On Fourier Transforms of Radial Functions and Distributions

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.

[1]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[2]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[3]  Hassler Whitney,et al.  Differentiable even functions , 1943 .

[4]  J. F. Treves.  Lectures on linear partial differential equations with constant coefficients , 1961 .

[5]  A. Zemanian.  A Distributional Hankel Transformation , 1966 .

[6]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[7]  M. Reed,et al.  Fourier Analysis, Self-Adjointness , 1975 .

[8]  M. Reed,et al.  Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness , 1975 .

[9]  Barry Simon,et al.  Methods of modern mathematical physics. III. Scattering theory , 1979 .

[10]  Z. Szmydt.  On homogeneous rotation invariant distributions and the Laplace operator , 1979 .

[11]  A. Zemanian.  Generalized Integral Transformations , 1987 .

[12]  O. P. Singh,et al.  The Fourier-Bessel series representation of the pseudo-differential operator (-⁻¹)^{} , 1992 .

[13]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[14]  Robert Schaback,et al.  Operators on radial functions , 1996 .

[15]  E. Liflyand,et al.  On asymptotics for a class of radial Fourier transforms , 1998 .

[16]  Gerald Teschl,et al.  Mathematical Methods in Quantum Mechanics , 2009 .

[17]  L. Grafakos.  Classical Fourier Analysis , 2010 .

[18]  L. Evans,et al.  Partial Differential Equations , 1941 .

[19]  Gerald Teschl,et al.  Weyl–Titchmarsh Theory for Schrödinger Operators with Strongly Singular Potentials , 2011 .

[20]  新國 裕昭,et al.  書評 Gerald Teschl : Mathematical Methods in Quantum Mechanics : With Applications to Schrodinger Operators , 2013 .