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Title: Bilinear biorthogonal expansions and the spectrum of an integral operator
Authors: Abreu, Luís Daniel 
Ciaurri, Óscar 
Varona, Juan Luis 
Keywords: Bilinear expansion; Biorthogonal expansion; Plane wave expansion; Sampling theorem; Fourier-Neumann expansion; Dunkl transform; Special functions; Q-special functions
Issue Date: 2009
Publisher: Centro de Matemática da Universidade de Coimbra
Citation: Pré-Publicações DMUC. 09-32 (2009)
Serial title, monograph or event: Pré-Publicações DMUC
Issue: 09-32
Place of publication or event: Coimbra
Abstract: We study an extension of the classical Paley-Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier- Neumann type series as special cases. Concerning applications, several new results are obtained. From the Dunkl analogue of Gegenbauer’s expansion of the plane wave, we derive sampling and Fourier-Neumann type expansions and an explicit closed formula for the spectrum of a right inverse of the Dunkl operator. This is done by stating the problem in such a way it is possible to use the technique due to Ismail and Zhang. Moreover, we provide a q-analogue of the Fourier-Neumann expansions in q-Bessel functions of the third type. In particular, we obtain a q-linear analogue of Gegenbauer’s expansion of the plane wave by using q-Gegenbauer polynomials defined in terms of little q-Jacobi polynomials.
Rights: openAccess
Appears in Collections:FCTUC Matemática - Vários

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