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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Divergent Jacobi polynomial series
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by Christopher Meaney
Proc. Amer. Math. Soc. 87 (1983), 459-462
DOI: https://doi.org/10.1090/S0002-9939-1983-0684639-4

Abstract:

Fix real numbers $\alpha \geqslant \beta \geqslant - \tfrac {1}{2}$, with $\alpha > - \tfrac {1}{2}$, and equip $[ - 1,1]$ with the measure $d\mu (x) = {(1 - x)^\alpha }{(1 + x)^\beta }dx$. For $p = 4(\alpha + 1)/(2\alpha + 3)$ there exists $f \in {L^p}(\mu )$ such that $f(x) = 0$ a.e. on $[ - 1,0]$ and the appropriate Jacobi polynomial series for $f$ diverges a.e. on $[ - 1,1]$. This implies failure of localization for spherical harmonic expansions of elements of ${L^{2d/(d + 1)}}(X)$, where $X$ is a sphere or projective space of dimension $d > 1$.
References
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Bibliographic Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 87 (1983), 459-462
  • MSC: Primary 42C10; Secondary 43A25, 58G25
  • DOI: https://doi.org/10.1090/S0002-9939-1983-0684639-4
  • MathSciNet review: 684639