Divergent Jacobi polynomial series

Abstract

Fix real numbers α ⩾ β ⩾ − 1 2 \alpha \geqslant \beta \geqslant - \tfrac {1}{2} , with α > − 1 2 \alpha > - \tfrac {1}{2} , and equip [ − 1 , 1 ] [ - 1,1] with the measure d μ ( x ) = ( 1 − x ) α ( 1 + x ) β d x d\mu (x) = {(1 - x)^\alpha }{(1 + x)^\beta }dx . For p = 4 ( α + 1 ) / ( 2 α + 3 ) p = 4(\alpha + 1)/(2\alpha + 3) there exists f ∈ L p ( μ ) f \in {L^p}(\mu ) such that f ( x ) = 0 f(x) = 0 a.e. on [ − 1 , 0 ] [ - 1,0] and the appropriate Jacobi polynomial series for f f diverges a.e. on [ − 1 , 1 ] [ - 1,1] . This implies failure of localization for spherical harmonic expansions of elements of L 2 d / ( d + 1 ) ( X ) {L^{2d/(d + 1)}}(X) , where X X is a sphere or projective space of dimension d > 1 d > 1 .