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Massopust, P. ; Zayed, A.I.*

On the invalidity of fourier series expansions of fractional order.

Fractional Calc. Appl. Anal. 18, 1507-1517 (2015)
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Open Access Green
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 einωx are replaced by the Mittag-Leffler functions Eα (i(nωx)α) , over the interval [0,Mα/ω] where 0 < ω < ∞ and Mα > 0 is the period of the function Eα (ixα) , i.e., Eα (ixα) = Eα (i(x +Mα)α) . He showed that any smooth periodic function f with period Mα/ω can be expanded in a Fourier-type series. We will show that the only possible period of the function Eα (ixα) is Mα = 0; hence the invalidity of any Fourier-type series expansion of f.
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Publication type Article: Journal article
Document type Scientific Article
Keywords Fourier Series Of Fractional Order ; Fractional Derivative ; Mittag-leffler Function
ISSN (print) / ISBN 1311-0454
e-ISSN 1314-2224
Quellenangaben Volume: 18, Issue: 6, Pages: 1507-1517 Article Number: , Supplement: ,
Publisher Versita
Publishing Place Warsaw
Reviewing status