Related papers: On Bernstein- and Marcinkiewicz-type inequalities …
The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz--Zygmund inequalities provide answers to both the recovery and the…
Kolesnikov-Milman [9] established a local $L_p$-Brunn-Minkowski inequality for $p\in(1-c/n^{\frac{3}{2}},1).$ Based on their local uniqueness results for the $L_p$-Minkowski problem, we prove in this paper the (global) $L_p$-Brunn-Minkowski…
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. This problem is very important in applications but there is no systematic study of it. We present here a new technique, which…
We study the relationship between sampling sequences in infinite dimensional Hilbert spaces of analytic functions and Marcinkiewicz-Zygmund inequalities in subspaces of polynomials. We focus on the study of the Hardy space and the Bergman…
We consider various inequalities for polynomials, with an emphasis on the most fundamental inequalities of approximation theory. In the sequel a key role is played by the generalized Minkowski functional \alpha(K,x), already being used by…
Let $\mu_p$ be the generalized Gaussian distribution on $\mathbb{R}^n$ with density $e^{-\frac{|x|^p}{p}}$ multiplied by a constant depending on $p\ge 1$ and $n$, and $\alpha_p(n)$ be the largest number such that the Brunn-Minkowski type…
We prove limit relations between the sharp constants in the multivariate Bernstein-Nikolskii type inequalities for trigonometric polynomials and entire functions of exponential type with the spectrum in a centrally symmetric convex body.
Due to the invalidation of the Gauss-Lucas type result for quaternionic polynomials, we first give in this paper an alternative proof of the Bernstein inequality in $L^{p} (1\leq p \leq+\infty)$ for slice regular polynomials by the Fej\'er…
We find necessary density conditions for Marcinkiewicz-Zygmund inequalities and interpolation for spaces of spherical harmonics with respect to the L^p norm. Moreover, we prove that there are no complete interpolation families for p\neq 2.
In this paper, we extend the Marcinkiewicz--Zygmund inequality to the setting of Orlicz and Lorentz spaces. Furthermore, we generalize a Kadec--Pe{\l}czy\'nski-type result -- originally established by the first and third authors for $L^p$…
We construct the extension of the curvilinear summation for bounded Borel measurable sets to the $L_p$ space for multiple power parameter $\bar{\alpha}=(\alpha_1, \cdots, \alpha_{n+1})$ when $p>0$. Based on this…
A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{\alpha,\beta}(x)$, which is uniform for all degrees $n\ge0$, all real $\alpha,\beta\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of…
We show that a strong version of the Brascamp--Lieb inequality for symmetric log-concave measure with $\alpha$-homogeneous potential $V$ is equivalent to a $p$-Brunn--Minkowski inequality for level sets of $V$ with some $p(\alpha,n)<0$. We…
In this paper, we investigate the wavelet coefficients for function spaces $\mathcal{A}_k^p=\{f: \|(i \omega)^k\hat{f}(\omega)\|_p\leq 1\}, k\in N, p\in(1,\infty)$ using an important quantity $C_{k,p}(\psi)$. In particular, Bernstein type…
Two types of Bernstein inequalities are established on the unit ball in $\mathbb{R}^d$, which are stronger than those known in the literature. The first type consists of inequalities in $L^p$ norm for a fully symmetric doubling weight on…
In this paper we give sharp $L_p$ Markov type inequalities for derivatives of multivariate polynomials for a wide family of UPC domains.
The paper addresses a problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under a standard assumption formulated in terms of…
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun recently. In this paper we…
We consider a general way to obtain Pr\'ekopa-Leindler and Borell-Brascamp-Lieb type inequalities from Brunn-Minkowski type inequalities and provide numerous examples. We use the same heuristic to prove a discrete version of the…
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, $||p'||$…