Minimal projections onto spaces of polynomials on real euclidean spheres

Abstract

We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous polynomials as well as for spaces of polynomials of finite degree on the unit sphere. We establish a connection between these quantities and certain weighted $L_1$-norms of specific Jacobi polynomials. As a consequence, we present exact formulas, computable expressions and asymptotically accurate estimates for them. The real case we address is considerably more nuanced than its complex counterpart.

Small corrections - Accepted in the Journal of the London Mathematical Society

References in corpus (5)

• Sampling projections in the uniform norm
• On the value of the fifth maximal projection constant
• A simple proof of the Grunbaum conjecture
• Projecting Lipschitz functions onto spaces of polynomials
• The projection constant for the trace class