Divergent Jacobi polynomial series

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 $d > 1$.