Fast computation of highly oscillatory Bessel transforms

Abstract

This study focuses on the efficient and precise computation of Bessel transforms, defined as ${\int }_a^{b}f\left(x\right)J_{\nu }\left(\omega x\right)dx$. Exploiting the integral representation of $J_{\nu }\left(\omega x\right)$, these Bessel transformations are reformulated into the oscillatory integrals of Fourier-type. When $a>0$, these Fourier-type integrals are transformed through distinct complex integration paths for cases with $b<+\infty$ and $b=+\infty$. Subsequently, we approximate these integrals using the generalized Gauss–Laguerre rule and provide error estimates. This approach is further extended to situations where $a=0$ by partitioning the integral’s interval into two separate subintervals. Several numerical experiments are provided to demonstrate the efficiency and accuracy of the proposed algorithms.