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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the singularities of Gegenbauer (ultraspherical) expansions
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by Ahmed I. Zayed PDF
Trans. Amer. Math. Soc. 262 (1980), 487-503 Request permission

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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 262 (1980), 487-503
  • MSC: Primary 33A50; Secondary 46F10
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0586730-1
  • MathSciNet review: 586730