On the Invalidity of Fourier Series Expansions of Fractional Order

TL;DR

This paper demonstrates that the proposed Fourier series expansion of fractional order using Mittag-Leffler functions, as suggested by G. Jumarie, is invalid, challenging prior claims in fractional Fourier analysis.

Contribution

It critically analyzes and refutes the validity of G. Jumarie's fractional Fourier series expansion based on Mittag-Leffler functions.

Findings

The fractional Fourier series expansion is mathematically invalid.

The replacement of exponential functions with Mittag-Leffler functions does not preserve periodicity.

The paper clarifies limitations in fractional Fourier analysis methods.

Abstract

The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions $e^{in\omega x}$ are replaced by the Mittag-Leffler functions $E_\alpha \left (i (n\omega x)^\alpha\right) ,$ over the interval $[0, M_\alpha/ \omega]$ where $0< \omega<\infty $ and $M_\alpha$ is the period of the function $E_\alpha \left( ix^\alpha\right),$ i.e., $E_\alpha \left( ix^\alpha\right)=E_\alpha \left( i(x+M_\alpha)^\alpha\right).$