Related papers: Jackson's inequality on the hypercube

This is a conitunation of [1] and [2]. We prove that if function $f$ belongs to the class $\Lambda_{\omega} \overset{\text{def}}{=} \{f: \omega_{f}(\delta)\leq \text{const} \omega(\delta)\} $ for an arbitrary modulus of continuity $\omega$,…

The classical Jackson-Stechkin inequality estimates the value of the best uniform approximation of a periodic function by trigonometric polynomials of degree $\le n-1$ in terms of its $r$-th modulus of smoothness $\omega_r(f,\delta)$. The…

In this paper we deal with improvement of Jensen, Jensen-Steffensen's and Jensen's functionals related inequalities for uniformly convex, phi-convex and superquadratic functions.

We consider the inequality $f \geqslant f\star f$ for real integrable functions on $d$ dimensional Euclidean space where $f\star f$ denotes the convolution of $f$ with itself. We show that all such functions $f$ are non-negative, which is…

This note is concerned with the Bianchi-Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents $s \in (0, \frac{d}{2})$. We prove that in dimension $d \geq 2$ the best…

This paper studies two classical inequalities, namely the Hausdorff-Young inequality and equal-exponent Young's convolution inequality, for discrete functions supported in the binary cube $\{0,1\}^d\subset\mathbb{Z}^d$. We characterize the…

Let $G(k)=\int_0^1g(x)e^{kx}dx$, $g\in L^1(0,1)$. The main result of this paper is the following theorem. {\bf Theorem}. {\it If $\limsup_{k\to +\infty}|G(k)|<\infty$, then $g=0$. There exists $g\not\equiv 0$, $g\in L^1(0,1)$, such that…

Let $R_j$ denote the $j^{\text{th}}$ Riesz transform on $\mathbb{R}^n$. We prove that there exists an absolute constant $C>0$ such that \begin{align*} |\{|R_jf|>\lambda\}|\leq C\left(\frac{1}{\lambda}\|f\|_{L^1(\mathbb{R}^n)}+\sup_{\nu}…

In $1977$, G. Bennett proved, by means of non-deterministic methods, an inequality which plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for $p_{1},p_{2} \in\lbrack1,\infty]$…

Let $u$ and $v$ be harmonic in $ \Omega \subset \mathbb{R}^n$ functions with the same zero set $Z$. We show that the ratio $f$ of such functions is always well-defined and is real analytic. Moreover it satisfies the maximum and minimum…

Let $S \subseteq \mathbb{R}^n$ be a compact semialgebraic set and let $f$ be a polynomial nonnegative on $S$. Schm\"udgen's Positivstellensatz then states that for any $\eta > 0$, the nonnegativity of $f + \eta$ on $S$ can be certified by…

We show several variants of concentration inequalities on the sphere stated as subgaussian estimates with optimal constants. For a Lipschitz function, we give one-sided and two-sided bounds for deviation from the median as well as from the…

Let $\mathscr{F}_{n,d}$ be the class of all functions $f:\{-1,1\}^n\to[-1,1]$ on the $n$-dimensional discrete hypercube of degree at most $d$. In the first part of this paper, we prove that any (deterministic or randomized) algorithm which…

We establish sharp forms of Young's convolution inequality and its reverse on the discrete hypercube $\{0,1\}^d$ in the diagonal case $p=q$. As applications, we derive bounds for additive energies and sumsets. We also investigate the…

We establish new upper and lower bounds on the number of queries required to test convexity of functions over various discrete domains. 1. We provide a simplified version of the non-adaptive convexity tester on the line. We re-prove the…

Classically, Jensen's Inequality asserts that if $X$ is a compact convex set, and $f:K\to \mathbb{R}$ is a convex function, then for any probability measure $\mu$ on $K$, that $f(\text{bar}(\mu))\le \int f\;d\mu$, where $\text{bar}(\mu)$ is…

We study the sharp constant $W_{n}(D)$ in Wiener's inequality for positive definite functions \[ \int_{\mathbb{T}^{n}}|f|^{2}\,dx\le W_{n}(D)|D|^{-1}\int_{D}|f|^{2}\,dx,\quad D\subset \mathbb{T}^{n}. \] N. Wiener proved that…

We prove that for certain positive operators $T$, such as the Hardy-Littlewood maximal function and fractional integrals, there is a constant $D>1$, depending only on the dimension $n$, such that the two weight norm inequality…

We show that the best constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal function associated to some finite radial measures, such as the standard gaussian measure, grow exponentially…

Log-Sobolev inequalities (LSIs) upper-bound entropy via a multiple of the Dirichlet form (i.e. norm of a gradient). In this paper we prove a family of entropy-energy inequalities for the binary hypercube which provide a non-linear…