Differential and falsified sampling expansions

TL;DR

This paper investigates differential and falsified sampling expansions, providing new approximation order results for band-limited functions with discontinuous Fourier transforms and comparing these with differential expansions.

Contribution

It extends approximation order analysis to band-limited functions with discontinuous Fourier transforms and compares falsified sampling expansions with differential expansions.

Findings

Approximation order results for band-limited functions with discontinuous Fourier transforms.

Error estimates in $L_p$-norm based on Fourier transform of the function.

Comparison of falsified sampling expansions with differential expansions.

Abstract

Differential and falsified sampling expansions $\sum_{k\in \mathbb{Z}^d}c_k\phi(M^jx+k)$, where $M$ is a matrix dilation, are studied. In the case of differential expansions, $c_k=Lf(M^{-j}\cdot)(-k)$, where $L$ is an appropriate differential operator. For a large class of functions $\phi$, the approximation order of differential expansions was recently studied. Some smoothness of the Fourier transform of $\phi$ from this class is required. In the present paper, we obtain similar results for a class of band-limited functions $\phi$ with the discontinuous Fourier transform. In the case of falsified expansions, $c_k$ is the mathematical expectation of random integral average of a signal $f$ near the point $M^{-j}k$. To estimate the approximation order of the falsified sampling expansions we compare them with the differential expansions. Error estimations in $L_p$-norm are given in terms of the Fourier transform of $f$.