A Multiplier Version of the Bernstein Inequality on the Complex Sphere

TL;DR

This paper establishes a new multiplier version of the Bernstein inequality on the complex sphere, linking polynomial spaces invariant under unitary and orthogonal groups through reproducing kernels.

Contribution

It introduces a novel multiplier Bernstein inequality on the complex sphere and relates sums of Jacobi and Gegenbauer polynomials to invariant polynomial spaces.

Findings

Derived a new inequality relating polynomial spaces under different symmetry groups.

Connected sums of reproducing kernels with invariant polynomial spaces.

Provided a new relation between Jacobi and Gegenbauer polynomial sums.

Abstract

We prove a multiplier version of the Bernstein inequality on the complex sphere. Included in this is a new result relating a bivariate sum involving Jacobi polynomials and Gegenbauer polynomials, which relates the sum of reproducing kernels on spaces of polynomials irreducibly invariant under the unitary group, with the reproducing kernel of the sum of these spaces, which is irreducibly invariant under the action of the orthogonal group.