Application of the Wavelet Transform with a Piecewise Linear Basis to the Evaluation of the Hankel Transform

TL;DR

This paper introduces a novel method for computing the Hankel transform by combining wavelet decomposition with a piecewise linear basis, enabling analytical calculation of the scalar product with Bessel functions.

Contribution

The paper presents a new approach that uses wavelet decomposition to efficiently evaluate the Hankel transform, improving accuracy and local adaptability over traditional methods.

Findings

Effective in representing smooth functions with wavelet basis

Allows analytical computation of scalar products with Bessel functions

Demonstrated with practical example and plots

Abstract

A method for computing the Hankel transform is proposed whereby the letter is reduced to a sum by representing the integrand as a smooth function times a Bessel function. The smooth function is replaced by its wavelet decomposition with a basis such that its scalar product with the Bessel function is calculated analytically. The result is represented as a series, with the coefficients strongly depending on the local behavior of the function being transformed. The application of the method is demonstrated by an example illustrated with plots.