Approximation by refinement masks

TL;DR

This paper introduces a method to construct a compactly supported Parseval wavelet frame with a refinement mask that can uniformly approximate any continuous periodic function satisfying certain bounds, ensuring stable integer shifts.

Contribution

It presents a novel approach to design refinement masks for wavelet frames that approximate continuous functions while maintaining stability and compact support.

Findings

Refinement mask uniformly approximates arbitrary continuous periodic functions.

Constructs a Parseval wavelet frame with compact support.

Refinable function has stable integer shifts.

Abstract

In the paper we design a Parseval wavelet frame with a compact support. The corresponding refinement mask uniformly approximates an arbitrary continuous periodic function $f$, $f(0)=1$, $|f(x)|^2+|f(x+\pi)|^2\le 1$. The refinable function has stable integer shifts.