Strict positive definiteness on spheres via disk polynomials.
We present a solution to the (strict) Bernstein-Nachbin approximation problem in the general complex case. As a corollary, we get proofs of the analytic, the quasi-analytic, and the bounded criteria for localizability in the general complex case. This generalizes the known results of the real or self-adjoint complex cases, in the same way that Bishop’s Theorem generalizes the Weierstrass-Stone Theorem. However, even in the real or self-adjoint complex cases, the results that we obtain are stronger...
Due to the fact that in the case the -Bernstein polynomials are no longer positive linear operators on the study of their convergence properties turns out to be essentially more difficult than that for In this paper, new saturation theorems related to the convergence of -Bernstein polynomials in the case are proved.