## 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
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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 $d > 1$.## 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