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On the Invalidity of Fourier Series Expansions of Fractional Order

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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 ein?x are replaced by the Mittag-Leffler functions Ea (i(n?x)a), over the interval [0,Ma/?] where 0 < ? < 8 and Ma > 0 is the period of the function Ea (ixa), i.e., Ea (ixa) = Ea (i(x +Ma)a)

He showed that any smooth periodic function f with period Ma/? can be expanded in a Fourier-type series. We will show that the only possible period of the function Ea (ixa) is Ma = 0; hence the invalidity of any Fourier-type series expansion of f.

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Authors and Affiliations

  1. Center of Mathematics, Lehrstuhl M6 Technische Universität München, Boltzmannstr. 3, 85747, Garching b. Munich, Germany

    Peter R. Massopust

  2. Helmholtz Zentrum München, 85764, Neuherberg, Germany

    Peter R. Massopust

  3. Department of Mathematical Sciences, DePaul University, 2320 N. Kenmore Ave., SAC 524, Chicago, IL, 60614, USA

    Ahmed I. Zayed

Authors
  1. Peter R. Massopust
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  2. Ahmed I. Zayed
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Correspondence to Peter R. Massopust.

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Massopust, P.R., Zayed, A.I. On the Invalidity of Fourier Series Expansions of Fractional Order. FCAA 18, 1507–1517 (2015). https://doi.org/10.1515/fca-2015-0087

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  • DOI: https://doi.org/10.1515/fca-2015-0087

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