Symmetric Decompositions of $f\in L^2(\mathbb{R})$ Via Fractional Riemann-Liouville Operators

TL;DR

This paper establishes a unique symmetric decomposition of square-integrable functions using fractional Riemann-Liouville operators, characterizes the regularity of the decomposing function, and provides insights into the Fourier transform of such functions.

Contribution

It introduces a novel symmetric decomposition framework for functions in L^2(R) using fractional Riemann-Liouville operators, including regularity analysis and Fourier transform characterization.

Findings

Unique decomposition of f in L^2(R) into fractional derivatives of u.

Regularity properties of the decomposing function u.

Characterization of Fourier transforms of L^2(R) functions.

Abstract

It is proved that given $-1/2<s<1/2$, for any $f\in L^2(\mathbb{R})$, there is a unique $u\in \widehat{H}^{|s|}(\mathbb{R})$ such that $$ f=\boldsymbol{D}^{-s}u+\boldsymbol{D}^{s*}u\,, $$ where $\boldsymbol{D}^{-s}, \boldsymbol{D}^{s*}$ are fractional Riemann-Liouville operators and the fractional derivatives are understood in the weak sense. Furthermore, the regularity of $u$ is discussed, and other versions of the results are established. As an interesting consequence, the Fourier transform of elements of $L^2(\mathbb{R})$ is characterized.