A note on the Fourier transform of generalized functions

1971 ◽  
Vol 70 (1) ◽  
pp. 49-51
Author(s):  
B. Fisher
Keyword(s):  
Fourier Transform ◽  
Generalized Functions ◽  
Generalized Function ◽  
Convolution Product ◽  
Counter Example ◽  
The Fourier Transform ◽  
Image Position

If F(f) denotes the Fourier transform of a generalized function f and f * g denotes the convolution product of two generalized functions f and g then it is known that under certain conditionsJones (2) states that this is not true in general and gives as a counter-example the case when f = g = H, H denoting Heaviside's function. In this caseand the product (x−1 – iπδ)2 is not defined in his development of the product of generalized functions.

scholarly journals On dilation equations and the Hölder continuity of the de Rham functions

1994 ◽  
Vol 36 (3) ◽  
pp. 309-311
Author(s):  
Yibiao Pan
Keyword(s):  
Fourier Transform ◽  
Approximation Method ◽  
Hölder Continuity ◽  
Simple Approximation ◽  
Holder Continuity ◽  
Integrable Solution ◽  
De Rham ◽  
The Fourier Transform ◽  
Image Position ◽  
Dilation Equations

AbstractWe use a simple approximation method to prove the Holder continuity of the generalized de Rham functions.1. Consider the following dilatation equationwhere |α|<l/2. Suppose that f is an integrable solution of (1); then f must satisfywhere is the Fourier transform of f, andwhich immediately leads to

On the solution of certain integral equations by generalized functions

1961 ◽  
Vol 57 (4) ◽  
pp. 767-777 ◽  
Author(s):  
J. B. Miller
Keyword(s):  
Integral Equations ◽  
Generalized Functions ◽  
Generalized Solutions ◽  
Generalized Function ◽  
Usual Sense ◽  
Inverse Transformation ◽  
Extended Sense ◽  
Image Position

This note is concerned with a method by which generalized solutions can be shown to exist for certain types of integral equationswhere f(x) ∈ L2(0, ∞). The method is briefly this. An extended meaning is given to such equations by using generalized functions of a particular type. Then, if (1) denotes a transformation which has no everywhere-defined inverse in the usual sense, it may be possible to define in the extended sense an inverse transformationso that if f is given, a generalized function g = can be determined as a solution of (1).

Fourier restriction theorems for curves with affine and Euclidean arclengths

1985 ◽  
Vol 97 (1) ◽  
pp. 111-125 ◽  
Author(s):  
S. W. Drury ◽  
B. P. Marshall
Keyword(s):  
Fourier Transform ◽  
Smooth Manifold ◽  
Fourier Restriction ◽  
Survey Article ◽  
Restriction Theorems ◽  
Schwartz Class ◽  
The Fourier Transform ◽  
Image Position

Let M be a smooth manifold in . One may ask whether , the restriction of the Fourier transform of f to M makes sense for every f in . Since, for does not make sense pointwise, it is natural to introduce a measure σ on M and ask for an inequalityfor every f in (say) the Schwartz class. Results of this kind are called restriction theorems. An excellent survey article on this subject is to be found in Tomas[13].

The a-Local Operator Problem

1959 ◽  
Vol 11 ◽  
pp. 583-592 ◽  
Author(s):  
Louis de Branges
Keyword(s):  
Fourier Transform ◽  
Total Variation ◽  
Convergence Theorem ◽  
Borel Measure ◽  
Local Operator ◽  
Operator Problem ◽  
Dominated Convergence ◽  
The Fourier Transform ◽  
Image Position ◽  
The Right

Let be the Fourier transform of a Borel measure of finite total variation. The formulacan be justified if the integral on the right converges absolutely. ForwhereNow let h → 0 in both sides of this equation and use the Lebesgue dominated convergence theorem.

scholarly journals On fourier transforms of radial functions

1965 ◽  
Vol 5 (3) ◽  
pp. 289-298 ◽  
Author(s):  
James L. Griffith
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Radial Functions ◽  
Cartesian Space ◽  
The Fourier Transform ◽  
Image Position

The Fourier transformF(y) of a functionf(t) inL1(Ek) whereEkis thek-dimensional cartesian space will be defined by.

Fourier Transforms of Unbounded Measures

1979 ◽  
Vol 31 (6) ◽  
pp. 1281-1292 ◽  
Author(s):  
James Stewart
Keyword(s):  
Fourier Transform ◽  
Real Line ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Stieltjes Transform ◽  
The Real ◽  
Bounded Complex ◽  
The Fourier Transform ◽  
Image Position

1. Introduction. One of the basic objects of study in harmonic analysis is the Fourier transform (or Fourier-Stieltjes transform) μ of a bounded (complex) measure μ on the real line R:(1.1)More generally, if μ is a bounded measure on a locally compact abelian group G, then its Fourier transform is the function(1.2)where Ĝ is the dual group of G and One answer to the question “Which functions can be represented as Fourier transforms of bounded measures?” was given by the following criterion due to Schoenberg [11] for the real line and Eberlein [5] in general: f is a Fourier transform of a bounded measure if and only if there is a constant M such that(1.3)for all ϕ ∈ L1(G) where

The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1

1986 ◽  
Vol 38 (2) ◽  
pp. 328-359 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansion ◽  
Unit Sphere ◽  
Gaussian Curvature ◽  
Fourier Transforms ◽  
Surface Measure ◽  
Positive Gaussian Curvature ◽  
Smooth Measures ◽  
The Fourier Transform ◽  
Image Position

The Fourier transform of the surface measure on the unit sphere in Rn + 1, as is well-known, equals the Bessel functionIts behaviour at infinity is described by an asymptotic expansionThe purpose of this paper is to obtain such an expression for surfaces Σ other than the unit sphere. If the surface Σ is a sufficiently smooth compact n-surface in Rn + 1 with strictly positive Gaussian curvature everywhere then with only minor changes in the main term, such an asymptotic expansion exists. This result was proved by E. Hlawka in [3]. A similar result concerned with the minimal smoothness of Σ was later obtained by C. Herz [2].

A Mixed Parseval's Equation And a Generalized Hankel Transformation of Distributions

1989 ◽  
Vol 41 (2) ◽  
pp. 274-284 ◽  
Author(s):  
J. J. Betancor
Keyword(s):  
Dual Space ◽  
Integral Transform ◽  
Generalized Functions ◽  
Generalized Function ◽  
Hankel Transformation ◽  
Formula One ◽  
Image Position ◽  
Complex Valued

Let an integral transform T﹛f﹜ of a complex valued function f(x) defined over the interval (0, ∞) be defined as One of the most usual procedures to extend the classical transform (l.a) to generalized functions consists in constructing a space A of testing functions over (0, ∞) which is closed with respect to the classical transform (l.a) and then the corresponding transform of the generalized function/ of the dual space of A is defined through This approach has been followed by L. Schwartz [13] and A. H. Zemanian [20], amongst others.

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&rsquo;s summation formula to Heisenberg&rsquo;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&rsquo;s summation formula expresses a duality between discretization and periodization, Heisenberg&rsquo;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.

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&rsquo;s summation formula to Heisenberg&rsquo;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&rsquo;s summation formula expresses a duality between discretization and periodization, Heisenberg&rsquo;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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