On the Invalidity of Fourier Series Expansions of Fractional Order
On the Invalidity of Fourier Series Expansions of Fractional Order
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DOI：
10.1515/fca-2015-0087
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发表时间：
2015-07
影响因子：
3
通讯作者：
P. Massopust;A. Zayed
中科院分区：
文献类型：
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作者：
P. Massopust;A. Zayed
摘要 这篇短文的目的是证明 G. Jumarie 在一系列论文中导出的分数阶傅里叶级数展开的无效性。在他的工作中，指数函数 einωx 被 Mittag-Leffler 函数 Eα (i(nωx)α) 取代，区间为 [0,Mα/ω]，其中 0 0 是函数 Eα (ixα) 的周期，即 Eα (ixα) = Eα (i(x +Mα)α) 。他证明了任何具有周期 Mα/ω 的平滑周期函数 f 都可以展开为傅立叶级数。我们将证明函数 Eα (ixα) 唯一可能的周期是 Mα = 0；因此 f 的任何傅里叶型级数展开都是无效的。
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 einωx are replaced by the Mittag-Leffler functions Eα (i(nωx)α) , over the interval [0,Mα/ω] where 0 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.