# 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.

### On Radial Functions and Distributions and Their Fourier Transforms

We give formulas relating the Fourier transform of a radial function in $\mathbb{R}^{n}$ and the Fourier transform of the same function in $\mathbb{R}^{n+1}$, completing the analysis of Grafakos and

### Weighted Hankel Transform and Its Applications to Fourier Transform

• Mathematics
• 2021
The purpose of the present paper is to discuss the integrability of the Fourier transform of $$L^\infty$$ L ∞ -functions subject to a very weak decay condition. This will include the negative power

### Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

• Mathematics
Journal of Fourier Analysis and Applications
• 2016
The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier

### Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

• Mathematics
• 2014
The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier

### The Funk-Hecke formula, harmonic polynomials, and derivatives of radial distributions

We give a version of the Funk-Hecke formula that holds with minimal assumptons and apply it to obtain formulas for the distributional derivatives of radial distributions in Rn of the type Yk 􀀀 r j

### Fourier interpolation from spheres

• Martin Stoller
• Mathematics
Transactions of the American Mathematical Society
• 2020
In every dimension $d \geq 5$ we give an explicit formula that expresses the values of any Schwartz function on $\mathbb{R}^d$ only in terms of its restrictions, and the restrictions of its Fourier

### Diagonal spherical means

We introduce a mean for functions and distributions of two vector variables, , the diagonal spherical mean K, defined as We study several properties of these means as well as identities satisfied by

### On the Fourier transform of rotationally invariant distributions

• Mathematics
Bollettino dell'Unione Matematica Italiana
• 2018
We present an extension of the Poisson–Bochner formula for the Fourier transform of rotationally invariant distributions by analytic continuation “with respect to the dimension”. As application of

### What is the Fourier Transform of a Spatial Point Process?

• Mathematics
• 2020
This paper determines how to deﬁne a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; ﬁrst what is the

## 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