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Orthogonality of Certain Functions with Respect to Complex Valued Weights

Published online by Cambridge University Press:  20 November 2018

George Gasper
George Gasper*
Affiliation:
Northwestern University, Evanston, Illinois
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In his work on the Dirichlet problem for the Heisenberg group Greiner [5] showed that each Lα-spherical harmonic is a unique linear combination of functions of the form

with k = 0, 1,2, … and n = 0, ±l, ±2 , …, where Hk(α, n)(θiθ) is defined by the generating function

Since Hk(0,0)(eiθ) = Pk(cos θ), where Pk(x) is the Legendre polynomial of degree k, and these functions satisfy the orthogonality relation

Greiner raised the question of whether the functions Hk(0,0)(eiθ) are orthogonal or biorthogonal with respect to some complex valued weight function.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1261 - 1270
Copyright
Copyright © Canadian Mathematical Society 1981

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