On the Invalidity of Fourier Series Expansions of Fractional Order

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 $E_{\alpha }\left(i{x}^{\alpha }\right),$ i.e., $E_{\alpha }\left(i{x}^{\alpha }\right)=E_{\alpha }\left(i(x+M_{\alpha }{)}^{\alpha }\right).$