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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Divergent Jacobi polynomial series


Author: Christopher Meaney
Journal: Proc. Amer. Math. Soc. 87 (1983), 459-462
MSC: Primary 42C10; Secondary 43A25, 58G25
MathSciNet review: 684639
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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$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0684639-4
PII: S 0002-9939(1983)0684639-4
Keywords: Jacobi polynomial, localization, compact two-point homogeneous space, Laplace-Beltrami operator, spherical harmonic, zonal
Article copyright: © Copyright 1983 American Mathematical Society



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