Ryll-Wojtaszczyk Formulas for bihomogeneous polynomials on the sphere

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.

💡 Research Summary

The paper studies projection constants for spaces of bihomogeneous polynomials and bihomogeneous harmonic polynomials on the unit sphere of a finite‑dimensional complex Hilbert space. After recalling the classical notion of a projection constant λ(X,Y) for a subspace X of a Banach space Y and the absolute projection constant λ(X), the authors focus on the subspaces
(P_{p,q}(S^{n-1})) – the space of (p,q)‑bihomogeneous polynomials restricted to the sphere, and
(H_{p,q}(S^{n-1})) – the subspace of those that are also harmonic.
Both families are invariant under the unitary group U(n) and are mutually orthogonal for distinct (p,q) pairs.

The central methodological innovation is an abstract averaging framework modelled on Rudin’s technique. The authors introduce a “Rudin triple” ((K,\mu),(G,m),\varphi) where (K=S^{n-1}), (\mu) is the normalized rotation‑invariant measure, (G=U(n)) with Haar measure (m), and (\varphi_g(z)=gz). The triple satisfies three natural conditions: transitivity of the group action, invariance of the measure under the action, and density of a family of finite‑dimensional invariant subspaces. Within this setting they define the reproducing kernel (k_S) of a finite‑dimensional invariant subspace (S\subset C(K)) and prove that the orthogonal projection (\pi_S) onto (S) is the unique (G)‑equivariant projection.

A key concept is “accessibility”. A subspace is called accessible if the restriction of (\pi_S) to (C(K)) is the unique (G)‑equivariant projection onto (S). A stronger notion, “strong accessibility”, requires that any function in (S) fixed by the stabiliser of a point (x_0) must be a scalar multiple of the kernel section (k_S(x_0,\cdot)). The authors prove that strong accessibility implies accessibility. Consequently, for any accessible subspace, the projection constant equals the (L^1)‑norm of the kernel section:
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