Hardy-type theorem for orthogonal functions with respect to their zeros. The Jacobi weight case

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:341

Article

Abreu, L. D.^2,3  Marcellan, F.¹  Yakubovich, S. B.⁴ 

[1] Univ Carlos III Madrid, Dept Math, Leganes 28911, Spain

[2] Univ Coimbra, Dept Math, FCTUC, P-3001454 Coimbra, Portugal

[3] Univ Vienna, Dept Math, NuHag, A-1090 Vienna, Austria

[4] Univ Porto, Fac Sci, Dept Pure Math, P-4169007 Oporto, Portugal

关键词: zeros of special functions;    orthogonality;    Jacobi weights;    Mellin transform on distributions;    entire functions;    Bessel functions;    hyperbessel functions;   

DOI  :  10.1016/j.jmaa.2007.10.050

来源: Elsevier

【 摘 要 】

Motivated by the G.H. Hardy's 1939 results [G.H. Hardy, Notes on special systems of orthogonal functions 11: On functions orthogonal with respect to their own zeros, J. London Math. Soc. 14 (1939) 37-44] on functions orthogonal with respect to their real zeros lambda(n), n = 1, 2,..., we will consider, under the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval (0, 1), that is, the functions f (z) = z(nu) F(z), nu is an element of R, where F is entire and integral(1)(0) f(lambda(n)t)f(lambda(m)t)t(alpha)(1-t)(beta) dt = 0, alpha > -1-2 nu, beta > -1, when n not equal m. Considering all possible functions on this class we obtain a new family of generalized Bessel functions including Bessel and hyperbessel functions as special cases. (C) 2007 Elsevier Inc. All rights reserved.

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