Related papers: On Bernstein's inequality for polynomials

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…

In this note we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $GL_2(\mathbb{Z}_p)$-type $V$ in $\pi$ and proof…

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…

Bernstein inequalities and inverse theorems are a recent development in the theory of radial basis function(RBF) approximation. The purpose of this paper is to extend what is known by deriving $L^p$ Bernstein inequalities for RBF networks…

Plunnecke's inequality is the standard tool to obtain estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that the proof is completed with no reference to…

For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with…

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…

We develop a new kind of nonnegativity certificate for univariate polynomials on an interval. In many applications, nonnegative Bernstein coefficients are often used as a simple way of certifying polynomial nonnegativity. Our proposed…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

We give, using higher Bernstein polynomials defined in our paper [2], a stronger version of our previous result in [1] whose converse is proved in [2] and we give some complements to the results in [2] which help to compute these higher…

We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that \[ \binom{a p}{b p} \equiv \binom{a}{b} \] modulo $p^3$ for primes $p\ge5$. This congruence subsumes and builds on earlier congruences by Babbage,…

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number…

If $P(z)=\sum_{j=0}^{n}a_jz^j$ is a polynomial of degree $n$ having no zero in $|z|<1,$ then it was recently proved that for every $p\in[0,+\infty]$ and $s=0,1,\ldots,n-1,$ \begin{align*} \left\|a_nz+\frac{a_s}{\binom{n}{s}}\right\|_{p}\leq…

To prove that a polynomial is nonnegative on R^n one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than…

A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\le t\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved…

We show that standard deviation $\s$ satisfies the Leibniz inequality $\s(fg) \leq \s(f)\|g\| + \|f\|\s(g)$ for bounded functions f, g on a probability space, where the norm is the supremum norm. A related inequality that we refer to as…

The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix…

We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$…

In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of…

The problem of minimizing a polynomial over the standard simplex is one of the basic NP-hard nonlinear optimization problems --- it contains the maximum clique problem in graphs as a special case. It is known that the problem allows a…