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Ryll-Wojtaszczyk Formulas for bihomogeneous polynomials on the sphere

Andreas Defant, Daniel Galicer, Martín Mansilla, Mieczysław Mastyło, Santiago Muro

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.

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 \linebreak -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.

Paper Structure

This paper contains 13 sections, 30 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Brief Preliminaries on Bihomogeneous Polynomial Spaces
  3. Rudin's Framework: Paving the Way to an Integral Formula
  4. Reproducing kernels
  5. Accessability
  6. A concrete Rudin triple and its application to our setting
  7. Jacobi Polynomials as a Foundation for Reproducing Kernels
  8. Integral Formulas
  9. Asymptotic Behavior and Various Bounds
  10. Equidistant parameters
  11. Upper estimates
  12. The case $\pmb{q=1}$
  13. Non-compact inclusions: Variants of the Ryll-Wojtaszczyk results

Key Result

Lemma 2.1

For all $p, q \geq 0$, $f \in \mathfrak{P}_{p,q}(\mathbb{S}^{n-1})$, and $U \in \mathcal{U}_n$, we have that $f \circ U \in \mathfrak{P}_{p,q}(\mathbb{S}^{n-1})$. The same result holds if $\mathfrak{P}_{p,q}(\mathbb{S}^{n-1})$ is replaced by $\mathcal{H}_{p,q}(\mathbb{S}^{n-1})$.

Theorems & Definitions (56)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 46 more