Bernstein-Nikolskii Inequality: Optimality with Respect to the Smoothness Parameter

TL;DR

The paper determines sharp, s-dependent constants in Bernstein–Nikolskii inequalities for both trig polynomials and entire functions of exponential type, establishing precise asymptotics in terms of $n$, $s$, $p$, and $q$. It proves matching upper and lower bounds, including an enhanced bound for trigonometric polynomials with concave coefficient sequences, and extends the analysis to fractional derivatives. The authors employ a blend of Jackson kernels, Paley–Wiener spaces, and discrete Hardy-space techniques, together with concave-sequence decompositions, to obtain optimal growth in $s$ and $n$ and to illuminate extremal polynomials. The results have implications for approximation theory and harmonic analysis by clarifying how smoothness constraints govern the size of derivatives in $L_q$ norms. The work also provides a rigorous framework for related inequalities in the discrete setting and deepens understanding of extremal polynomials and their zeros.

Abstract

In this paper, we study the form of the constant $C$ in the Bernstein--Nikolskii inequalities $\|f^{(s)}\|_q \lesssim C(s, p, q)\left\|f\right\|_p,\,0<p<q \leq\infty$, for trigonometric polynomials and entire functions of exponential type. We obtain the optimal behavior of the constant with respect to the smoothness parameter $s$.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof