A four dimensional Bernstein Theorem

TL;DR

This paper extends Bernstein's theorem to four dimensions using quaternionic polynomials, establishing a new inequality and connecting it with zonal harmonics and Gegenbauer polynomials.

Contribution

It introduces a four-dimensional Bernstein theorem for quaternionic polynomials and derives a related Bernstein inequality with harmonic analysis connections.

Findings

Proved a four-dimensional Bernstein theorem for quaternionic polynomials

Derived a quaternionic Bernstein inequality

Connected the results with zonal harmonics and Gegenbauer polynomials

Abstract

We prove a four dimensional version of the Bernstein Theorem, with complex polynomials being replaced by quaternionic polynomials. We deduce from the theorem a quaternionic Bernstein's inequality and give a formulation of this last result in terms of four-dimensional zonal harmonics and Gegenbauer polynomials.