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On the Bessel-Wright transform. (English) Zbl 1438.33026

Summary: In the present paper, we consider a class of second-order singular differential operators which generalize the well-known Bessel differential operator. The associated eigenfunctions are the Bessel-Wright functions. These functions can be obtained by the action of the Riemann-Liouville operator on the normalized Bessel functions. We introduce a Bessel-Wright transform with Bessel-Wright functions as kernel which is connected to the classical Bessel-Fourier transform via the dual of the Riemann-Liouville operator. The Bessel-Wright transform leaves invariant the Schwartz space and sends the set of functions indefinitely differentiable with compact support into the Paley-Wiener space. We conclude the paper by proving two variants of the inversion formulas.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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