Publication: Hardy-type theorem for orthogonal functions with respect to their zeros. The Jacobi weight case
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2008-05-15
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Elsevier
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Abstract
Motivated by the G.H. Hardy's 1939 results [G.H. Hardy, Notes on special systems of orthogonal functions II: 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 λn, 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^ν F(z), ν in R, where F is entire and,
$\int_0 1 f(λ_n t)f(λ_m t)t (1-t) dt=0, α>-1-2ν, β>-1
when n≠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.
$\int_0 1 f(λ_n t)f(λ_m t)t (1-t) dt=0, α>-1-2ν, β>-1
when n≠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.
Description
10 pages, no figures.
MR#: MR2398249 (2009d:46074)
Zbl#: Zbl 1139.42005
MR#: MR2398249 (2009d:46074)
Zbl#: Zbl 1139.42005
Keywords
Zeros of special functions, Orthogonality, Jacobi weights, Mellin transform on distributions, Entire functions, Bessel functions, Hyperbessel functions
Bibliographic citation
Journal of Mathematical Analysis and Applications, 2008, vol. 341, n. 2, p. 803-812