Real Paley‐Wiener theorem for the octonion Fourier transform

2021 ◽  
Author(s):  
Yong Li ◽  
Guangbin Ren
Keyword(s):  
Fourier Transform ◽  
Wiener Theorem

Tempered Distributions Supported on a Half-Space of RN and Their Fourier Transforms

1991 ◽  
Vol 43 (1) ◽  
pp. 61-88
Author(s):  
Jean-Pierre Gabardo
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Half Space ◽  
Fourier Transforms ◽  
Fundamental Problem ◽  
Half Line ◽  
Tempered Distributions ◽  
Difficult Question ◽  
Wiener Theorem ◽  
Special Cases

A fundamental problem in Fourier analysis is to characterize the behaviour of a function (or distribution) whose Fourier transform vanishes in some particular set. Of course, this is, in general, a very difficult question and little seems to be known, except in some special cases. For example, a theorem of Paley-Wiener (Theorem XII in [6]) characterizes exactly the behaviour of the modulus of a function in L2(R) whose Fourier transform vanishes on a half-line.

A weighted version of the Paley–Wiener theorem

1989 ◽  
Vol 105 (2) ◽  
pp. 389-395 ◽  
Author(s):  
T. G. Genchev
Keyword(s):  
Fourier Transform ◽  
Exponential Type ◽  
Entire Functions ◽  
Weighted Version ◽  
Cauchy Theorem ◽  
Wiener Theorem ◽  
Functions Of Exponential Type ◽  
The Fourier Transform

A generalization of the classical theorems of Paley and Wiener[5] and Plancherel and Polya[6] concerning entire functions of exponential type is obtained. The proof relies only on the Cauchy theorem and the Hardy–Littlewood inequality for the Fourier transform (see [8, 9]). Since the functions under consideration are supposed to be defined only in two opposite octants in ℂn, a version of the edge of the wedge theorem [7] is derived as a by-product.

scholarly journals On some extension of Paley Wiener theorem

2020 ◽  
Vol 7 (1) ◽  
pp. 81-90
Author(s):  
Ettien Yves-Fernand N’Da ◽  
Kinvi Kangni
Keyword(s):  
Fourier Transform ◽  
Lie Group ◽  
Compact Support ◽  
Fourier Transforms ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Classical Case ◽  
Decay Properties ◽  
Wiener Theorem ◽  
Unitary Dual

AbstractPaley Wiener theorem characterizes the class of functions which are Fourier transforms of ℂ∞ functions of compact support on ℝn by relating decay properties of those functions or distributions at infinity with analyticity of their Fourier transform. The theorem is already proved in classical case : the real case with holomorphic Fourier transform on L2(ℝ), the case of functions with compact support on ℝn from Hörmander and the spherical transform on semi simple Lie groups with Gangolli theorem.Let G be a locally compact unimodular group, K a compact subgroup of G, and δ an element of unitary dual ̑K of K. In this work, we’ll give an extension of Paley-Wiener theorem with respect to δ, a class of unitary irreducible representation of K, where G is either a semi-simple Lie group or a reductive Lie group with nonempty discrete series after introducing a notion of δ-orbital integral. If δ is trivial and one dimensional, we obtain the classical Paley-Wiener theorem.

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