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 $e^{inωx}$ are replaced by the Mittag-Leffler functions $E_α\left (i (nωx)^α\right) ,$ over the interval $[0, M_α/ ω]$ where $0< ω<\infty $ and $M_α$ is the period of the function $E_α\left( ix^α\right),$ i.e., $E_α\left( ix^α\right)=E_α\left( i(x+M_α)^α\right).$