Related papers: Certain Bernstein-type $L_p$ inequalities for poly…

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…

Let $ P(z) $ be a polynomial of degree $ n $ having all zeros in $|z|\leq k$ where $k\leq 1,$ then it was proved by Dewan \textit{et al} that for every real or complex number $\alpha$ with $|\alpha|\geq k$ and each $r\geq 0$ $$…

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 some sufficient conditions implying $l^p$ inequalities of the form $||x||_p \leq ||y||_p$ for vectors $ x, y \in [0,\infty)^n$ and for $p$ in certain positive real intervals. Our sufficient conditions are strictly weaker than the…

Denoting by $E_{n}^{(+2)}(f)$ the best uniform approximation of $f$ by convex polynomials of degree $\le n$, there is an open question if there exists the limit $\lim_{n\to \infty}n^{\lambda}E_{n}^{(+2)}(|x|^{\lambda})$ for $\lambda \ge 1$.

Higher order Bernstein- and Markov-type inequalities are established for trigonometric polynomials on compact subsets of the real line and algebraic polynomials on compact subsets of the unit circle. In the case of Markov-type inequalities…

In this paper, we present certain new $L_p$ inequalities for $\mathcal B_{n}$-operators which include some known polynomial inequalities as special cases.

In this paper we obtain estimates for certain transcendence measures of an entire function $f$. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial $P(z,w)$ in ${\Bbb C}^2$ along the graph of $f$.…

We prove that, over a field of characteristic $0$, the degrees of factors of a binomial $t^n-\alpha$ are divisible by the least such degree. As a consequence, we deduce that for relatively prime natural numbers $m,n$, a polynomial has the…

In this article we consider a modification of the Stein's spherical maximal operator of complex order $\alpha$ on ${\mathbb R^n}$: $$ {\mathfrak M}^\alpha_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over \Gamma(\alpha) } \int_{|y|\leq…

In this paper we study the average $\NL_{2\alpha}$-norm over $T$-polynomials, where $\alpha$ is a positive integer. More precisely, we present an explicit formula for the average $\NL_{2\alpha}$-norm over all the polynomials of degree…

We shall study non-linear extremal problems in Bergman space $\mathcal{A}^2(\mathbb{D})$. We show the existence of the solution and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials,…

We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.

In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.

A recent conjecture by I. Ra\c{s}a asserts that the sum of the squared Bernstein basis polynomials is a convex function in $[0,1]$. This conjecture turns out to be equivalent to a certain upper pointwise estimate of the ratio…

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…

We consider almost-primes of the form $f(p)$ where $f$ is an irreducible polynomial over $\mathbb Z$ and $p$ runs over primes. We improve a result of Richert for polynomials of degree at least $3$. In particular we show that, when the…

Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\lambda_{0},...,\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex…

It is shown that the $L^\alpha$-norms polynomials Rudin conjecture fails. Our counterexample is inspired by Bourgain's work on NLS. Precisely, his study of the Strichartz's inequality of the $L^6$-norm of the periodic solutions given by the…

Recently, Nagy-To\'okos and Totik-Varga proved an asymptotically sharp $L^{p}$ Bernstein type inequality on union of finitely many intervals. We extend this inequality to the case when the power $p$ is between $0$ and $1$; such sharp…