# 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} }

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

- Mathematics
- 2010

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

- Mathematics
- 1943

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

- Physics
- 2009

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

- Mathematics
- 1941

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…