The Fourier transform: Tempered distributions, the wave equation and Laplace’s equation

2015 ◽  
pp. 133-146
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Laplace’S Equation ◽  
Tempered Distributions ◽  
Laplace's Equation ◽  
The Fourier Transform

scholarly journals The Fourier transform of thick distributions

2020 ◽  
pp. 1-26
Author(s):  
Ricardo Estrada ◽  
Jasson Vindas ◽  
Yunyun Yang
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Canonical Projection ◽  
Test Functions ◽  
Tempered Distributions ◽  
Delta Functions ◽  
The Fourier Transform ◽  
The One ◽  
One Point Compactification ◽  
Logarithmic Type

We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a canonical projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the canonical projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

scholarly journals Strong boundedness of analytic functions in tubes

1979 ◽  
Vol 2 (1) ◽  
pp. 15-28
Author(s):  
Richard D. Carmichael
Keyword(s):  
Fourier Transform ◽  
Analytic Function ◽  
Analytic Functions ◽  
Weak Topology ◽  
Complex Space ◽  
Direct Proof ◽  
Tempered Distributions ◽  
Boundedness Property ◽  
Tube Domains ◽  
The Fourier Transform

Certain classes of analytic functions in tube domainsTC=ℝn+iCinn-dimensional complex space, whereCis an open connected cone inℝn, are studied. We show that the functions have a boundedness property in the strong topology of the space of tempered distributionsg′. We further give a direct proof that each analytic function attains the Fourier transform of its spectral function as distributional boundary value in the strong (and weak) topology ofg′.

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.

The support of tempered distributions

2008 ◽  
Vol 144 (2) ◽  
pp. 495-498 ◽  
Author(s):  
Colin C. Graham
Keyword(s):  
Fourier Transform ◽  
Polynomial Growth ◽  
Test Functions ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Schwartz Functions ◽  
Technical Lemma ◽  
Approximate Identities ◽  
The Fourier Transform

AbstractWe identify the support of a tempered distribution by evaluation of a sequence of test functions against the Fourier transform of the distribution. This improves previous results by removing the restriction that the distribution's Fourier transform be in $L^1_{loc}$ and be of polynomial growth. We use an apparently new technical lemma that implies that certain bounded approximate identities for $L^1(\R^n)$ are also topological approximate identities for elements of the space $\Sl$ of Schwartz functions.

XII.—Paraboloidal Co-ordinates and Laplace's Equation

1963 ◽  
Vol 66 (3) ◽  
pp. 129-139
Author(s):  
F. M. Arscott
Keyword(s):  
Wave Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Subsequent Paper ◽  
Laplace’S Equation ◽  
Mathieu Functions ◽  
Laplace's Equation ◽  
Normal Surfaces ◽  
Special Cases ◽  
Hyperbolic Paraboloids

SynopsisIn this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.

On the diffraction of an acoustic pulse by a wedge

1952 ◽  
Vol 212 (1111) ◽  
pp. 543-547 ◽  
Keyword(s):  
Wave Equation ◽  
Homogeneous Function ◽  
Acoustic Pulse ◽  
Scattered Wave ◽  
Acoustic Velocity ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Domain Cut ◽  
Pressure Discontinuity

The diffraction of a pressure discontinuity by a wedge is attacked by posing the pressure as a homogeneous function of r/ct , where r , c and t are the distance from the edge, acoustic velocity and time. The scattered wave is then determined by reducing the original wave equation to Laplace’s equation with the aid of a Tschplygin transformation and applying Poisson’s formula to the sectoral domain cut out of the circle r≤ct by the wedge. The result appears in simpler form than that previously obtained by Sommerfeld.

scholarly journals Analysis of Sound Waves with Semi Perforated Pipe

2019 ◽  
Vol 2 (3) ◽  
pp. 704-710
Author(s):  
Burhan Tiryakioglu
Keyword(s):  
Boundary Condition ◽  
Fourier Transform ◽  
Wave Equation ◽  
Pipe Wall ◽  
Far Field ◽  
Sound Waves ◽  
Hopf Equation ◽  
Hopf Method ◽  
The Fourier Transform ◽  
Perforated Pipe

The paper presents analytical results of radiation phenomena at the far field and solution of the wave equation with adequate boundary condition imposed by the pipe wall. An infinite pipe with perforated part is considered. The solution is obtained by using the Fourier transform technique in conjunction with the Wiener-Hopf Method. Applying the Fourier transform technique, the boundary value problem is described by Wiener Hopf equation and then solved analytically.

scholarly journals On the Duality of Regular and Local Functions

2017 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Summation Formula ◽  
Generalized Functions ◽  
Tempered Distributions ◽  
Regular Functions ◽  
Local Functions ◽  
The Fourier Transform

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are furthermore the inverses of one another. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

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