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November 2009 Manifestations of the Parseval identity
Kalyan Chakraborty, Shigeru Kanemitsu, Jinhon Li, Xiaohan Wang
Proc. Japan Acad. Ser. A Math. Sci. 85(9): 149-154 (November 2009). DOI: 10.3792/pjaa.85.149

Abstract

In this paper, we make structural elucidation of some interesting arithmetical identities in the context of the Parseval identity.

In the continuous case, following Romanoff [R] and Wintner [Wi], we study the Hilbert space of square-integrable functions L2(0,1) and provide a new complete orthonormal basis-the Clausen system-, which gives rise to a large number of intriguing arithmetical identities as manifestations of the Parseval identity. Especially, we shall refer to the identity of Mikolás-Mordell.

Secondly, we give a new look at enormous number of elementary mean square identities in number theory, including H. Walum's identity [Wa] and Mikolás' identity (1.16). We show that some of them may be viewed as the Parseval identity. Especially, the mean square formula for the Dirichlet L-function at 1 is nothing but the Parseval identity with respect to an orthonormal basis constructed by Y. Yamamoto [Y] for the linear space of all complex-valued periodic functions.

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Kalyan Chakraborty. Shigeru Kanemitsu. Jinhon Li. Xiaohan Wang. "Manifestations of the Parseval identity." Proc. Japan Acad. Ser. A Math. Sci. 85 (9) 149 - 154, November 2009. https://doi.org/10.3792/pjaa.85.149

Information

Published: November 2009
First available in Project Euclid: 5 November 2009

zbMATH: 1251.11060
MathSciNet: MR2573965
Digital Object Identifier: 10.3792/pjaa.85.149

Subjects:
Primary: 11R58
Secondary: 11R29

Keywords: Dirichlet L-function , orthnormal basis , Parseval identity

Rights: Copyright © 2009 The Japan Academy

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Vol.85 • No. 9 • November 2009
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