On eigenfunctions of the Fourier transform

In \cite{LM} we considered a nontrivial example of eigenfunction in the sense of distribution for the planar Fourier transform. Here a method to obtain other eigenfunctions is proposed. Moreover we consider positive homogeneous distributions in \(\R^n\) of order \(-n/2\). It is shown that \({F(\om)}{|\bx|^{-n/2}}, |\om|=1\) is an eigenfunction in the sense of distribution of the Fourier transform if and only if \(F(\om)\) is an eigenfunction of a certain singular integral operator on the unit sphere of \(\R^n\). Since \(Y_{m,n}^{(k)}(\om)|\bx|^{-n/2}\) are eigenfunctions of the Fourier transform, we deduce that \(Y_{m,n}^{(k)}\) are eigenfunctions of the above mentioned singular integral operator. Here \(Y_{m,n}^{(k)}\) denote the spherical functions of order \(m\) in \(\R^n\). In the planar case, we give a description of all eigenfunctions of the Fourier transform of the form \({F(\om)}{|\bx|^{-1}}\) by means of the Fourier coefficients of \(F(\om)\).