Summability of product Jacobi expansions

Orthogonal expansions in product Jacobi polynomials with respect to the weight function W alpha,beta(x)= Pi(j=1)(d) (1 - x(j))(alpha j)(1 + x(j))(beta j) on [-1, 1](d) are studied. For alpha(j), beta(j) > -1 and alpha(j)+ beta(j) greater than or equal to -1, the Crsdro (C,delta) means of the product Jacobi expansion converge in the norm of LP(W-alpha. beta, [-1, 1](d)) , 1 less than or equal to p < infinity, and C([-1, 1]d) if delta > Sigma(j=1)(d) max {alpha(j), beta(j)} + d/2 + max {0, -Sigma(j=t)(d) min {alpha(j), beta(j)} - d+2/2}. Moreover, for alpha(j), beta(j) greater than or equal to - 1/2, the (C, delta) means define a positive linear operator if and only if delta greater than or equal to Sigma(i=1)(d) (alpha(i)+beta(i)) + 3d - 1. (C) 2000 Academic Press.