Ryll-Wojtaszczyk Formulas for bihomogeneous polynomials on the sphere

TL;DR

This paper extends classical formulas for projection constants to bihomogeneous polynomials on complex spheres, providing explicit bounds, practical computation methods, and asymptotic estimates using harmonic analysis and Jacobi polynomials.

Contribution

It generalizes the Ryll-Wojtaszczyk formula to bihomogeneous polynomials, connecting projection constants with Jacobi polynomial norms and establishing explicit bounds and asymptotic estimates.

Findings

Derived explicit bounds for projection constants.

Connected projection constants with weighted L1-norms of Jacobi polynomials.

Extended classical formulas to bihomogeneous polynomial setting.

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 \linebreak $L_1$-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.