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.

scholarly journals CLASSICAL SOLUTION OF THE CAUCHY PROBLEM FOR BIWAVE EQUATION: APPLICATION OF FOURIER TRANSFORM

2012 ◽  
Vol 17 (5) ◽  
pp. 630-641 ◽  
Author(s):  
Victor Korzyuk ◽  
Nguyen Van Vinh ◽  
Nguyen Tuan Minh
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Fourier Transform ◽  
Fourier Analysis ◽  
Classical Solution ◽  
Half Space ◽  
Analysis Techniques ◽  
The Cauchy Problem

In this paper, we use some Fourier analysis techniques to find an exact solution to the Cauchy problem for the n-dimensional biwave equation in the upper half-space ℝ n × [0, +∞).

Tempered distributions with spectral gaps

1989 ◽  
Vol 106 (1) ◽  
pp. 143-162 ◽  
Author(s):  
Jean-Pierre Gabardo
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
The Other ◽  
Convolution Operators ◽  
Spectral Gaps ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Other Hand

AbstractA tempered distribution on ℝ whose Fourier transform is supported in an interval [−Ω,Ω], where Ω>0, can be characterized by the behaviour of its successive derivatives. On the other hand, a tempered distribution on ℝ whose Fourier transform vanishes in an interval (−Ω,Ω), where Ω>0, can be characterized by the behaviour of a particular sequence of successive antiderivatives. Similar considerations apply to general convolution operators acting on J′(ℝn) and yield characterizations for tempered distributions having their Fourier transforms supported in sets of the form or , where and Ω>0.

scholarly journals Uncertainty principles invariant under the fractional Fourier transform

1991 ◽  
Vol 33 (2) ◽  
pp. 180-191 ◽  
Author(s):  
David Mustard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Intrinsic Property ◽  
Continuous Group ◽  
Uncertainty Principles ◽  
New Family ◽  
Special Cases ◽  
Fractional Fourier Transforms ◽  
The Fourier Transform

AbstractUncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

scholarly journals Fourier transforms in generalized Fock spaces

1990 ◽  
Vol 13 (3) ◽  
pp. 431-441
Author(s):  
John Schmeelk
Keyword(s):  
Fourier Transform ◽  
Taylor Series ◽  
Fourier Transforms ◽  
Fock Space ◽  
Natural Generalization ◽  
Fock Spaces ◽  
Test Functions ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
The Fourier Transform

A classical Fock space consists of functions of the form,Φ↔(ϕ0,ϕ1,…,ϕq,…),whereϕ0∈Candϕq∈L2(R3q),q≥1. We will replace theϕq,q≥1withq-symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter,s, which sweeps out a scale of generalized Fock spaces.

Impact and Moving Loads on a Slightly Curved Elastic Half Space

1959 ◽  
Vol 26 (4) ◽  
pp. 491-498
Author(s):  
A. C. Eringen ◽  
J. C. Samuels
Keyword(s):  
Half Space ◽  
Fourier Transforms ◽  
Normal Load ◽  
Elastic Half Space ◽  
Two Dimensional ◽  
Moving Loads ◽  
Special Cases ◽  
Indefinite Integral ◽  
Wavy Boundary ◽  
Boundary Curves

Abstract Two-dimensional Fourier transforms are employed to treat the two-dimensional dynamic problem of elastic half space having a slightly wavy boundary. The various boundary curves considered include square and triangular bumps and holes, and sinusoidal and periodic boundaries. The number of different types of surface loadings considered are: (a) Normal tractions and zero shear, (b) impulsive normal tractions and zero shear, (c) suddenly applied normal tractions and zero shear, (d) concentrated normal load and zero shear, (e) concentrated impulsive load and zero shear, (f) pulsating normal load and zero shear, (g) moving loads, (h) pulsating moving loads, (i) vertical and horizontal loads, (j) moving vertical loads. Stress and displacement components for special cases of the loads described in (a, c, f, and g) acting on a sinusoidal boundary lead to a solution which requires evaluation of a single indefinite integral. Closed-form results are given for a uniform pulsating pressure load.

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.

scholarly journals Theoretical foundations of digital vector Fourier analysis of two-dimensional signals padded with zero samples

2021 ◽  
pp. 55-65
Author(s):  
Olga Ponomareva ◽  
Aleksey Ponomarev
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Discrete Fourier Transform ◽  
Fourier Transforms ◽  
Information Control ◽  
Two Dimensional ◽  
Negative Effects ◽  
Variable Parameters ◽  
Discrete Fourier Transforms ◽  
Fourier Matrix

Introduction: The practice of using Fourier-processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. A well-known method of dealing with negative effects of Fourier-processing is padding signals with zeros. However, the use of this operation leads to the need to provide information control systems with additional memory and perform unproductive calculations. Purpose: To develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals padded with zero samples. Method: We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. The method is based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. Results: New discrete Fourier transforms with variable parameters were developed, being a generalization of the classical discrete Fourier transform. The article investigates the properties of Fourier transform bases with variable parameters. In respect to these transforms, the validity has been proved for the theorems of linearity, shift, correlation and Parseval's equality. In the digital spectral Fourier analysis, the concepts of a parametric shift of a two-dimensional signal, and a parametric periodicity of a two-dimensional signal have been introduced. We have estimated the reduction of the required memory size and the number of calculations when applying the proposed transforms, and compared them with the discrete Fourier transform. Practical relevance: The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier processing of two-dimensional signals (including images) padded with zeros.

Measures with Fourier Transforms in L2 of a Half-space

2011 ◽  
Vol 54 (1) ◽  
pp. 172-179
Author(s):  
Bassam Shayya
Keyword(s):  
Fourier Transform ◽  
Lebesgue Measure ◽  
Half Space ◽  
Fourier Transforms ◽  
Geometric Information ◽  
Absolutely Continuous ◽  
Compactly Supported ◽  
The Fourier Transform

AbstractWe prove that if the Fourier transform of a compactly supported measure is in L2 of a half-space, then the measure is absolutely continuous to Lebesgue measure. We then show how this result can be used to translate information about the dimensionality of a measure and the decay of its Fourier transform into geometric information about its support.

scholarly journals Four Particular Cases of the Fourier Transform

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 335 ◽  
Author(s):  
Jens Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform ◽  
Poisson’S Summation Formula

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

scholarly journals Four Particular Cases of the Fourier Transform

2018 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

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