# Fractional operators via analytic interpolation of integer powers

## TL;DR

This paper introduces a novel interpolation framework for defining fractional powers of operators using integer powers, enhancing understanding and computational approaches in fractional calculus and related fields.

## Contribution

It presents a new method for constructing fractional operator powers through analytic interpolation of integer powers, offering both theoretical insights and practical tools.

## Key findings

- Provides a framework for approximating fractional powers by finite sums of integer powers.
- Enables new approaches to numerical analysis of fractional PDEs.
- Offers an interpolation procedure for the Riemann zeta function.

## Abstract

Although the study of functional calculus has already established necessary and sufficient conditions for operators to be fractionalized, this paper aims to use our well-conceived notion of integer powers of operators to construct non-integer powers of operators. In doing so, we not only provide a more intuitive understanding of fractional theories, but also provide a framework for producing new fractional theories. Such interpolations allow one to approximate fractional powers by finite sums of integer powers of operators, and thus may find much use in numerical analysis of fractional PDEs and the time-frequency analysis of the fractional Fourier transform. Further, these results provide an interpolation procedure for the Riemann zeta function.