Related papers: Bernstein- and Markov-type inequalities

We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. We build the change-of-basis matrices between the generalized Tschebyscheff of the second kind polynomial basis…

This paper is devoted to Markov's extremal problems of the form $M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\le k\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree at most $n$ and $X$ is a…

A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply…

The aim of this paper is to investigate generating functions for modification of the Milne-Thomson's polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. By applying the Umbral algebra to these generating…

We introduce a series of numbers which serve as a generalization of Bernoulli, Euler numbers and binomial coefficients. Their properties are applied to solve a probability problem and suggest a statistical test for independence and…

We give the sufficient condition on coefficients $a_k$ of an algebraic polynomial $P(z)=\sum_{k=0}^{n}a_kz^k$, $a_n\not=0,$ for the pointwise Bernstein inequality $|P'(z)|\le n|P(z)|$ to be true for all $z\in\overline{\mathbb…

In this paper, we investigate the properties of q-Hermite polynomials related to q-Bernstein polynomials. From these properties, we derive some interesting relations between q-Berstein polynomials and q-Hermite polynomials.

Multipartite Bell-type inequalities are derived for general systems. They involve up to eight observables with arbitrary spectra on each site. These inequalities are closely related to the algebras of quaternions and octonions.

Recently Rather et al. \cite{NT} considered the generalized derivative and the generalized polar derivative and studied the relative position of zeros of generalized derivative and generalized polar derivative with respect to the zeros of…

We present a short proof of a conjecture proposed by I. Ra\c{s}a (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Ra\c{s}a (2017). The methods of our proof…

We collect here elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange \& Hermite…

We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities…

In this paper, we consider higher-order Bernoulli and poly-Bernoulli mixed type polynomials and we give some interesting identities of those polynomials arising from umbral calculus.

Bernstein's inequality is a central result in the theory of $D$-modules on smooth varieties. While Bernstein's inequality fails for rings of differential operators on general singularities, recent work of \`{A}lvarez Montaner, Hern\'andez,…

In this paper, we consider Barnes' multiple Bernoulli and poly-Bernoulli mixed-type polynomials. From the properties of Sheffer sequences of these polynomials arising from umbrral calculus, we derive new and interesting identities.

In this paper, we consider the degenerate poly-Bernoulli polynomials and present new and explicit formulas for computing them in terms of the degenerate Bernoulli polynomials and Stirling numbers of the second kind.

We establish an inequality of different metrics for algebraic polynomials.

Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. Both old and new results on these invariants are collected.

We derive the non-asymptotical non-uniform sharp error estimation for Bernstein's approximation of continuous function based on the modern probabilistic apparatus. We investigate also the convergence of derivative of these polynomials and…

We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…