Representation of Reproducing Kernels and the Lebesgue Constants on the Ball

Abstract

For the weight function (1−‖x‖2)μ−1/2 on the unit ball, a closed formula of the reproducing kernel is modified to include the case −1/2<μ<0. The new formula is used to study the orthogonal projection of the weighted L2 space onto the space of polynomials of degree at most n, and it is proved that the uniform norm of the projection operator has the growth rate of n(d−1)/2 for μ<0, which is the smallest possible growth rate among all projections, while the rate for μ⩾0 is nμ+(d−1)/2.