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.

A Remark on Fourier Transforms

1936 ◽  
Vol 32 (2) ◽  
pp. 321-327 ◽  
Author(s):  
A. Zygmund
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Complex Function ◽  
The Fourier Transform ◽  
Image Position ◽  
The Right

1. Let f(x) be a complex function belonging to LP (−∞, ∞); i.e. let f(x) be measurable, and |f(x)|p integrable, over (−∞, ∞). The functionis called the Fourier transform of f(x), if the integral on the right exists, in some sense, for almost every value of y. It is well known that, if 1 ≤ p ≤ 2, the integral (1) converges in mean, with index p′ = p/(p – l)† i.e. thatwhere

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

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].

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.

The λ-cosine transforms with odd kernel and the hemispherical transform

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Boris Rubin
Keyword(s):  
Fourier Transform ◽  
Unit Sphere ◽  
Borel Measure ◽  
Fractional Integrals ◽  
Inversion Method ◽  
Spherical Means ◽  
Finite Borel Measure ◽  
Basic Facts ◽  
Hemispherical Transform ◽  
The Fourier Transform

AbstractWe review some basic facts about the λ-cosine transforms with odd kernel on the unit sphere S n−1 in ℝn. These transforms are represented by the spherical fractional integrals arising as a result of evaluation of the Fourier transform of homogeneous functions. The related topic is the hemispherical transform which assigns to every finite Borel measure on S n−1 its values for all hemispheres. We revisit the known facts about this transform and obtain new results. In particular, we show that the classical Funk- Radon-Helgason inversion method of spherical means is applicable to the hemispherical transform of L p-functions.

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].

Non-Relativistic Dispersion Relations for a Two Channel Scattering Process

1962 ◽  
Vol 58 (2) ◽  
pp. 363-376 ◽  
Author(s):  
J. Underhill
Keyword(s):  
Fourier Transform ◽  
Dispersion Relation ◽  
Scattering Amplitude ◽  
Bound State ◽  
Potential Scattering ◽  
Scattering Process ◽  
Relativistic Dispersion ◽  
The Fourier Transform ◽  
Image Position ◽  
Relativistic Potential

In (l) Khuri has proved the validity of a dispersion relation for non-relativistic potential scattering. More precisely, he has shown that if the potential V(r) is central and satisfies: then the scattering amplitude f(k, τ) = M(E, τ) (where E = k2 is the energy) satisfies the following dispersion relation for fixed momentum transfer τ ≤ 2α: In (2), Rj(τ) is the (real) residue of the scattering amplitude at the bound state Ej and is the Fourier transform of the potential .

scholarly journals A change of scale formula for Wiener integrals of cylinder functions on the abstract Wiener space II

2001 ◽  
Vol 25 (4) ◽  
pp. 231-237 ◽  
Author(s):  
Young Sik Kim
Keyword(s):  
Fourier Transform ◽  
Borel Measure ◽  
Feynman Integral ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Analytic Feynman Integral ◽  
Change Of Scale ◽  
The Fourier Transform ◽  
Complex Valued

We show that for certain bounded cylinder functions of the formF(x)=μˆ((h1,x)∼,...,(hn,x)∼),x∈Bwhereμˆ:ℝn→ℂis the Fourier-transform of the complex-valued Borel measureμonℬ(ℝn), the Borelσ-algebra ofℝnwith‖μ‖<∞, the analytic Feynman integral ofFexists, although the analytic Feynman integral,limz→−iqIaw(F;z)=limz→−iq(z/2π)n/2∫ℝnf(u→)exp{−(z/2)|u→|2}du→, do not always exist for bounded cylinder functionsF(x)=f((h1,x)∼,...,(hn,x)∼),x∈B. We prove a change of scale formula for Wiener integrals ofFon the abstract Wiener space.

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