Fast computation of highly oscillatory Bessel transforms

Abstract

This study focuses on the efficient and precise computation of Bessel transforms, defined as ∫abf(x)Jν(ωx)dx. Exploiting the integral representation of Jν(ωx), 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<+∞ and b=+∞. 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.

Keywords

• Bessel transform
• Highly oscillatory integrals
• Steepest descent method