Fractional calculus of tempered distributions. A new approach

TL;DR

This paper introduces a novel method for defining fractional derivatives within the framework of tempered distributions, overcoming limitations of previous approaches and enabling fractional differentiation of periodic functions.

Contribution

It presents a new approach to fractional calculus using tempered distributions, ensuring desirable properties like the semigroup property.

Findings

The new method is free from previous drawbacks.

Fractional derivatives of periodic functions are definable.

The approach maintains the semigroup property.

Abstract

In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such as e.g. the absence of a semigroup property. It is shown that the proposed definition is free from such drawbacks. In order to achieve such a generalization one has to work in the context of tempered distributions (generalized functions) where the concept works nicely. The last part of the article shows the possibility of defining fractional derivatives of periodic functions (Fourier series).