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We explore the possibilities of reaching the characterization ofeigenfunction of Laplacian as a degenerate case of the inverse Paley-Wienertheorem (characterizing functions whose Fourier transform is supported on acompact annulus) for the Riemannian symmetric spaces of noncompact type. Mostdistinguished prototypes of these spaces are the hyperbolic spaces. Thestatement and the proof of the main result work mutatis-mutandis for a numberof spaces including Euclidean spaces and Damek-Ricci spaces.

Function with its Fourier transform supported on annulus and eigenfunction of Laplacian