A note on the L1 discretization error for the Caputo derivative in H\"older spaces

TL;DR

This paper derives uniform error bounds for the L1 discretization of the Caputo derivative applied to H"older continuous functions, linking the error to the function's smoothness and the derivative's order, supported by an elementary proof and numerical validation.

Contribution

It provides a new, simple proof of error bounds for L1 discretization of the Caputo derivative in H"older spaces, highlighting the relationship between smoothness and discretization error.

Findings

Error bound proportional to smoothness minus derivative order

Elementary proof of the error estimate

Numerical examples confirm optimality

Abstract

We establish uniform error bounds of the L1 discretization of the Caputo derivative of H\"older continuous functions. The result can be understood as: error = (degree of smoothness - order of the derivative). We present an elementary proof and illustrate its optimality with numerical examples.