Summability of Product Jacobi Expansions

Abstract

Orthogonal expansions in product Jacobi polynomials with respect to the weight function Wα, β(x)=∏dj=1(1−xj)αj (1+xj)βj on [−1,1]d are studied. For αj,βj>−1 and αj+βj⩾−1, the Cesàro (C,δ) means of the product Jacobi expansion converge in the norm of Lp(Wα,β,[−1,1]d), 1⩽p<∞, and C([−1,1]d) if δ>∑j=1dmax{αj,βj}+d2+max0,−∑j=1dmin{αj,βj}−d+22 .Moreover, for αj,βj⩾−1/2, the (C,δ) means define a positive linear operator if and only if δ⩾∑di=1 (αi+βi)+3d−1.