Local density of Caputo-stationary functions of any order

TL;DR

This paper demonstrates that any function can be approximated arbitrarily closely by solutions to linear time-fractional equations of any order, extending previous results to a broader class of fractional derivatives.

Contribution

It generalizes existing approximation results from fractional derivatives of order less than one to any fractional order, including the $ ext{psi}$-Caputo-stationary case.

Findings

Any function can be approximated by solutions of fractional equations of any order.

Extension of approximation results to $ ext{psi}$-Caputo-stationary functions.

Foundation for future work on operators involving anisotropic superpositions.

Abstract

We show that any given function can be approximated with arbitrary precision by solutions of linear, time-fractional equations of any prescribed order. This extends a recent result by Claudia Bucur, which was obtained for time-fractional derivatives of order less than one, to the case of any fractional order of differentiation. In addition, our result applies also to the $\psi$-Caputo-stationary case, and it will provide one of the building blocks of a forthcoming paper in which we will establish general approximation results by operators of any order involving anisotropic superpositions of classical, space-fractional and time-fractional diffusions.