Proceedings of the Japan Academy, Series A, Mathematical Sciences

Manifestations of the Parseval identity

Kalyan Chakraborty, Shigeru Kanemitsu, Jinhon Li, and Xiaohan Wang

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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.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 85, Number 9 (2009), 149-154.

First available in Project Euclid: 5 November 2009

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Zentralblatt MATH identifier

Primary: 11R58: Arithmetic theory of algebraic function fields [See also 14-XX]
Secondary: 11R29: Class numbers, class groups, discriminants

Parseval identity orthnormal basis Dirichlet L-function


Chakraborty, Kalyan; Kanemitsu, Shigeru; Li, Jinhon; Wang, Xiaohan. Manifestations of the Parseval identity. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 9, 149--154. doi:10.3792/pjaa.85.149.

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