A Matricial Algorithm for Polynomial Refinement

TL;DR

This paper proves that all single-variable polynomials are finitely refinable using a linear algebra approach, addressing a question posed by David Larson about polynomial refinability in wavelet analysis.

Contribution

It provides a concise proof that all polynomials are finitely refinable, filling a gap in the understanding of wavelet scaling functions.

Findings

All single-variable polynomials are finitely refinable.

The proof uses basic linear algebra techniques.

Addresses a longstanding open question in wavelet theory.

Abstract

In order to have a multiresolution analysis, the scaling function must be refinable. That is, it must be the linear combination of 2-dilation, $\mathbb{Z}$-translates of itself. Refinable functions used in connection with wavelets are typically compactly supported. In 2002, David Larson posed the question in his REU site, "Are all polynomials (of a single variable) finitely refinable?" That summer the author proved that the answer indeed was true using basic linear algebra. The result was presented in a number of talks but had not been typed up until now. The purpose of this short note is to record that particular proof.