Related papers: Jackson's inequality on the hypercube

We give a distribution-dependent concentration inequality for functions of independent variables. The result extends Bernstein's inequality from sums to more general functions, whose variation in any argument does not depend too much on the…

In this note we study the error term R_{n,L}(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary…

We extend the main result of Harrow, Kolla, and Schulman -- the existence of dimension-free $L^2$-bounds for the spherical maximal function in the hypercube -- to all $L^p, p > 1$. Our approach is motivated by the spectral technique…

Let $A^2(D)$ be the Bergman space over the open unit disk $D$ in the complex plane. Korenblum conjectured that there is an absolute constant $c \in (0,1)$ such that whenever $|f(z)|\le |g(z)|$ in the annulus $c<|z|<1$ then $||f(z)|| \le…

For arrays $(S_{i,j})_{1\leq i\leq j}$ of random variables that are stationary in an appropriate sense, we show that the fluctuations of the process $(S_{1,n})_{n=1}^{\infty}$ can be bounded in terms of a measure of the ``mean…

Let $(\{f_j\}_{j=1}^n, \{\tau_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^n, \{\omega_k\}_{k=1}^n)$ be two p-orthonormal bases for a finite dimensional Banach space $\mathcal{X}$. Let $M,N\subseteq \{1, \dots, n\}$ be such that \begin{align*}…

Talagran's correlation inequality provides quantitative lower bounds on the covariance of two increasing Boolean functions in terms of their coordinate influences, but, in general, a logarithmic loss is necessary. Motivated by a question of…

This paper is devoted to the equivalence of two type direct theorems in Approximation Theory: a) for smooth functions (Favard's estimates). b) for arbitrary continuous function (Jackson--Stechkin estimates). Specifically, we will show that…

We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $\alpha>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot…

In this paper we prove that there exists a constant $C$ such that, if $S,\Sigma$ are subsets of $\R^d$ of finite measure, then for every function $f\in L^2(\R^d)$, $$\int_{\R^d}|f(x)|^2 dx \leq C e^{C \min(|S||\Sigma|, |S|^{1/d}w(\Sigma),…

We study the problem of estimating a monotone function $f:\{0,1\}^d\to[0,1]$ from noisy observations at uniformly random vertices of the Boolean hypercube. As a measure of complexity for the target~$f$, we use the total $L^1$-influence…

We consider a type of Hardy-Sobolev inequality, whose weight function is singular on the whole domain boundary. We are concerned with the attainability of the best constant of such inequality. In dimension two, we link the inequality to a…

Let $\mathcal{A}$ be a $C^*$-algebra and $\phi:\cA\to L(H)$ be a positive unital map. Then, for a convex function $f:I\to \mathbb{R}$ defined on some open interval and a self-adjoint element $a\in \mathcal{A}$ whose spectrum lies in $I$, we…

Suppose that $\alpha \in (0,2)$ and that $X$ is an $\alpha$-stable-like process on $\R^d$. Let $F$ be a function on $\R^d$ belonging to the class $\bf{J_{d,\alpha}}$ (see Introduction) and $A_{t}^{F}$ be $\sum_{s \le t}F(X_{s-},X_{s}), t>…

This paper studies estimation of and inference on a distribution function $F$ that is concave on the nonnegative half line and admits a density function $f$ with potentially unbounded support. When $F$ is strictly concave, we show that the…

The classical Davis inequality $\mathbb{E} Mf\simeq \mathbb{E} Sf$, where $(Sf)^2=\sum_{k}\left|f_{k}-f_{k-1}\right|^2$ is the square function and $Mf= \sup_n \left|f_n\right|$ is the maximal function, is true with a universal constant for…

If $\mu$ is a distribution over the $d$-dimensional Boolean cube $\{0,1\}^d$, our goal is to estimate its mean $p\in[0,1]^d$ based on $n$ iid draws from $\mu$. Specifically, we consider the empirical mean estimator $\hat p_n$ and study the…

A classical result due to Frank and Seiringer asserts that for $1\leq p<\frac Ns$, there exists a sharp constant $\mathcal{C}_{N,s,p}>0$ such that $$…

We generalize Gr\"unbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to $\mathrm{RCD}(0,N)$-spaces with $N \in (1,\infty)$ as well as weighted Riemannian manifolds of…

The problem of testing monotonicity for Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$ is a classic topic in property testing. When $n=2$, the domain is the hypercube. For the hypercube case, a breakthrough result of…