Preservation of Positive-Definiteness by Bernstein Operators on the Circle

Abstract

We prove that, for every $n\ge1$, the degree-$n$ Bernstein operator on $[0,π]$ preserves positive-definiteness on the circle $S^1$. Equivalently, if a continuous function on $[0,π]$ defines a positive-definite isotropic kernel on $S^1$, then its Bernstein polynomial approximation of any fixed degree does as well. The proof reduces the problem to the nonnegativity of the cosine coefficients of the Bernstein images $Q_{n,m}=B_n[\cos(mx)]$, which we prove using an explicit coefficient formula and a two-regime positivity argument. We also discuss the higher-dimensional sphere analogue and show that the naive affine Bernstein operator fails to preserve the positive-definite cone already on $S^2$.