Bilinear biorthogonal expansions and the spectrum of an integral operator

• Author(s): Abreu, Luís Daniel ; Ciaurri, Óscar ; Varona, Juan Luis
• Date: 2009
• Persistent ID: http://hdl.handle.net/10316/13647
• Origin: Estudo Geral - Universidade de Coimbra
• Subject(s): Bilinear expansion; Biorthogonal expansion; Plane wave expansion; Sampling theorem; Fourier-Neumann expansion; Dunkl transform; Special functions; Q-special functions
• Document Type: Preprint
• Language: English

Description

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.