ON EIGENFUNCTIONS OF THE FOURIER TRANSFORM

摘要

A nontrivial example of an eigenfunction in the sense of the theory of distributions for the planar Fourier transform was described by the authors in their previous work. In this paper, a method for obtaining other eigenfunctions is proposed. Positive homogeneous distributions in ℝ_( n )of order − n /2 are considered, and it is shown that F ( ω )| x |_(− n /2), | ω | = 1, is an eigenfunction in the sense of the theory of distributions of the Fourier transform if and only if F ( ω ) is an eigenfunction of a certain singular integral operator on the unit sphere of ℝ_( n ). Since Y m , n k ω x − n / 2 $$ {Y}_{m,n}^{(k)}left(omega right){left|mathbf{x}right|}^{-n/2} $$ , where Y m , n k $$ {Y}_{m,n}^{(k)} $$ denote the spherical functions of order m in ℝ_( n ), are eigenfunctions of the Fourier transform, it follows that Y m , n k $$ {Y}_{m,n}^{(k)} $$ are eigenfunctions of the above-mentioned singular integral operator. In the planar case, all eigenfunctions of the Fourier transform of the form F ( ω )| x |_(−1)are described by means of the Fourier coefficients of F ( ω ).