Error Bounds for Numerical Integration of Oscillatory Bessel Transforms with Algebraic or Logarithmic Singularities

TL;DR

This paper develops and rigorously analyzes Clenshaw-Curtis-Filon methods for oscillatory Bessel transforms with singularities, providing sharp error bounds that optimize accuracy with respect to frequency and discretization parameters.

Contribution

It introduces new error bounds for these quadrature rules, demonstrating their optimality and improved accuracy at higher frequencies.

Findings

Error bounds are sharp and optimal for fixed N or fixed ω.

Accuracy improves significantly as ω increases for fixed N.

The methods effectively handle singularities in oscillatory Bessel transforms.

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.