Representation of reproducing kernels and the Lebesgue constants on the ball

For the weight function (1 - parallel tox parallel to (2))(mu -1/2) on the unit ball, a closed formula of the reproducing kernel is modified to include the case -1/2 < mu < 0. The new formula is used to study the orthogonal projection of the weighted L-2 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 for mu < 0, which is the smallest possible growth rate among all projections, while the rate for mu greater than or equal to 0 is n(mu+(d-1)/2). (C) 2001 Academic Press.