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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the singularities of Gegenbauer (ultraspherical) expansions


Author: Ahmed I. Zayed
Journal: Trans. Amer. Math. Soc. 262 (1980), 487-503
MSC: Primary 33A50; Secondary 46F10
MathSciNet review: 586730
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Abstract: The results of Gilbert on the location of the singular points of an analytic function $ f(z)$ given by Gegenbauer (ultraspherical) series expansion $ f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^\mu (z)$ are extended to the case where the series converges to a distribution. On the other hand, this generalizes Walter's results on distributions given by Legendre series: $ f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^{1/2}(z)$. The singularities of the analytic representation of $ f(z)$ are compared to those of the associated power series $ g(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}{z^n}$. The notion of value of a distribution at a point is used to study the boundary behavior of the associated power series. A sufficient condition for Abel summability of Gegenbauer series is also obtained in terms of the distribution to which the series converges.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0586730-1
PII: S 0002-9947(1980)0586730-1
Keywords: Gegenbauer expansions, generalized functions, value of a distribution at a point, Abel summability
Article copyright: © Copyright 1980 American Mathematical Society