Manifestations of the Parseval identity

作者：Chakraborty Kalyan^*; Kanemitsu Shigeru; Li Jinhon; Wang Xiaohan

来源：Proceedings of the Japan Academy Series A-Mathematical Sciences, 2009, 85(9): 149-154.

DOI：10.3792/pjaa.85.149

• Parseval identity
• orthnormal basis
• Dirichlet L-function

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 L(2)(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 Mikolas-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 Mikolas' 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.

• 出版日期2009-11
• 单位山东大学