Markov Inequality Research Articles

Markov inequality on sets with polynomial parametrization

The main result of this paper is the following: if a compact subset E of $ℝ^n$ is UPC in the direction of a vector $v ∈ S^{n-1}$ then E has the Markov property in the direction of v. We present a method which permits us to generalize as well as to improve

Elementary Proof of the Remez Inequality

The famous Russian mathematician Pafnutii Lvovich Tchebycheff (1821-1894) introduced T1(x) as the polynomial of least uniform norm on [-1, 1] amid the polynomials of degree n with fixed leading coefficient. The Tchebycheff polynomials appear prominently in various extremal problems posed in irn (the set of all polynomials of degree n). An illuminating example is the classical Markov inequality, Which shows that

Markov's property of the Cantor ternary set

We prove that the Cantor ternary set E satisfies the classical Markov inequality (see [Ma]): for each polynomial p of degree at most n (n = 0, 1, 2,...) (M) $|p'(x)| ≤ Mn^{m} sup_{E}|p|$ for x ∈ E, where M and m are positive constants depending only on E.

Analogs of the Markov and Bernstein inequalities for polynomials in Banach spaces

Analogs of the Markov and Bernstein inequalities for polynomials in Banach spaces

Multivariate Bernstein and Markov inequalities

For a polynomial P n of total degree n and a bounded convex set S it will be shown that for 0 < p ⩽ ∞ ‖ ∂ ∂ξ P n‖ L p(s) ⩽ Cn 2∦P n∦L p(s) with C independent of n and of P n ϵ Π n . The Bernstein inequality ‖√1−x 2 d dx P n(x)‖L p[−1,1] ⩽ Cn ∦P n∦L p[−1,1] will also be generalized and that generalization will be the crucial result. Theorems for higher and mixed derivatives will be achieved.

Counter-examples to Markov and Bernstein inequalities

We show that for any function ϑ: N → R + one can find a Cantor set C and a trigonometric polynomial T of order d( T) such that the generalized Bernstein inequality ∥ T′∥ c ⩽ ϑ( d( T)) ∥ T∥ c does not hold. Furthermore, if ϑ( n) ⩽ Mn ϱ (for some M and ϱ > 1), the set C can be chosen to be regular with respect to the Green's function of C \\ C with pole at ∞. Analogous results are established for algebraic polynomials and Markov's inequality.

Dimension and geometry of sets defined by polynomial inequalities

Markov's inequality for the maxima of the derivatives of polynomials over cubes is replaced by an inequality where the cubes are changed to certain cubes intersected by a given subset F of R n . This new inequality is true for certain sets F and false for others. We are interested in the sets F for which this inequality is true and we prove that these sets must have positive Hausdorff dimension. Our inequality is not true if F is the closure of a domain with an outgoing cusp. We introduce a generalized inequality which holds for these sets and prove that this new inequality allows sets F with Hausdorff dimension zero.

Weighted Markov and Bernstein type inequalities for generalized non-negative polynomials

Weighted Markov and Bernstein type inequalities are established for generalized non-negative polynomials and generalized polynomial weight functions. The novelty of the results lies in the fact that in these estimates only the generalized degrees of the generalized polynomial and the generalized polynomial weight function, respectively, and a multiplicative absolute constant show up. To prove such inequalities was motivated by studying systems of orthogonal polynomials simultaneously, associated with generalized Jacobi weight functions.

Some remarks on Bernstein's theorems

On compact sets preserving Markov's inequality, Bernstein-type conditions for a continuous function to be of class C k are discussed. Also, relationships between the distribution of zeros of polynomials of best uniform or L p approximation to a given function and its differential properties are established.

Estimates for the Lorentz degree of polynomials

In 1916 S. N. Bernstein observed that every polynomial p having no zeros in (−1, 1) can be written in the form ∑ j = 0 d a j (1 − x) j (1 + x) d − j with all a j ⩾ 0 or all a j ⩽ 0. The smallest natural number d for which such a representation holds is called the Lorentz degree of p and it is denoted by d( p). The Lorentz degree d( p) can be much larger than the ordinary degree deg( p). In this paper lower bounds are given for the Lorentz degree of certain special polynomials which show the sharpness of some earlier results of T. Erdélyi and J. Szabados. As a by-product, we give a short proof of Markov and Bernstein type inequalities for the derivatives of polynomials of the above form, established by G. G. Lorentz in 1963. The Lorentz degree of trigonometric polynomials is also introduced and some analogues of our algebraic results are established.

Bounds on the derivatives of polynomials on Banach spaces

AbstractWe generalize the classical Bernstein's and Markov's Inequalities for polynomials on any real Banach space. We also give estimates for the derivatives of homogeneous polynomials on real Banach spaces.

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

A Markov-type inequality for the derivatives of constrained polynomials

Markov's inequality asserts that max −1⩽x⩽1 |p′(x)|⩽n 2 max −1⩽x⩽1 |p(x)| (1) for every polynomial of degree at most n. The magnitude of sup p∈S max −1⩽x⩽1|p′(x)| max −1⩽x⩽1|p(x)| (2) was examined by several authors for certain subclasses S of Π n . In this paper we introduce S = S n m ( r) (0 ⩽ m ⩽ n, 0 < r ⩽ 1), the set of those polynomials from Π n which have all but at most m zeros outside the circle with center 0 and radius r, and establish the exact order of the above expression up to a multiplicative constant depending only on m.

On the Markov inequality in L P-spaces

This paper gives new admissible values for the constant in Markov inequality in the p-metric. We improve a classical theorem of Hille, Szegö, and Tamarkin. For p = 2, sharp numerical values are obtained.

Markov's inequality and the existence of an extension operator for C∞ functions

It is shown that if E is a C ∞ determining compact set in R n , then Markov's inequality for derivatives of polynomials holds on E iff there exists a continuous linear extension operator L: C ∞( E)→ C ∞( R n). Other equivalent statements (e.g., Bernstein's approximation theorem for C ∞ functions, topological linear embedding of C ∞( E) into the space of rapidly decreasing sequences of real numbers) are also given. As an application, we prove that each of those properties (of the set E) is invariant under regular analytic mappings.

Markov's inequality for typically real polynomials

On etablit une inegalite du type Markov pour la classe des polynomes complexes qui sont typiquement reels sur le disque unite ouvert. Ces polynomes sont aussi croissants sur l'intervalle unite. Pour des points particuliers de x, on trouve la valeur exacte de Sup(|p'(x)|/∥p∥) pour tous les polynomes de cette classe, ∥• etant la norme uniforme

L∞ Markov and Bernstein inequalities for Erdös weights

Recently, weighted Markov and Bernstein inequalities have been established for large classes of Freud weights, that is, weights of the form W( x) ≔ e − Q( x) , where Q( x) is even and of smooth polynomial growth at infinity. In this paper, we consider Erdős weights, which have the form W( x) ≔ e − Q( x ), where Q( x) is even and of faster than polynomial growth at infinity. For a large class of Erdős weights, we establish the Markov type inequality ‖P ′‖ R⩽CD ′(a n)‖PW‖ R , (1) for n ⩾ 1 and P any polynomial of degree at most n. Here the norm is the sup norm, and C is independent of n and P, while a n is the Mhaskar-Rahmanov-Saff number, that is, it is the positive root of the equation n = 2 n ∝ 0 1 a ntQ′(a nt)dt √1 − t 2 For example, we consider Q( x) ≔exp k (|| α ), where α > 0, and where exp k denotes the kth iterated exponential, and give a more explicit formulation of (1). We also establish Bernstein type inequalities that for part of the range (− ∞, ∞) improve on (1).

Markov inequalities for polynomials on triangles

Markov inequalities for polynomials on triangles

On the Complexity of Computing the Volume of a Polyhedron

We show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix.

Extremal polynomials for weighted Markov inequalities

Extremal polynomials for weighted Markov inequalities