Local density of Caputo-stationary functions in the space of smooth functions

TL;DR

This paper proves that functions with zero Caputo fractional derivative are dense in the space of smooth functions, meaning any smooth function can be approximated by Caputo-stationary functions.

Contribution

It establishes the density of Caputo-stationary functions in the space of smooth functions, extending understanding of fractional derivatives and their approximation properties.

Findings

Caputo-stationary functions are dense in C^k_{loc}(ℝ)

Any C^k([0,1]) function can be approximated by Caputo-stationary functions

Caputo derivatives of these functions are zero in the specified domain

Abstract

We consider the Caputo fractional derivative and say that a function is Caputo-stationary if its Caputo derivative is zero. We then prove that any $C^k\big([0,1]\big)$ function can be approximated in $[0,1]$ by a a function that is Caputo-stationary in $[0,1]$, with initial point $a<0$. Otherwise said, Caputo-stationary functions are dense in $C^k_{loc}(\mathbb{R})$.