STRONG AND WEAK WEIGHTED CONVERGENCE OF JACOBI SERIES

Given alpha,beta>-1, let p(n)(x)=p(n)((alpha,beta))(x), n=0,1,2, ... b e the sequence of Jacobi polynomials orthonormal on (-1,1) with respec t to the weight u(x)=(1-x)(alpha)(1+x)(beta). Denote by (S(N)f)(x) the Nth partial sum of the Fourier-Jacobi series of the function f on (-1 ,1), so that (S(N)f)(x)=Sigma(n=0)(N)a(n)p(n)(x), with a(n)=integral(- 1)(1)f(x)p(n)(x)u(x)dx. For fixed P is an element of(1,infinity), we c haracterize the weights w such that inity)integral(-1)(-)\[S(N)f)(x)-f (x)]w(x)\(p)u(x) dx=0 whenever integral(-1)(1)\f(x)w(x)\(p)dx<infinity , the weights w such that lim(N-->infinity)sup(lambda>0)lambda[integra l (N)(E lambda)w(x)(p)u(x)dx](1/p)=0 whenever integral(0)(infinity)[in tegral(F lambda)w(x)dx](1/p) d lambda<infinity; here, E(lambda)(N)={x is an element of (-1,1);\(S(N)f(x)-f(x)\>lambda} and F-lambda={x is an element of(-1,1):\f(x)\>lambda}. (C) 1997 Academic Press