Lower bounds for quasianalytic functions, II. The Bernstein quasianalytic functions

Reverse Bernstein Inequality on the Circle

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 …

Oscillation of Fourier Integrals with a spectral gap

Suppose that Fourier transform of a function f is zero on the interval [-a,a]. We prove that the lower density of sign changes of f is at least a/pi, provided that f is a locally integrable temperate distribution in the sense of Beurling, with non-quasianalytic weight. We construct an example showing that the last condition cannot be omitted.

二维实值多项式的无维马尔可夫-伯恩斯坦型不等式

Dimension-free Markov--Bernstein-type inequality for $\mathbb R^2$-valued polynomials

该研究证明了$\boldsymbol{R}^n$到$\boldsymbol{R}^2$的次数不超过$d$的多项式映射的无维马尔可夫-伯恩斯坦型不等式，所得高斯不等式解答了Nourdin等人关于马利亚万行列式期望下界的问题。

加权切比雪夫多项式和正交多项式的下界

Lower Bounds for Weighted Chebyshev and Orthogonal Polynomials

本文推导了实直线紧致子集上加权切比雪夫和正交多项式Widom因子的最优渐近与非渐近下界，并推广了Bernstein和Widom的经典结果至更广权重类。

A lower bound for the norm of the minimal residual polynomial

Let $S$ be a compact infinite set in the complex plane with $0\notin{S}$, and let $R_n$ be the minimal residual polynomial on $S$, i.e., the minimal polynomial of degree at most $n$ on $S$ with respect to the supremum norm provided that $R_n(0)=1$. For the norm $L_n(S)$ of the minimal residual polynomial, the limit $κ(S):=\lim_{n\to\infty}\sqrt[n]{L_n(S)}$ exists. In addition to the well-known and widely referenced inequality $L_n(S)\geqκ(S)^n$, we derive the sharper inequality $L_n(S)\geq2κ(S)^n/(1+κ(S)^{2n})$ in the case that $S$ is the union of a finite number of real intervals. As a conseq…

On a problem of Ramachandra and approximation of functions by Dirichlet polynomials with bounded coefficients

We prove effective results on when a function can be approximated by a Dirichlet polynomial with bounded coefficients. Assuming that Φ(n) is an increasing function we prove that the set of polynomials {\sum_{n=2}^N a_n n^{it-1}: N \geq 2, |a_n| \leq Φ(n)}, is dense in L^2(0,H) if and only if \sum_{n=2}^\infty \frac{\log Φ(n)} {n \log^2 n} = \infty. We also prove variants of this result for generalized Dirichlet polynomials. The main tools are theorems of Paley and Wiener related to quasianalyticity and the Pechersky rearrangement theorem. We use this result to give precise conditions on when a…

Right order Turan-type converse Markov inequalities for convex domains on the plane

For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$ can be arbitrarily small as compared to $||p||$. The situation changes when we assume that the polynomials in question have all their zeroes in the convex body $K$. This was first investigated by Turán, who showed the lower bounds $||p'|| \ge (n/2) ||p||$ for the unit disk $D$ and $||p'|| > c \sqrt{n} ||p||$ for the unit interval $I:=[-1,1]$. Although parti…

Polynomial inequalities on the Hamming cube

Let $(X,\|\cdot\|_X)$ be a Banach space. The purpose of this article is to systematically investigate dimension independent properties of vector valued functions $f:\{-1,1\}^n\to X$ on the Hamming cube whose spectrum is bounded above or below. Our proofs exploit contractivity properties of the heat flow, induced by the geometry of the target space $(X,\|\cdot\|_X)$, combined with duality arguments and suitable tools from approximation theory and complex analysis. We obtain a series of improvements of various well-studied estimates for functions with bounded spectrum, including moment compariso…