On Eigenfunctions of the Fourier Transform

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