Plancherel and Paley-wiener Theorems for an Index Integral Transform Vu Kim Tuan, Ali Ismail

where Jν(x) is the Bessel function of the first kind of order ν [1], and =z denotes the imaginary part of z. An extensive table of integral transforms involving the Bessel functions in the kernels is collected in [6]. Since the integration in (2) is with respect to the order of the Bessel function, such a pair of integral transforms is called index transform. Details about many other index transforms can be found in [15]. In Section 2 we will show that the pair of transforms (1)-(2) arises from a singular SturmLiouville problem on a half line. As a consequence, a Plancherel’s theorem and a Parseval’s formula for the pair of transforms (1)-(2) will be established. In Section 3 we will characterize function g(t) as the transform (2) of a function G(τ) with a compact support. The classical Paley-Wiener theorem [5] for the Fourier transform gives a characterization of the space of square integrable functions with compact support in terms of its image under the Fourier transform by showing that f ∈ L2(R) has a compact support if and only if its Fourier transform f̂ can be continued analytically to the whole complex plane