Fractional Multiresolution Analysis and Associated Scaling Functions in $L^2(\mathbb R)$

TL;DR

This paper develops a method to construct orthonormal bases from Riesz bases using fractional translates in fractional multiresolution analysis, providing new insights into the structure of scaling functions in $L^2( eal)$.

Contribution

It introduces a novel approach to generate orthonormal bases from Riesz bases in fractional multiresolution analysis, including conditions for intersection triviality and union density.

Findings

Constructed orthonormal bases from Riesz bases using fractional translates.

Showed that intersection triviality follows from other conditions.

Provided a complete characterization of scaling functions in fractional multiresolution analysis.

Abstract

In this paper, we show how to construct an orthonormal basis from Riesz basis by assuming that the fractional translates of a single function in the core subspace of the fractional multiresolution analysis form a Riesz basis instead of an orthonormal basis. In the definition of fractional multiresolution analysis, we show that the intersection triviality condition follows from the other conditions. Furthermore, we show that the union density condition also follows under the assumption that the fractional Fourier transform of the scaling function is continuous at $0$. At the culmination, we provide the complete characterization of the scaling functions associated with fractional multiresolutrion analysis.