Summability of Fourier orthogonal series for Jacobi weight on a ball in $\mathbb {R}^d$

Abstract:

Fourier orthogonal series with respect to the weight function $(1-|\mathbf {x} |^{2})^{\mu - 1/2}$ on the unit ball in $\mathbb {R}^{d}$ are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourier orthogonal series with respect to $(1-|\mathbf {x} |^{2})^{\mu -1/2}$ is uniformly $(C, \delta )$ summable on the ball if and only if $\delta > \mu + (d-1)/2$.