Average Nikolskii factors for random trigonometric polynomials

TL;DR

This work quantifies the average $(p,q)$-Nikolskii factor for random trigonometric polynomials with Gaussian coefficients, demonstrating substantial improvement over the worst-case bounds in most regimes. A general finite-dimensional framework is developed, establishing basis invariance and connecting moment estimates to structural tools like the Christoffel function. In the univariate setting, the authors derive precise orders: N^{ave}_{2,q}(T_n) remains constant for finite q, while N^{ave}_{2,∞}(T_n) scales as (ln N)^{1/2}, and they bound E[1/||T_a||_∞^r] to enable full characterization. The results extend to multivariate polynomials on T^d, with N=(2n+1)^d, and crucially the constants are dimension-free, revealing that average Nikolskii factors stay near constant even as the ambient dimension grows, whereas worst-case factors can grow with N.

Abstract

For $1\le p,q\le \infty$, the Nikolskii factor for a trigonometric polynomial $T_{\bf a}$ is defined by $$\mathcal N_{p,q}(T_{\bf a})=\frac{\|T_{\bf a}\|_{q}}{\|T_{\bf a}\|_{p}},\ \ T_{\bf a}(x)=a_{1}+\sum\limits^{n}_{k=1}(a_{2k}\sqrt{2}\cos kx+a_{2k+1}\sqrt{2}\sin kx).$$ We study this average Nikolskii factor for random trigonometric polynomials with independent $N(0,σ^{2})$ coefficients and obtain that the exact order. For $1\leq p<q<\infty$, the average Nikolskii factor is order degree to the 0, as compared to the degree $1/p-1/q$ worst case bound. We also give the generalization to random multivariate trigonometric polynomials.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Corollary 2.4