On Bernstein type quantitative estimates for Ornstein non-inequalities

Abstract

For the sequence of multi-indices $\{{\alpha }_j{\}}_{j=1}^m$ and $\beta$, we study the inequality

where $f$ is a trigonometric polynomial of degree at most $N$ on the $d$-dimensional torus. Assuming some natural geometric property of the set $\{{\alpha }_j\}\cup \{\beta \}$, we show that

where $\varphi <1$ depends only on the set $\{{\alpha }_j\}\cup \{\beta \}$.