Generalization of a Theorem of Boas to a Class of Integral

A characterization of the space L2[−σ, σ] in terms of its image under the Fourier transformation has been given by the Paley-Wiener theorem [5]. The theorem asserts that f ∈ L2[−σ, σ] if and only if its Fourier transform f̂(ω) can be continued analytically to the whole complex plane as an entire function of exponential type at most σ whose restriction to the real axis belongs to L2(R). This characterization uses complex-variable techniques and is not easy to extend to other more complicated integral transforms. Alternative approaches using real analysis techniques have been developed to characterize the images of spaces of the form L2[I, dρ], for some measure dρ and an interval I (finite or infinite), under various integral transformations, such as the Mellin [10], Hankel [9], Y [8], and Airy transforms [11]. One of the first such results was discovered by H. Bang [1]. It can be rephrased as follows: let f ∈ C∞(R) be such that all its derivatives belong to L2(R). Then