On Fourier Transforms of Radial Functions and Distributions

@article{Grafakos2013OnFT,
title={On Fourier Transforms of Radial Functions and Distributions},
author={Loukas Grafakos and Gerald Teschl},
journal={Journal of Fourier Analysis and Applications},
year={2013},
volume={19},
pages={167-179}
}
• Published 22 December 2011
• Mathematics
• Journal of Fourier Analysis and Applications
We find a formula that relates the Fourier transform of a radial function on Rn with the Fourier transform of the same function defined on Rn+2. This formula enables one to explicitly calculate the Fourier transform of any radial function f(r) in any dimension, provided one knows the Fourier transform of the one-dimensional function t↦f(|t|) and the two-dimensional function (x1,x2)↦f(|(x1,x2)|). We prove analogous results for radial tempered distributions.

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References

SHOWING 1-10 OF 23 REFERENCES

On asymptotics for a class of radial Fourier transforms

• Mathematics
• 1998
A connection is established between the multidimensional Fourier transform of a radial function f from a given class and the one-dimensional Fourier transform of a related function. This is applied

Classical Fourier Analysis

Preface.- 1. Lp Spaces and Interpolation.- 2. Maximal Functions, Fourier Transform, and Distributions.- 3. Fourier Series.- 4. Topics on Fourier Series.- 5. Singular Integrals of Convolution Type.-

The Fourier-Bessel series representation of the pseudo-differential operator (-⁻¹)^{}

• Mathematics
• 1992
For a certain Frechet space F consisting of complex-valued C ∞ functions defined on I=(0, ∞) and characterized by their asymptotic behaviour near the boundaries, we show that : (I) The

Fractional Integrals and Derivatives: Theory and Applications

• Mathematics
• 1993
Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of

Differentiable even functions

An even function f (x) = f (−x) (defined in a neighborhood of the origin) can be expressed as a function g(x 2); g(u) is determined for u ≥ 0, but not for u < 0. We wish to show that g may be defined

Weyl–Titchmarsh Theory for Schrödinger Operators with Strongly Singular Potentials

• Mathematics
• 2011
We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in

Mathematical Methods in Quantum Mechanics

Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements

Methods of modern mathematical physics. III. Scattering theory

• Physics, Mathematics
• 1979
Topics covered include: overview; classical particle scattering; principles of scattering in Hilbert space; quantum scattering; long range potentials; optical and acoustical scattering; the linear

PARTIAL DIFFERENTIAL EQUATIONS

Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear