Orthogonal polynomials and summability in Fourier orthogonal series on spheres and on balls (Q2743681)

The fact that the orthogonal polynomials with respect to \((1-|x|^2)^{(m- 2)/2}\) on the unit ball \(B^d\), where \(|x|\) is the usual Euclidean norm, and the spherical harmonics on the unit sphere \(S^{d+m-1}\) are related has already been observed. The present author reveals that the relation between orthogonal polynomials on \(S^{d+m-1}\) and orthogonal polynomials on \(B^d\) holds for a large family of weight functions. As a consequence, he derives relations between reproducing kernels on \(S^{d+m-1}\) and on \(B^d\), and proves that summability of Fourier orthogonal series on the unit ball follows from summability on the unit sphere in the case of a number of weight functions.