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

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 $ω$ for fixed $N$, while other error bounds are optimal on $N$ for fixed $ω$. Furthermore, we prove from the presented error bounds in inverse powers of $ω$ that the accuracy improves greatly, for fixed $N$, as $ω$ increases.