be solved as a boundary value problem with six initial values and two end values. The solution of this boundary value problem requires the solution of a linear system of equations having a band structure. The end value can be estimated by the asymptotic expansions in [@Kythe] or can be set equal to zero. The Lozier’s algorithm incorporates a numerical test for determining the optimum location of the endpoint, when the end value is set to be zero. The advantage is that a user-required accuracy is automatically obtained, without computation of the asymptotic expansion. For details one can refer to [@Kythe; @Lozier; @Oliver; @Piessens]. To start the recurrence relation with $k=0,1,2,3,\ldots$, we only need $M_0, M_1, M_2$, and $M_3$. By plugging the shifted Chebyshev polynomials $T^*_0(x)=1$, $T^*_1(x)=2x-1$, $T^*_2(x)=8x^2-8x+1$ and $T^*_3(x)=32x^3-48x^2+18x-1$ on $[0,1]$ into , we obtain $$\begin{aligned}
\nonumber M_0&=&G(\omega,m,\alpha),\\
  \nonumber M_1&=&2G(\omega,m,\alpha+1)-M_0,\\
  \nonumber  M_2&=&8G(\omega,m,\alpha+2)-4M_1-3M_0,\\
\nonumber  M_3&=&32G(\omega,m,\alpha+3)-6M_2-15M_1-10M_0.\end{aligned}$$ Then, it is apparent from the above equalities that $$\begin{aligned}
\label{2.111} \widetilde{M}_0&=&\frac{\partial}{\partial\alpha}G(\omega,m,\alpha),\\
\widetilde{M}_1&=&2\frac{\partial}{\partial\alpha}G(\omega,m,\alpha+1)-\widetilde{M}_0,\\
\widetilde{M}_2&=&8\frac{\partial}{\partial\alpha}G(\omega,m,\alpha+2)-4\widetilde{M}_1-3\widetilde{M}_0,\\
\label{2.114}\widetilde{M}_3&=&32\frac{\partial}{\partial\alpha}G(\omega,m,\alpha+3)-6\widetilde{M}_2-15\widetilde{M}_1-10\widetilde{M}_0,\end{aligned}$$ where, from \[1, p.480\], \[15, p.676\] and \[23, p.44\], we find several moments formulae as follows, for $\Re(m+\alpha)>-1,$ $$\begin{aligned}
\nonumber G(\omega,m,\alpha)&=&\int_0^1x^\alpha J_m(r x)dx\\
\label{2.18}&=&\frac{2^\alpha\Gamma(\frac{m+\alpha+1}{2})}{r^{\alpha+1}\Gamma(\frac{m-\alpha+1}{2})}+\frac{1}{r^\alpha}[(\alpha+m-1)J_m(r)S_{\alpha-1,m-1}(r)-J_{m-1}(r)S_{\alpha,m}(r)],\ \ \ \ \ \\
\label{2.19} G(\omega,m,\alpha)
&=&\frac{r^m}{2^m(\alpha+m+1)\Gamma(m+1)}{_1F_2}(\frac{\alpha+m+1}{2};\frac{\alpha+m+3}{2},m+1;-\frac{r^2}{4}),\\
\label{2.20} G(\omega,m,\alpha)
&=&\frac{\Gamma(\frac{m+\alpha+1}{2})}{r\Gamma(\frac{m-\alpha+1}{2})}\sum_{j=0}^\infty\frac{(m+2j+1)\Gamma(\frac{m-\alpha+1}{2}+j)}{\Gamma(\frac{m+\alpha+3}{2}+j)}J_{m+2j+1}(r),\end{aligned}$$ where $S_{\mu,\nu}(z),\Gamma(z),{_1F_2}(\mu;\nu,\lambda;z)$ denote a Lommel function of the second kind, the gamma function, a class of generalized hypergeometric function, respectively. Moreover, ${_1F_2}(\mu;\nu,\lambda;z)$ converges for all $|z|$. From \[32, p.346\], $S_{\mu,\nu}(z)$ can be expressed in terms of ${_1F_2}(\mu;\nu,\lambda;z)$, namely, $$\begin{aligned}
\label{2.21}
\nonumber
S_{\mu,\nu}(z)&=&\frac{z^{\mu+1}}{(\mu+\nu+1)(\mu-\nu+1)}{_1F_2}(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};-\frac{z^2}{4})\\
&&-\frac{2^{\mu-1}\Gamma(\frac{\mu+\nu+1}{2})}{\pi\Gamma(\frac{\nu-\mu}{2})}(J_\nu(z)-\cos(\pi(\mu-\nu)/2)Y_\nu(z)),\end{aligned}$$ where $Y_\nu(z)$ is a Bessel function of the second kind of order $\nu$. The right-hand sides of (2.11-2.14) involve the derivatives of the generalized hypergeometric function with respect to the parameter $\alpha$, which have been shown in [@Kang3]. The required derivatives of the gamma function are also described in terms of the Psi (Digamma) function $\psi_0(z)$, such as [@Abram] $$\begin{aligned}
\label{2.221}\Gamma'(z)=\Gamma(z)\psi_0(z).\end{aligned}$$

The efficient implementation of the modified moments is based on the fast computation of the Lommel functions $S_{\mu,\nu}(z)$ and the hypergeometric function ${_1F_2}(\mu;\nu,\lambda;z)$. Excellent references in this area are [@GTS; @Watson]. Obviously, when programming the proposed algorithm in a language like Matlab, we can calculate the values of $\Gamma(z), J_m(z)$ and ${_1F_2}(\mu;\nu,\lambda;z)$ by invoking the biult-in functions ‘ gamma$(z)$’, ‘ besselj$(m,z)$’ and calling mfun(‘ hypergeom’, $[\mu], [\nu,\lambda ], z$) from Maple, respectively.

**The computation of** $S_{\mu,\nu}(z)$:

\(1) For large $|z|$ and $|\arg z|<\pi,$ we can calculate efficiently $S_{\mu,\nu}(z)$ by truncating the following asymptotic expansion (see [@Watson pp. 351-352]) in inverse powers of $z$: $$\begin{aligned}
\nonumber S_{\mu,\nu}(z)&=&z^{\mu-1}\left\{1-\frac{(\mu-1)^2-\nu^2}{z^2}+\frac{[(\mu-1)^2-\nu^2][(\mu-3)^2-\nu^2]}{z^4}-\ldots\right.\\
\nonumber&&\left.+(-1)^p\frac{[(\mu-1)^2-\nu^2]\ldots[(\mu-2p+1)^2-\nu^2]}{z^{2p}}\right\}+O(z^{\mu-2p-2}),\end{aligned}$$

\(2) For small $|z|$, we prefer to compute $S_{\mu,\nu}(z)$ using .

So, when $r=b\omega$ is large, such as $r\geq50,$ we prefer to compute the moments using . when $r=b\omega$ is small, for example $r<50,$ the moments are available. This may be due to the property that $J_m(r)$ is a fast decreasing function of $m$ when $m>r$. Practical experiments also demonstrate that $J_m(r)$ can decrease to zero quite rapidly when $m$ is a little larger than $r$. Fortunately, the moments is available for all $r$.

Error bounds of the CCF methods and 
====================================

To obtain results that are absolutely reliable for numerical computations, it is necessary to construct a upper bound for the corresponding error. In the following, we will consider new and computational error bounds. These new error bounds share that for fixed $N$, the error bounds are optimal on $\omega$, while for fixed $\omega$ the error bounds are optimal on $N$.

In the following, in order to derive these new error bounds in inverse powers of $\omega$, we first give Lemmas 3.1 and 3.2.

For every $t\in[0,b]$ $(b>0)$ and $\alpha>-1$, it is true that, for $\omega\geq1$, $$\begin{aligned}
\label{3.23}\int_0^tx^\alpha J_m(\omega x)dx&=&{\displaystyle\left\{
\begin{array}{ll}O(\frac{1}{\omega^{\alpha+1}}), & \textrm{if}\ -1<\alpha<0,
\\O(\frac{1}{\omega}), & \textrm{if}\ \
\alpha\geq0,\end{array}\right.}\\
\label{3.24}\int_0^tx^\alpha \ln (x) J_m(\omega
x)dx&=&{\displaystyle\left\{
\begin{array}{ll}O(\frac{1+\ln(\omega)}{\omega^{\alpha+1}}), & \textrm{if}\ -1<\alpha\leq0,
\\O(\frac{1}{\omega}), & \textrm{if}\ \
\alpha>0.\end{array}\right.}\end{aligned}$$

We divide our proof in three steps.

\(1) For $-1<\alpha<0,$ setting $y=\omega x$ yields that $$\begin{aligned}
\nonumber \int_0^tx^\alpha J_m(\omega
x)dx&=&\frac{1}{\omega^{\alpha+1}}\int_0^{\omega t}y^\alpha
J_m(y)dy,\\
\nonumber \int_0^tx^\alpha \ln (x) J_m(\omega
x)dx&=&\frac{1}{\omega^{\alpha+1}}\left[\int_0^{\omega t}y^\alpha
\ln(y) J_m(y)dy-\ln(\omega)\int_0^{\omega t}y^\alpha
J_m(y)dy\right].\end{aligned}$$ Obviously, whether the integral upper limit $\omega t$ in the right-side of the above two formulae is finite or not, by convergence tests for improper integrals (*Cauchy’s test* or *Dirichelet’s test*), we know that the resulting defect or infinite integrals are convergent. It leads to the first identities in and .

\(2) For $\alpha=0,$ combining $\int_0^\infty J_m(t)dt=1$ [@Abram p.486] and the moments formula , we have $$\int_0^t J_m(\omega
x)dx=\frac{1}{\omega}\int_0^{\omega
t}J_m(y)dy=O\bigg(\frac{1}{\omega}\bigg).$$

If $0<\omega t\leq1,$ from [@Abram p.362] and [@Frank], we have $$\begin{aligned}
\label{3.251}|J_m(x)|\leq1,m\geq0,x\in\Re.\end{aligned}$$ So, the first identity in follows that $$\begin{aligned}
\nonumber \left|\int_0^t \ln (x) J_m(\omega
x)dx\right|&=&\frac{1}{\omega}\left|\int_0^{\omega
t}\ln(\frac{y}{\omega})
J_m(y)dy\right|\\
  \nonumber&\leq&\frac{1}{\omega}\int_0^{\omega
t}|\ln(y)-\ln(\omega)||J_m(y)|dy\\
\nonumber&\leq&\frac{1}{\omega}\int_0^1 \big(-\ln(y)+\ln(\omega)
\big) dy\\
\label{3.25}&=&\frac{1+\ln(\omega)}{\omega}.\end{aligned}$$ If $\omega t>1,$ from the proof of , we then obtain $$\begin{aligned}
\nonumber \left|\int_0^t \ln (x) J_m(\omega
x)dx\right|&=&\frac{1}{\omega}\left|\int_0^{\omega
t}\ln(\frac{y}{\omega})
J_m(y)dy\right|\\
  \nonumber&\leq&\frac{1}{\omega}\left|\int_0^1(\ln(y)-\ln(\omega))J_m(y)dy\right|+\frac{1}{\omega}\left|\int_1^{\omega t}(\ln(y)-\ln(\omega))J_m(y)dy\right|\\
\label{3.26}&\leq&\frac{1+\ln(\omega)}{\omega}+\frac{1}{\omega}\left|\int_1^{\omega
t}(\ln(y)-\ln(\omega))J_m(y)dy\right|.\end{aligned}$$ Using the mean value theorem for integrals, we have $$\int_1^{\omega t}(\ln(y)-\ln(\omega))J_m(y)dy=-\ln(\omega)\int_1^\xi
J_m(y)dy+\ln(t)\int_\xi^{\omega t} J_m(y)dy,\ \text{for}\ 1\leq
\xi\leq\omega t.$$ Then, it follows that $$\begin{aligned}
\nonumber \left|\int_1^{\omega
t}(\ln(y)-\ln(\omega))J_m(y)dy\right|&\leq&\ln(\omega)\left|\int_1^\xi
J_m(y)dy\right|+|\ln(t)|\left|\int_\xi^{\omega t} J_m(y)dy\right|\\
\label{3.27}&=&O(1+\ln(\omega)),\end{aligned}$$ which is due to the fact that both $\int_1^\xi J_m(y)dy$ and $\int_\xi^{\omega t} J_m(y)dy$ converge by referring to the identity $\int_0^\infty J_m(t)dt=1$ and the moments formula . Thus, combining and yields the first identity in .

\(3) For $\alpha>0$, by integrating by parts and noting the differential relation \[1,pp.361,439\] $$\begin{aligned}
\label{3.281}d[x^{m+1}J_{m+1}(\omega x)]=\omega x^{m+1}J_m(\omega x)dx,\end{aligned}$$ and together with the first identities in and , we find $$\begin{aligned}
\nonumber \int_0^{t}x^\alpha J_m(\omega
x)dx&=&\frac{1}{\omega}\int_0^t
x^{\alpha-m-1}d[x^{m+1}J_{m+1}(\omega x)]\\
\nonumber&=&\frac{1}{\omega}\left[x^\alpha J_{m+1}(\omega
x)\big|_0^t-(\alpha-m-1)\int_0^t x^{\alpha-1}J_{m+1}(\omega
x)dx\right]\\
\nonumber&=&O\bigg(\frac{1}{\omega}\bigg).\end{aligned}$$ Similarly, we obtain $$\int_0^{t}x^\alpha \ln(x) J_m(\omega
x)dx=O\bigg(\frac{1}{\omega}\bigg).$$ This completes the proof.

From Lemma 3.1, we prove Lemma 3.2.

For $f\in C[0,b], \alpha>-1$ and $\omega\geq1$, it is true that, $$\begin{aligned}
\label{3.28}\left|\int_0^bt^\alpha f(t)J_m(\omega
t)dt\right|&\leq&C_1(\omega)(|f(b)|+\int_0^b|f'(t)|dt),\\
\label{3.29}\left|\int_0^bt^\alpha \ln(t)f(t)J_m(\omega
t)dt\right|&\leq&C_2(\omega)(|f(b)|+\int_0^b|f'(t)|dt),\end{aligned}$$ where $$\begin{aligned}
\nonumber C_1(\omega)&=&{\displaystyle\left\{
\begin{array}{ll}\frac{c_1}{\omega^{\alpha+1}}, & \textrm{if}\ -1<\alpha<0,
\\\frac{c_2}{\omega}, & \textrm{if}\ \
\alpha\geq0,\end{array}\right.}\\
\nonumber C_2(\omega)&=&{\displaystyle\left\{
\begin{array}{ll}\frac{c_3(1+\ln(\omega))}{\omega^{\alpha+1}}, & \textrm{if}\ -1<\alpha<0,
\\\frac{c_4}{\omega}, & \textrm{if}\ \
\alpha\geq0,\end{array}\right.}\end{aligned}$$ and $c_k(k=1,2,3,4)$ are four constants independent of $\omega$ and $f$.

Setting $F(t)=\int_0^tx^\alpha J_m(\omega t)dt,t\in[0,b]$, together with Lemma 3.1, we then have $$\begin{aligned}
\nonumber C_1(\omega)=||F(t)||_\infty={\displaystyle\left\{
\begin{array}{ll}\frac{c_1}{\omega^{\alpha+1}}, & \textrm{if}\ -1<\alpha<0,
\\\frac{c_2}{\omega}, & \textrm{if}\ \
\alpha\geq0,\end{array}\right.}\end{aligned}$$ These together implies that, by integrating by parts, $$\begin{aligned}
\nonumber \left|\int_0^bt^\alpha f(t)J_m(\omega r
)dt\right|&=&\left|\int_0^b f(t)dF(t)\right|\\
\nonumber&=&\left|f(t)F(t)\big|_0^b-\int_0^b F(t)f'(t)dt\right|\\
\nonumber&\leq&|f(b)||F(b)|+\int_0^b |F(t)||f'(t)|dt\\
\nonumber&\leq&|f(b)|||F(t)||_\infty+\int_0^b |f'(t)|dt||F(t)||_\infty\\
\nonumber&=&C_1(\omega)(|f(b)|+\int_0^b|f'(t)|dt).\end{aligned}$$ It is now obvious that the assertion holds. The proof of can be completed by the method analogous to that used above.

Meanwhile, it should also be noted that the following Lemma 3.3 also plays an important role in the construction of error bounds.

(see [@Xiang2]) Let $n$ be a nonnegative integer. If $f$ is analytic with $|f(z)|\leq M$ in the region $\mathscr{E}_{\rho}$ bounded by the ellipse with foci $\pm 1$ and major and minor semi-axes whose lengths sum to $\rho>1,$ then for $x\in[-1,1]$, $$\|f^{(n)}(x)-P_{N}^{(n)}f(x)\|_\infty\leq
\frac{2M(N+1)^{2n}}{({\rho}^N-{\rho}^{-N})(2n-1)!!}\sum_{j=0}^n
\biggl(\frac{2\rho}{(\rho-1)^2}\biggr)^{n+1-j},$$ where $(2n-1)!!=1\cdot3\cdot5 \cdots (2n-1)$ and $(-1)!!=1$.

Based on the above Lemmas 3.1-3.3, we derive error bounds in inverse powers of $\omega$ in the following Theorems 3.1-3.2. For $f\in
C^2[0,b],$ the error bound of the CCF methods is shown as follows.

Assume that $f\in C^2[0,b].$ Then the absolute error of the CCF methods for $\omega\geq1, \alpha>-1$ and every fixed $N$, satisfies $$\begin{aligned}
\label{3.31}|I_1[f]-I_1^{CCF}[f]|\leq \min{\displaystyle\left\{
\begin{array}{lll}\frac{b^{\alpha+1}}{\alpha+1}||f(x)-P_Nf(x)||_\infty,\\
bC_1(\omega)||f'(x)-P'_Nf(x)||_\infty,\\
\frac{C_1(\omega)}{\omega}[|f'(b)-P'_Nf(b)|+b(1+\frac{3}{2}|\alpha-m-1|)||f''(x)-P''_Nf(x)||_\infty].
\end{array}\right\}}\ \ \\end{aligned}$$

From the definition of $P_Nf(x)$, it is obvious that $$\label{3.32}f(0)-P_Nf(0)=f(b)-P_Nf(b)=0.$$ In the following the proof will be split into three parts.

\(1) For the first inequality in , it follows at once from that $$\begin{aligned}
\nonumber
|I_1[f]-I_1^{CCF}[f]|&=&\bigg|\int_0^bx^\alpha(f(x)-P_Nf(x))J_m(\omega
x)dx\bigg|\\
\nonumber&\leq&\int_0^b|x^\alpha(f(x)-P_Nf(x))J_m(\omega
x) |dx\\
\nonumber&\leq&\int_0^bx^\alpha dx||f(x)-P_Nf(x)||_\infty\\
\nonumber&=&\frac{b^{\alpha+1}}{\alpha+1}||f(x)-P_Nf(x)||_\infty.\end{aligned}$$

\(2) By using Lemma 3.2 and the identities , the second inequality in follows that $$\begin{aligned}
\nonumber
|I_1[f]-I_1^{CCF}[f]|&=&\bigg|\int_0^bx^\alpha(f(x)-P_Nf(x))J_m(\omega
x)dx\bigg|\\
\nonumber&\leq&C_1(\omega)(|f(b)-P_Nf(b)|+\int_0^b|f'(x)-P'_Nf(x)|dx)\\
\nonumber&\leq&bC_1(\omega)||f'(x)-P'_Nf(x)||_\infty.\end{aligned}$$

\(3) Since $f(x)-P_Nf(x)$ and $f'(x)-P'_Nf(x)$ can be expanded in terms of [*Maclaurin expansions*]{}, as follows, $$\begin{aligned}
\nonumber
f(x)-P_Nf(x)&=&f(0)-P_Nf(0)+(f'(0)-P'_Nf(0))x+\frac{f''(\eta_1)-P''_Nf(\eta_1)}{2}x^2\\
\nonumber&=&(f'(0)-P'_Nf(0))x+\frac{f''(\eta_1)-P''_Nf(\eta_1)}{2}x^2,\
\ 0<\eta_1<x,\\
\nonumber
f'(x)-P'_N(x)&=&f'(0)-P'_Nf(0)+(f''(\eta_2)-P''_Nf(\eta_2))x, \ \
0<\eta_2<x,\end{aligned}$$ then we have $$\begin{aligned}
\nonumber
\bigg|\bigg(\frac{f(x)-P_Nf(x)}{x}\bigg)'\bigg|&=&\bigg|\frac{x(f'(x)-P'_Nf(x))-(f(x)-P_Nf(x))}{x^2}\bigg|\\
\nonumber&=&\big|(f''(\eta_2)-P''_Nf(\eta_2))-\frac{1}{2}(f''(\eta_1)-P''_Nf(\eta_1))\big|\\
\nonumber&\leq&\big|f''(\eta_2)-P''_Nf(\eta_2)\big|+\frac{1}{2}\big|f''(\eta_1)-P''_Nf(\eta_1)\big|\\
\label{3.34}&\leq&\frac{3}{2}||f''(x)-P''_Nf(x)||_\infty.\end{aligned}$$

By integrating by parts and using Lemma 3.2, together with , , , and due to the limit $$\lim_{x\rightarrow0+}\frac{f(x)-P_Nf(x)}{x}=f'(0)-P'_Nf(0),$$ we then obtain $$\begin{aligned}
\nonumber
|I_1[f]-I_1^{CCF}[f]|&=&\bigg|\int_0^bx^{\alpha+1}\frac{f(x)-P_Nf(x)}{x}J_m(\omega
x)dx\bigg|\\
\nonumber&=&\frac{1}{\omega}\bigg|\int_0^bx^{\alpha-m}\frac{f(x)-P_Nf(x)}{x}d[x^{m+1}J_{m+1}(\omega
x)]\bigg| \\
\nonumber&=&\bigg|\frac{1}{\omega}x^{\alpha+1}\frac{f(x)-P_Nf(x)}{x}J_{m+1}(\omega
x)\big|_0^b\\
\nonumber&&-\frac{1}{\omega}\int_0^bx^{m+1}J_{m+1}(\omega
x)d\bigg[x^{\alpha-m}\frac{f(x)-P_Nf(x)}{x}\bigg]\bigg| \\
\nonumber&\leq&\frac{1}{\omega}\big|\alpha-m-1\big|\bigg|\int_0^bx^{\alpha}\frac{f(x)-P_Nf(x)}{x}J_{m+1}(\omega
x)dx\bigg|\\
\nonumber&&+\frac{1}{\omega}\bigg|\int_0^bx^\alpha[f'(x)-P'_Nf(x)]J_{m+1}(\omega
x)dx\bigg|\\
\nonumber&\leq&\bigg|\alpha-m-1\bigg|\frac{C_1(\omega)}{\omega}\bigg(\bigg|\frac{f(b)-P_Nf(b)}{b}\bigg|+\int_0^b\bigg|\bigg(\frac{f(x)-P_Nf(x)}{x}\bigg)'\bigg|dx\bigg)\\
\nonumber&&+\frac{C_1(\omega)}{\omega}\big(|f'(b)-P'_Nf(b)|+\int_0^b|(f'(x)-P'_Nf(x))'|dx\big)\\
\label{3.351}&\leq&\frac{C_1(\omega)}{\omega}[|f'(b)-P'_Nf(b)|+b(1+\frac{3}{2}|\alpha-m-1|)||f''(x)-P''_Nf(x)||_\infty].
\ \ \ \ \ \ \ \ \\end{aligned}$$

We have thus proved the theorem.

For $f\in C^3[0,b],$ the error bound of the CCF methods is presented as follows.

Assume that $f\in C^3[0,b].$ Then the absolute error of the CCF methods for $\omega\geq1,\alpha>-1$ and every fixed $N$, satisfies $$\begin{aligned}
\label{3.35}
|I_2[f]-I_2^{CCF}[f]|\leq \min{\displaystyle\left\{
\begin{array}{lll}{\displaystyle\left\{\begin{array}{lll}\frac{b^{\alpha+1}(1-(\alpha+1)\ln
(b))}{(\alpha+1)^2}||f(x)-P_Nf(x)||_\infty,& \textrm{if}\ \ 0<b\leq 1,\\
\frac{2+b^{\alpha+1}((\alpha+1)\ln
(b)-1)}{(\alpha+1)^2}||f(x)-P_Nf(x)||_\infty,& \textrm{if}\ \ b>1,
\end{array}\right.}\\
bC_2(\omega)||f''(x)-P''_Nf(x)||_\infty,\\
\frac{3b}{2\omega}(C_1(\omega)+|\alpha-m|C_2(\omega))||f''(x)-P''_Nf(x)||_\infty\\
+\frac{C_2(\omega)}{\omega}[\frac{1}{b}|f'(b)-P'_Nf(b)|+\frac{7}{3}b||f'''(x)-P'''_Nf(x)||_\infty].
\end{array}\right\}}\end{aligned}$$

\(1) For the first inequalities in , it follows that $$\begin{aligned}
\nonumber |I_2[f]-I_2^{CCF}[f]|&=&\bigg|\int_0^bx^\alpha
\ln(x)(f(x)-P_Nf(x))J_m(\omega
x)dx\bigg|\\
\nonumber&\leq&\int_0^b|x^\alpha \ln(x)(f(x)-P_Nf(x))J_m(\omega
x) |dx\\
\nonumber&\leq&\int_0^b|x^\alpha \ln(x)|dx||f(x)-P_Nf(x)||_\infty\\
\nonumber&=&{\displaystyle\left\{\begin{array}{lll}\frac{b^{\alpha+1}(1-(\alpha+1)\ln
(b))}{(\alpha+1)^2}||f(x)-P_Nf(x)||_\infty,& \textrm{if}\ \ 0<b\leq 1,\\
\frac{2+b^{\alpha+1}((\alpha+1)\ln
(b)-1)}{(\alpha+1)^2}||f(x)-P_Nf(x)||_\infty,& \textrm{if}\ \ b>1.
\end{array}\right.}\end{aligned}$$

\(2) The second inequality in can be proved by the same method as employed in the proof of the second inequality in .

\(3) Since $f(x)-P_Nf(x), f'(x)-P'_Nf(x)$ and $f''(x)-P''_Nf(x)$ can be expanded in terms of [*Maclaurin expansions*]{}, as follows, $$\begin{aligned}
\nonumber
f(x)-P_Nf(x)&=&f(0)-P_Nf(0)+(f'(0)-P'_Nf(0))x+\frac{f''(0)-P''_Nf(0)}{2}x^2+\frac{f'''(\xi_1)-P''_Nf(\xi_1)}{6}x^3\\
\nonumber&=&(f'(0)-P'_Nf(0))x+\frac{f''(0)-P''_Nf(0)}{2}x^2+\frac{f'''(\xi_1)-P''_Nf(\xi_1)}{6}x^3,\
\ 0<\xi_1<x,\\
\nonumber
f'(x)-P'_Nf(x)&=&f'(0)-P'_Nf(0)+(f''(0)-P''_Nf(0))x+\frac{f'''(\xi_2)-P''_Nf(\xi_2)}{2}x^2,
\ \ 0<\xi_2<x,\\
\nonumber
f''(x)-P''_Nf(x)&=&f''(0)-P''_Nf(0)+(f'''(\xi_3)-P'''_Nf(\xi_3))x, \
\ 0<\xi_3<x,\end{aligned}$$ we then have $$\begin{aligned}
\nonumber
\bigg|\bigg(\frac{f(x)-P_Nf(x)}{x}\bigg)''\bigg|&=&\bigg|\frac{x^2(f''(x)-P''_Nf(x))-2x(f'(x)-P'_Nf(x))+2(f(x)-P_Nf(x))}{x^3}\bigg|\\
\nonumber&=&\big|(f'''(\xi_3)-P'''_Nf(\xi_3))-(f'''(\xi_2)-P'''_Nf(\xi_2))-\frac{1}{3}(f'''(\xi_1)-P'''_Nf(\xi_1))\big|\\
\label{3.36}&\leq&\frac{7}{3}||f'''(x)-P'''_Nf(x)||_\infty.\end{aligned}$$

By integrating by parts and using Lemma 3.2, together with , , , and noting that $$\lim_{x\rightarrow0+}x^{\alpha+1}\ln(x)\frac{f(x)-P_Nf(x)}{x}=0,$$ using the same argument as in the proof of , we can easily carry out the proof of the third inequality in .

The proof of the theorem is now complete.

By transferring the integra interval $[0, b]$ into $[-1,1]$, these estimates $||f(x)-P_Nf(x)||_\infty$, $||f'(x)-P'_Nf(x)||_\infty,||f''(x)-P''_Nf(x)||_\infty,||f'''(x)-P'''_Nf(x)||_\infty$ in Theorems 3.1-3.2, have been given by Lemma 3.3.

As shown in the above Theorems 3.1-3.2, for fixed $N$, we give these error bounds in inverse powers of $\omega$. In the sequel, for fixed $\omega$, we consider error bounds in inverse powers of $N$. Here, to derive these error bounds, we first establish the following Lemma 3.4.

For every $j\geq1$ and fixed $\omega$, it is true that $$\begin{aligned}
\label{3.38}\int_0^bx^\alpha T^{**}_j(x)J_m(\omega
x)dx&=&{\displaystyle\left\{
\begin{array}{ll}O(\frac{1}{j^{2\alpha+2}}), & \textrm{if}\ -1<\alpha<-\frac{1}{2},
\\O(\frac{1}{j^2}), & \textrm{if}\ \
\alpha\geq-\frac{1}{2},\end{array}\right.}\\
\label{3.39}\int_0^bx^\alpha \ln (x) T^{**}_j(x)J_m(\omega
x)dx&=&{\displaystyle\left\{
\begin{array}{ll}O(\frac{1+\ln(j)}{j^{2\alpha+2}}), & \textrm{if}\ -1<\alpha<-\frac{1}{2},
\\O(\frac{1}{j}), & \textrm{if}\ \
\alpha=-\frac{1}{2},\\
O(\frac{1}{j^2}), & \textrm{if}\ \ \alpha>-\frac{1}{2}.
\end{array}\right.}\end{aligned}$$

For $-1\leq
t\leq1$, three transformations $x=\frac{b}{2}+\frac{b}{2}t, $ $t=\cos\theta$ and $\theta=\pi-2\varphi$, yields that $$\label{3.40}
\int_0^bx^\alpha T^{**}_j(x)J_m(\omega
x)dx=(-1)^j2b^{\alpha+1}\int_0^{\frac{\pi}{2}}\cos(2j\varphi)\sin^{2\alpha+1}(\varphi)\cos(\varphi)J_m(b\omega
\sin^2(\varphi))d\varphi.$$

\(1) In the case of $\alpha\geq-\frac{1}{2}$: Based on differential relations $\cos(2j\varphi)d\varphi=\frac{1}{2j}d\sin(2j\varphi)$ and $\sin(2j\varphi)d\varphi=-\frac{1}{2j}d\cos(2j\varphi),$ we can derive the second identity in by integrating by parts twice.

\(2) In the case of $-1<\alpha<-\frac{1}{2}$: Setting $\varphi=\frac{u}{2j},$ we have

$$\begin{aligned}
\label{3.41}
\nonumber &&\int_0^bx^\alpha T^{**}_j(x)J_m(\omega x)dx\\
&=&(-1)^j\frac{b^{\alpha+1}}{2^{2\alpha+1}}\frac{1}{j^{2\alpha+2}}\int_0^{j\pi}u^{2\alpha+1}\cos(u)\left(\frac{\sin(\frac{u}{2j})}{\frac{u}{2j}}\right)^{2\alpha+1}\cos(\frac{u}{2j})J_m(b\omega
\sin^2(\frac{u}{2j}))du.\end{aligned}$$

Now that the right-side improper integral in is convergent, it is evident to see that the first identity in holds.

Similarly, according to the fact that $$\begin{aligned}
\nonumber
&&\int_0^bx^\alpha\ln(x) T^{**}_j(x)J_m(\omega x)dx\\
\nonumber&=&(-1)^j4b^{\alpha+1}\int_0^{\frac{\pi}{2}}\cos(2j\varphi)\sin^{2\alpha+1}(\varphi)\ln(\sin(\varphi))\cos(\varphi)J_m(b\omega
\sin^2(\varphi))d\varphi\\
\nonumber&&+[\ln(b)+(2^\alpha-1)\ln2]\int_0^bx^\alpha
T^{**}_j(x)J_m(\omega x)dx,\end{aligned}$$ and the logarithmic relation $$\ln\bigg(\sin\bigg(\frac{u}{2j}\bigg)\bigg)=\ln\bigg(\frac{\sin(\frac{u}{2j})}{\frac{u}{2j}}\bigg)+\ln(2j)-\ln(u),$$ together with the assertion , by the same procedure in the proof of , we then obtain the desired result . We have thus proved the lemma.

Using the asymptotic theory of Fourier integrals (see Erd$\acute{e}$lyi[@Erdelyi1; @Erdelyi]), Piessens [@Kythe; @Piessens] established this asymptotic expansion for $j\rightarrow\infty$: $$\begin{aligned}
\label{3.421}
\nonumber &&\int_0^1x^\alpha T^*_j(x)J_m(\omega x)dx\\
\nonumber&=&2^{-\alpha-1}\int_{-1}^1(1+x)^\alpha T_j(x)J_m(\omega
(x+1)/2)dx\\
&\sim&-2^{-\alpha-2}J_j(\omega)j^{-2}+(-1)^j2^{-3m-3\alpha-2}\frac{\omega^m}{\Gamma(m+1)}\cos((\alpha+1)\pi)\Gamma(2\alpha+2)j^{-2\alpha-2m-2}.\
\ \ \ \\end{aligned}$$ Then, by differentiating with respect to $\alpha$ and using , we have $$\begin{aligned}
\label{3.422}
\nonumber &&\int_0^1x^\alpha\ln(x) T^*_j(x)J_m(\omega x)dx\\
\nonumber&\sim&2^{-\alpha-2}\ln(2)J_j(\omega)j^{-2}+(-1)^{j+1}3\ln(2)2^{-3m-3\alpha-2}\frac{\omega^m}{\Gamma(m+1)}\cos((\alpha+1)\pi)\Gamma(2\alpha+2)j^{-2\alpha-2m-2}\\
\nonumber &&+(-1)^{j+1}\pi2^{-3m-3\alpha-2}\frac{\omega^m}{\Gamma(m+1)}\sin((\alpha+1)\pi)\Gamma(2\alpha+2)j^{-2\alpha-2m-2}\\
\nonumber &&+(-1)^{j}2^{-3m-3\alpha-1}\frac{\omega^m}{\Gamma(m+1)}\cos((\alpha+1)\pi)\Gamma(2\alpha+2)\psi_0(2\alpha+2)j^{-2\alpha-2m-2}\\
&&+(-1)^{j+1}2^{-3m-3\alpha-1}\frac{\omega^m}{\Gamma(m+1)}\cos((\alpha+1)\pi)\Gamma(2\alpha+2)j^{-2\alpha-2m-2}\ln(j).\end{aligned}$$ However, for each fixed $j$, the asymptotic expansions and are divergent for $\omega$. Moreover, the estimates in and can not be directly derived from these asymptotic formulas and particularly for $\alpha>0$.

With the aid of the above Lemma 3.4, we derive error bounds in inverse powers of $N$ as follows.

Assume that $f(x)$ has an absolutely continuous $(k-1)$st derivative $f^{(k-1)}(x)$ on $[0,b]$ and a $k$th derivative $f^{(k)}(x)$ of bounded variation $V_k$ for some $k\geq1.$ Then, for every fixed $\omega$, and $N\geq k,$ the absolute errors of the CCF methods and satisfy $$\begin{aligned}
\label{3.42}|I_1[f]-I_1^{CCF}[f]|&=&{\displaystyle\left\{
\begin{array}{ll}O(\frac{1}{N^{k+2\alpha+2}}), & \textrm{if}\ -1<\alpha<-\frac{1}{2},
\\O(\frac{1}{N^{k+1}}), & \textrm{if}\ \
\alpha\geq-\frac{1}{2},\end{array}\right.}\\
\label{3.43}|I_2[f]-I_2^{CCF}[f]|&=&{\displaystyle\left\{
\begin{array}{ll}O(\frac{\ln(N)}{N^{k+2\alpha+2}}), & \textrm{if}\ -1<\alpha<-\frac{1}{2},\\
O(\frac{\ln(N)}{N^{k+1}}), & \textrm{if}\ \alpha=-\frac{1}{2},\\
O(\frac{1}{N^{k+1}}), & \textrm{if}\ \ \alpha>-\frac{1}{2}.
\end{array}\right.}\end{aligned}$$

Recalling that $T_n^{**}(x)=T_n(\frac{2x}{b}-1)$ and , we have $$\begin{aligned}
\nonumber&&T_{2pN\pm j}^{**}(x_k)=T_{2pN\pm
j}\bigg(\frac{2x_k}{b}-1\bigg)=T_{2pN\pm
j}\bigg(\cos\bigg(\frac{k\pi}{N}\bigg)\bigg)=\cos\bigg((2pN\pm
j)\frac{k\pi}{N}\bigg)\\
\label{3.46}&&=\cos\bigg(\frac{jk\pi}{N}\bigg)=T_j\bigg(\cos\bigg(\frac{k\pi}{N}\bigg)\bigg)=T_{j}\bigg(\frac{2x_k}{b}-1\bigg)=T_{j}^{**}(x_k),\end{aligned}$$ for each $j$, $k=0,1,\ldots,N$ and $p=1,2,\ldots$.

From and , together with the discrete orthogonality of Chebyshev polynomials (see [@Mason Section 4.6]), we obtain $$\begin{aligned}
\label{3.47}
P_NT_{pN+j}^{**}(x)&=&{\displaystyle\left\{
\begin{array}{ll}T_{N-j}^{**}(x), & \textrm{if $p$ is odd},
\\T_{j}^{**}(x), & \textrm{if $p$ is even}.\end{array}\right.}\end{aligned}$$ Then, we can see directly from , and that $$\begin{aligned}
\label{3.48}I_1^{CCF}[T_{pN+j}^{**}(x)]&=&{\displaystyle\left\{
\begin{array}{ll}I_1[T_{N-j}^{**}(x)], & \textrm{if $p$ is odd},
\\I_1[T_{j}^{**}(x)], & \textrm{if $p$ is
even},\end{array}\right.}\\
\label{3.49}I_2^{CCF}[T_{pN+j}^{**}(x)]&=&{\displaystyle\left\{
\begin{array}{ll}I_2[T_{N-j}^{**}(x)], & \textrm{if $p$ is odd},
\\I_2[T_{j}^{**}(x)], & \textrm{if $p$ is even}.\end{array}\right.}\end{aligned}$$

Therefore, we can deduce from Lemma 3.4, and , that the sums of aliasing errors for the CCF methods and can be estimated for $p$ being a positive integer by $$\begin{aligned}
\nonumber\sum_{j=0}^NI_1^{CCF}[T_{pN+j}^{**}(x)]&=&{\displaystyle\left\{
\begin{array}{ll}\sum_{j=0}^NI_1[T_{N-j}^{**}(x)]=\sum_{j=0}^N\int_0^bx^\alpha T^{**}_{N-j}(x)J_m(\omega
x)dx, & \textrm{if $p$ is odd},
\\\sum_{j=0}^NI_1[T_{j}^{**}(x)]=\sum_{j=0}^N\int_0^bx^\alpha T^{**}_{0}(x)J_m(\omega
x)dx, & \textrm{if $p$ is
even},\end{array}\right.}\\
\label{3.50}&=&{\displaystyle\left\{\begin{array}{ll}O(\frac{1}{N^{2\alpha+1}}),
& \textrm{if}\ -1<\alpha<-\frac{1}{2},
\\O(1), & \textrm{if}\ \
\alpha\geq-\frac{1}{2},\end{array}\right\} \textrm{$p$ is either odd or even;}}\\
\nonumber\sum_{j=0}^NI_2^{CCF}[T_{pN+j}^{**}(x)]&=&{\displaystyle\left\{
\begin{array}{ll}\sum_{j=0}^NI_2[T_{N-j}^{**}(x)]=\sum_{j=0}^N\int_0^bx^\alpha\ln(x) T^{**}_{N-j}(x)J_m(\omega
x)dx, & \textrm{if $p$ is odd},
\\\sum_{j=0}^NI_2[T_{j}^{**}(x)]=\sum_{j=0}^N\int_0^bx^\alpha \ln(x)T^{**}_{j}(x)J_m(\omega
x)dx, & \textrm{if $p$ is
even},\end{array}\right.}\\
\label{3.51}&=&{\displaystyle\left\{\begin{array}{ll}O(\frac{\ln(N)}{N^{2\alpha+1}}),
& \textrm{if}\ -1<\alpha<-\frac{1}{2},\\
O(\ln(N)), & \textrm{if}\ \alpha=-\frac{1}{2},
\\O(1), & \textrm{if}\ \
\alpha>-\frac{1}{2},\end{array}\right\} \textrm{$p$ is either odd or
even.}}\end{aligned}$$

Moreover, we can find that Theorem 4.2 in [@Trefethen] implies the estimate $$\begin{aligned}
\label{3.491}|a_j|=O\bigg(\frac{1}{j^{k+1}}\bigg).\end{aligned}$$

Combining , and , and by a similar way as shown in the proof of Theorem 5.1 in [@Trefethen], we can deduce the desired results and .

Extension to a higher order method and error estimate
=====================================================

The choice of the extreme points as interpolation points for highly oscillatory integrals is not only a technical necessity but also can improve the accuracy significantly [@Olver; @Xiang; @Xiang2]. Moreover, only by adding derivative information of $f(x)$ at the endpoints $0$ and $b$ can the asymptotic order of the method be improved [@Kang1; @Kang; @Olver; @Xiang; @Xiang2]. By the above-mentioned particular observations, the higher order CCF methods for (1.1) and (1.2) can be defined, respectively, by $$\begin{aligned}
\nonumber I_1^{HCCF}[f]&=&\int_0^bx^\alpha P_{N+2s}f(x)J_m(\omega
x)dx\\
\label{4.55}&=&b^{\alpha+1}\sum_{j=0}^{N+2s}b_jM_j,\\
\nonumber I_2^{HCCF}[f]&=&\int_0^bx^\alpha \ln(x)P_{N+2s}f(x)J_m(\omega x)dx\\
\label{4.56}&=&b^{\alpha+1}\sum_{j=0}^{N+2s}b_j[\ln(b)M_j+\widetilde{M}_j],\end{aligned}$$ where $P_{N+2s}f(x)$ is the special *Hermite interpolation polynomial* [@Kang1; @Xiang2] of $f(x)$ at the Clenshaw-Curtis points $c_n =\frac{b}{2}+\frac{b}{2}\cos\left(\frac{n\pi}{N}\right)$ on $[0,b]$ satisfying $$\label{4.57}
P_{N+2s}^{(k)}f(0)=f^{(k)}(0),\quad P_{N+2s}f(c_n)=f(c_n),\quad
P_{N+2s}^{(k)}f(b)=f^{(k)}(b),$$ for $n=1,2,...,N-1$ and $k=0,1,...,s;$ it can be evaluated in $O(N\log N)$ operations [@Xiang2]. Moreover, the polynomial $P_{N+2s}f(x)$ can be expressed by $$P_{N+2s}f(x)=\sum_{j=0}^{N+2s} b_jT_j^{**}(x),$$ where the coefficients ${b_j}$ can be computed efficiently by an algorithm [@Xiang2].

Here, we establish error bounds in inverse powers of $N$ as follows.

Assume that $f\in C^{s+2}[0,b].$ Then the absolute errors of the higher order CCF methods and for $\alpha>-1$ and every fixed $N$, satisfy $$\begin{aligned}
|I_1[f]-I_1^{HCCF}[f]|&=&O\bigg(\frac{C_1(\omega)}{\omega^{s+1}}\bigg),\\
|I_2[f]-I_2^{HCCF}[f]|&=&O\bigg(\frac{C_2(\omega)}{\omega^{s+1}}\bigg).\end{aligned}$$

Based on the differential relation , we have $$\begin{aligned}
\nonumber I_1[f]-I_1^{HCCF}[f]&=&\int_0^bx^\alpha(f(x)-P_{N+2s}f(x))J_m(\omega x)dx\\
\label{4.61}&=&\frac{1}{\omega}\int_0^bx^{\alpha+s-m}\frac{f(x)-P_{N+2s}f(x)}{x^{s+1}}d[x^{m+1}J_{m+1}(\omega
x)],\\
\nonumber
I_2[f]-I_2^{HCCF}[f]&=&\int_0^bx^\alpha\ln(x)(f(x)-P_{N+2s}f(x))J_m(\omega
x)dx\\
\label{4.62}&=&\frac{1}{\omega}\int_0^bx^{\alpha+s-m}\ln(x)\frac{f(x)-P_{N+2s}f(x)}{x^{s+1}}d[x^{m+1}J_{m+1}(\omega
x)].\end{aligned}$$ By using and resorting to integrating and by parts $s+1$-time, respectively, together with Lemma 3.2, we establish the desired results as in the proof of the third inequalities in Theorems 3.1-3.2.

Conclusion
==========

This paper presents a series of new sharp error bounds of the Clenshaw-Curtis-Filon methods for two classes of oscillatory Bessel transforms with algebraic or logarithmic singularities. From the above error bounds, it is worth noting that the required accuracy can be obtained by using derivatives of $f(x)$ at the end-points or adding the number of the interior node points. Moreover, the Clenshaw-Curtis-Filon methods posses the advantageous property that for fixed $N$ the accuracy increases when oscillation becomes faster, which can be also obtained directly from these error bounds in inverse powers of $\omega$.

[1]{}

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[^1]: This work is supported by National Natural Science Foundation of China (Grant Nos.11301125, 11071260, 11226305,11301222), Scientific Research Startup Foundation of Hangzhou Dianzi University (KYS075613017), the Fundamental Research Funds for the Central Universities (No. 21612336) and NSF of Guangdong (No. S2012040007860).
