Ryll-Wojtaszczyk formulas for bihomogeneous polynomials on the sphere

Abstract

We investigate projection constants for spaces of bihomogeneous harmonic and bihomogeneous polynomials on the unit sphere in finite-dimensional complex Hilbert spaces. Using averaging techniques, we demonstrate that the minimal norm projection aligns with the natural orthogonal projection. This result enables us to establish a connection between these constants and weighted $L_1$-norms of specific Jacobi polynomials. Consequently, we derive explicit bounds, provide practical expressions for computation, and present asymptotically sharp estimates for these constants. Our findings extend the classical Ryll and Wojtaszczyk formula for the projection constant of homogeneous polynomials in finite-dimensional complex Hilbert spaces to the bihomogeneous setting.