Complex Analysis and Operator Theory

, Volume 6, Issue 1, pp 219–236 | Cite as

The Hermite Functions Related to Infinite Series of Generalized Convolutions and Applications

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Abstract

In this paper, we show that arbitrary Hermite function or appropriate linear combination of those functions is a weight-function of four explicit generalized convolutions for the Fourier cosine and sine transforms. With respect to applications, normed rings on \({L^1(\mathbb{R}^d)}\) are constructed, and sufficient and necessary conditions for the solvability and explicit solutions in \({L^1(\mathbb{R}^d)}\) of the integral equations of convolution type are provided by using the constructed convolutions.

Keywords

Hermite function Generalized convolution Normed ring Integral equations of convolution type 

Mathematics Subject Classification (2000)

Primary 42A85 45E10 Secondary 44A20 46J10 
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Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisUniversity of HanoiHanoiVietnam
  2. 2.Department of Basic ScienceUniversity of Phuong DongHanoiVietnam

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