A characterization of Fourier transform by Poisson summation formula

2010 ◽  
Vol 348 (7-8) ◽  
pp. 407-410 ◽  
Author(s):  
Dmitry Faifman
Keyword(s):  
Fourier Transform ◽  
Summation Formula ◽  
Poisson Summation Formula ◽  
Poisson Summation

Special affine wavelet transform and the corresponding Poisson summation formula

2021 ◽  
pp. 2050086
Author(s):  
Firdous A. Shah ◽  
Aajaz A. Teali ◽  
Azhar Y. Tantary
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Inversion Formula ◽  
Summation Formula ◽  
Poisson Summation Formula ◽  
Time Frequency ◽  
Fundamental Properties ◽  
Stationary Signals ◽  
Poisson Summation

In the article, “Windowed special affine Fourier transform” in J. Pseudo-Differ. Oper. Appl. (2020), we introduced the notion of windowed special affine Fourier transform (WSAFT) as a ramification of the special affine Fourier transform. Keeping in view the fact that the WSAFT is not befitting for in the context of non-stationary signals, we continue our endeavor and introduce the notion of the special affine wavelet transform (SAWT) by combining the merits of the special affine Fourier and wavelet transforms. Besides studying the fundamental properties of the SAWT including orthogonality relation, inversion formula and range theorem, we also demonstrate that the SAWT admits the constant [Formula: see text]-property in the time–frequency domain. Moreover, we formulate an analog of the well-known Poisson summation formula for the proposed SAWT.

scholarly journals Some Finite Analogues of the Poisson Summation Formula

1961 ◽  
Vol 12 (3) ◽  
pp. 133-138 ◽  
Author(s):  
L. Carlitz
Keyword(s):  
Summation Formula ◽  
Poisson Summation Formula ◽  
Euler’S Constant ◽  
Image Position ◽  
Positive Integers ◽  
Euler's Constant ◽  
Poisson Summation

1. Guinand (2) has obtained finite identities of the typewhere m, n, N are positive integers and eitherorwhere γ is Euler's constant and the notation ∑′ indicates that when x is integral the term r = x is multiplied by ½. Clearly there is no loss of generality in taking N = 1 in (1.1).

scholarly journals A strong version of Poisson summation

1997 ◽  
Vol 20 (1) ◽  
pp. 81-92 ◽  
Author(s):  
Nelson Petulante
Keyword(s):  
Summation Formula ◽  
Poisson Summation Formula ◽  
Strong Version ◽  
Poisson Summation

We establish a generalized version of the classical Poisson summation formula. This formula incorporates a special feature called “compression”, whereby, at the same time that the formula equates a series to its Fourier dual, the compressive feature serves to enable both sides of the equation to converge.

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