On Theta Functions and Weil's Generalized Poisson Summation Formula

1969 ◽  
Vol 141 ◽  
pp. 195
Author(s):  
Jun-Ichi Hano
Keyword(s):  
Theta Functions ◽  
Summation Formula ◽  
Poisson Summation Formula ◽  
Generalized Poisson ◽  
Poisson Summation

scholarly journals Some observations on algorithms of the Gauss-Borchardt type

1991 ◽  
Vol 34 (3) ◽  
pp. 415-431 ◽  
Author(s):  
Jaak Peetre
Keyword(s):  
Asymptotic Formula ◽  
Geometric Mean ◽  
Theta Functions ◽  
Summation Formula ◽  
Poisson Summation Formula ◽  
Parameter Family ◽  
Poisson Summation

A one parameter family of algorithms is studied, which contains both the arithmetic-geometric mean of Gauss and its generalization by Borchardt, recently studied by J. and P. Borwein. We prove that the presence of an asymptotic formula for such an algorithm is, in view of the Poisson summation formula, equivalent to the vanishing of certain integrals. In the case of Gauss and Borchardt the latter involve theta functions. Finally, we investigate the question of convergence of the algorithm for complex values, thereby generalizing the corresponding result of Gauss.

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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