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Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 1354129, 5 pages
https://doi.org/10.1155/2017/1354129
Research Article

Poisson Summation Formulae Associated with the Special Affine Fourier Transform and Offset Hilbert Transform

Beijing Information Science & Technology University, Beijing, China

Correspondence should be addressed to Zhi-Hai Zhuo; nc.ude.utsib@iahihzouhz

Received 7 May 2017; Accepted 16 July 2017; Published 15 August 2017

Academic Editor: Alessandro Lo Schiavo

Copyright © 2017 Zhi-Hai Zhuo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the generalized pattern of Poisson summation formulae from the special affine Fourier transform (SAFT) and offset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. Then, the generalized Poisson sum formulae are obtained based on above relationships. The novel results can be regarded as the generalizations of the classical results in several transform domains such as FT, fractional Fourier transform, and the linear canonical transform.