Moving Least Squares Approximation using Variably Scaled Discontinuous Weight Function

TL;DR

This paper introduces a moving least-squares method for approximating functions with discontinuities, using variably scaled discontinuous weight functions to improve accuracy in applications like image reconstruction and signal processing.

Contribution

It proposes a novel approach that incorporates discontinuities into the weight functions of the least-squares approximation, enhancing accuracy for functions with jumps.

Findings

Numerical experiments confirm the theoretical convergence rate.

The method effectively handles functions with discontinuities.

Error estimates are provided for piecewise Sobolev spaces.

Abstract

Functions with discontinuities appear in many applications such as image reconstruction, signal processing, optimal control problems, interface problems, engineering applications and so on. Accurate approximation and interpolation of these functions are therefore of great importance. In this paper, we design a moving least-squares approach for scattered data approximation that incorporates the discontinuities in the weight functions. The idea is to control the influence of the data sites on the approximant, not only with regards to their distance from the evaluation point, but also with respect to the discontinuity of the underlying function. We also provide an error estimate on a suitable {\it piecewise} Sobolev Space. The numerical experiments are in compliance with the convergence rate derived theoretically.