Peaks and jumps reconstruction with B-splines scaling functions

TL;DR

This paper introduces a B-spline based numerical method for inverting Fourier and Laplace transforms, especially effective for functions with discontinuities like peaks or jumps, demonstrating robustness and accuracy through numerical experiments.

Contribution

It presents a novel B-spline scaling function approach for stable inversion of transforms of functions with discontinuities, including explicit coefficient formulas.

Findings

Method effectively reconstructs functions with peaks and jumps.

Numerical experiments confirm robustness and high accuracy.

Approach is suitable for functions with domain discontinuities.

Abstract

We consider a methodology based in B-splines scaling functions to numerically invert Fourier or Laplace transforms of functions in the space $L^2(\mathbb{R})$. The original function is approximated by a finite combination of $j^{th}$ order B-splines basis functions and we provide analytical expressions for the recovered coefficients. The methodology is particularly well suited when the original function or its derivatives present peaks or jumps due to discontinuities in the domain. We will show in the numerical experiments the robustness and accuracy of the method.