On Eigenfunctions of the Fourier Transform

Abstract

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 ℝⁿ 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 ℝⁿ. Since \( {Y}_{m,n}^{(k)}\left(\omega \right){\left|\mathbf{x}\right|}^{-n/2} \), where \( {Y}_{m,n}^{(k)} \) denote the spherical functions of order m in ℝⁿ, are eigenfunctions of the Fourier transform, it follows that \( {Y}_{m,n}^{(k)} \) 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).