On quasi-periodicity properties of fractional integrals and fractional derivatives of periodic functions

TL;DR

This paper investigates the quasi-periodic properties of fractional integrals and derivatives of periodic functions, analyzing conditions under which these fractional operations preserve or relate to various generalized periodicity concepts.

Contribution

It provides new insights into the quasi-periodic behavior of fractional calculus operators applied to periodic functions, including boundedness and specific property conditions.

Findings

Fractional derivatives and integrals can exhibit quasi-periodic properties under certain conditions.

Boundedness of fractional derivatives and integrals of periodic functions is established.

Conditions for fractional operators to preserve or induce generalized periodicity are identified.

Abstract

This paper is devoted to the study of quasi-periodic properties of fractional order integrals and derivatives of periodic functions. Considering Riemann-Liouville and Caputo definitions, we discuss when the fractional derivative and when the fractional integral of a certain class of periodic functions satisfies particular properties. We study concepts close to the well known idea of periodic function, such as S-asymptotically periodic, asymptotically periodic or almost periodic function. Boundedness of fractional derivative and fractional integral of a periodic function is also studied.