Complex generalized Gauss-Radau quadrature rules for Hankel transforms of integer order

TL;DR

The paper addresses accurate quadrature for oscillatory Hankel transforms by developing complex generalized Gauss-Radau rules for integer-order transforms, incorporating left-endpoint values and derivatives and linking to generalized Prudnikov polynomials. It proves existence results for the associated orthogonal polynomials $P_n^{(\mu,\nu)}$ under parity conditions on $\mu-\nu$, derives exactness properties for the quadrature $\mathcal{Q}_{2n,\mu}^{\mathrm{HI}}$, and provides explicit asymptotic error rates that improve with the number of nodes. The methodology is validated numerically, with analysis of zero distributions and convergence behavior, and applied to oscillatory Hilbert transforms and electromagnetic field computations in layered media. The work offers a theoretically guaranteed, high-order approach for highly oscillatory Hankel integrals, with practical impact on simulations in physics and engineering where such transforms arise.

Abstract

Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to $[0,1/2]$. In this paper we consider the construction of generalized Gauss-Radau quadrature rules for Hankel transform. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss-Radau quadrature rules for Hankel transform of integer order can be constructed with theoretical guarantees. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to confirm our findings.

Figures (8)

• Figure 1: The zeros of $P_{n}^{(0,\nu)}(x)$ (blue) and the zeros of $P_{n}^{(\nu,\nu)}(x)$ (red). Top row shows $n=16$ for $\nu=1$ (left) and $\nu=2$ (right) and bottom row shows $n=36$ for $\nu=4$ (left) and $\nu=5$ (right). The vertical lines in the top row are $\Re(z)=\nu\pi/2$ and in the bottom row are $\Re(z)=(\nu-2)\pi/2$.
• Figure 2: The zeros of $P_n^{(\mu,\nu)}(x)$ for $\mu=1$ (left) and $\mu=2$ (right). Here $n=24$, $\nu=3/2$ (blue) and $\nu=7/2$ (red) and the vertical lines are $\Re(z)=(\nu-\mu)\pi/2$.
• Figure 3: Absolute errors of $(\mathcal{Q}_{2n,\mu}^{\mathrm{HI}}f)(\omega)$ with $n=1$ as a function of $\omega$ for $\nu=1$ (left) and $\nu=2$ (right). Here $f(x)= e^{-x}$ (top) and $f(x)=1/(1+x)^2$ (bottom), we choose different values of $\mu$ and the solid lines indicate the predicted rates $\mathcal{O}(\omega^{-4n-\mu-1})$ when $\mu-\nu$ is even and $\mathcal{O}(\omega^{-4n-\mu-2})$ when $\mu-\nu$ is odd.
• Figure 4: Absolute errors of $(\mathcal{Q}_{2n,\mu}^{\mathrm{HI}}f)(\omega)$ with $n=2$ as a function of $\omega$ for $\nu=1$ (left) and $\nu=2$ (right). Here $f(x)= e^{-x}$ (top) and $f(x)=1/(1+x)^2$ (bottom), we choose different values of $\mu$ and the solid lines indicate the predicted rates $\mathcal{O}(\omega^{-4n-\mu-1})$ when $\mu-\nu$ is even and $\mathcal{O}(\omega^{-4n-\mu-2})$ when $\mu-\nu$ is odd.
• Figure 5: Absolute errors of $(\mathcal{Q}_{2n,\mu}^{\mathrm{HI}}f)(\omega)$ as a function of $\omega$ for $n=1$, $\nu=2$ (left) and absolute and relative errors of $(\mathcal{Q}_{2n,\mu}^{\mathrm{HI}}f)(\omega)$ for $n=1$, $\nu=3$ (right). Here $f(x)=e^{-x^2}$, $\mu=\nu$ and the solid lines indicate the predicted rates $\mathcal{O}(\omega^{-4n-\mu-1})$.

Theorems & Definitions (15)

• Theorem 2.1
• proof
• Remark 2.2
• Remark 2.3
• Remark 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Remark 3.4