Reverse Bernstein Inequality on the Circle

TL;DR

This paper proves a reversed Bernstein inequality for functions orthogonal to fixed-degree polynomials on the circle, providing improved lower bounds on derivatives compared to traditional bounds.

Contribution

It introduces a reversed Bernstein inequality for orthogonal complements of polynomial spaces and generalizes it to higher derivatives, improving existing bounds.

Findings

Established a reversed Bernstein inequality for functions orthogonal to polynomials.

Derived better lower bounds for higher derivatives than iterative applications of first derivative bounds.

Extended classical Bernstein inequality to a broader function class on the circle.

Abstract

The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supremum norm of the polynomial itself. The reversed Bernstein inequality, that we prove in this note, says that the reverse inequality holds for functions in the orthogonal complement of the space of polynomials of fixed degree. In fact, we derived a more general result for the lower bounds on higher derivatives. These bounds are better then those obtained by applying bound for the first derivative successively several times.