Weighted Polynomial Inequalities Research Articles

Sharp Constants of Approximation Theory: VI. Weighted Polynomial Inequalities of Different Metrics on the Multidimensional Cube and Ball

We prove limit equalities between the sharp constants in weighted Nikolskii-type inequalities for multivariate polynomials on an m-dimensional cube and ball and the corresponding constants for entire functions of exponential type.

Weighted inequalities for generalized polynomials with doubling weights

Many weighted polynomial inequalities, such as the Bernstein, Marcinkiewicz, Schur, Remez, Nikolskii inequalities, with doubling weights were proved by Mastroianni and Totik for the case 1 leq p < infty, and by Tamás Erdélyi for 0< p leq1. In this paper we extend such polynomial inequalities to those for generalized trigonometric polynomials. We also prove the large sieve for generalized trigonometric polynomials with doubling weights.

Weighted polynomial inequalities in the complex plane

We establish weighted Lp,1≤p<∞ Bernstein-, Remez-, Nikolskii-, and Marcinkiewicz-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) arc in the complex plane.

Weighted Polynomial Inequalities with Doubling Weights on a Quasismooth Arc

We establish Lp,1≤p<∞ Markov–Bernstein-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) arc in the complex plane.

Markov type polynomial inequality for some generalized Hermite weight

ABSTRACT In this paper we study some weighted polynomial inequalities of Markov type in L2-norm. We use the properties of the system of generalized Hermite polynomials . The polynomials H(α)n (x) are orthogonal in ℝ = (−∞,∞) with respect to the weight function . The classical Hermite polynomials Hn(x) present the special case for α = 0.

A SURVEY OF POLYNOMIAL APPROXIMATION ON THE SPHERE

The main purpose of this paper is to survey some of the work on spherical approximation done by the BNU group under the direction of Professor Sun. The equiconvergent operators of Cesàro means, and their interesting applications are described. The Jackson inequality for spherical polynomials and some moduli of smoothness on the sphere are investigated. The equivalence between moduli of smoothness and K-functionals is also discussed. We also describe several weighted polynomial inequalities on the sphere, including the Remez-type and the Nikolskii-type inequalities, the Marcinkiewicz–Zygmund inequality, the Bernstein-type and the Schur-type inequalities. Positive cubature formulas on the sphere, and their relation to the Marcinkiewicz–Zygmund inequality are also discussed. A survey on recent results on asymptotic orders of the n-widths of Sobolev's classes on the sphere is also given.

Jackson-Type Inequality for Doubling Weights on the Sphere

In the one-dimensional case, Jackson's inequality and its converse for weighted algebraic polynomial approximation, as well as many important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumption on the weights. In most cases this minimal assumption is the doubling condition. In this paper, we establish Jackson's theorem and its Stechkin-type converse for spherical polynomial approximation with respect to doubling weights on the unit sphere.

Multivariate polynomial inequalities with respect to doubling weights and [formula omitted] weights

In one-dimensional case, various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz–Zygmund, Nikolskii, Schur, Remez, etc., have been proved under the doubling condition or the slightly stronger A ∞ condition on the weights by Mastroianni and Totik in a recent paper [G. Mastroianni, V. Totik, Weighted polynomial inequalities with doubling and A ∞ weights, Constr. Approx. 16 (1) (2000) 37–71]. The main purpose of this paper is to prove multivariate analogues of these results. We establish analogous weighted polynomial inequalities on some multivariate domains, such as the unit sphere S d − 1 , the unit ball B d , and the general compact symmetric spaces of rank one. Moreover, positive cubature formulae based on function values at scattered sites are established with respect to the doubling weights on these multivariate domains. Some of these multi-dimensional results are new even in the unweighted case. Our proofs are based on the investigation of a new maximal function for spherical polynomials.

Markov—Bernstein-Type Inequality for Trigonometric Polynomials with Respect to Doubling Weights on [-ω,ω

Abstract. Various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most cases this minimal assumption is the doubling condition. Here, based on a recently proved Bernstein-type inequality by D. S. Lubinsky, we establish Markov—Bernstein-type inequalities for trigonometric polynomials with respect to doubling weights on [-ω,ω] . Namely, we show the theorem below.

Extremal problems, inequalities, and classical orthogonal polynomials

In this survey paper we give a short account on characterizations for very classical orthogonal polynomials via extremal problems and the corresponding inequalities. Besides the basic properties of the classical orthogonal polynomials, we consider polynomial inequalities of Landau and Kolmogoroff type, some weighted polynomial inequalities in L 2-norm of Markov–Bernstein type, as well as the corresponding connections with the classical orthogonal polynomials.

Weighted Polynomial Inequalities with Doubling and A ∞ Weights

We consider weighted inequalities such as Bernstein, Nikolskii, Remez, etc., inequalities under minimal assumptions on the weights. It turns out that in most cases this mimimal assumption is the doubling condition. Sometimes, however, as for the Remez and Nikolskii inequalities, one needs the slightly stronger A ∈ fty condition. We shall consider both the trigonometric and the algebraic cases.

Notes on Inequalities with Doubling Weights

Various important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condition. Sometimes, however, as in the weighted Nikolskii inequality, the slightly stronger A ∞condition is used. Throughout their paper the L p norm is studied under the assumption 1⩽ p<∞. In this note we show that their proofs can be modified so that many of their inequalities hold even if 0< p<1. The crucial tool is an estimate for quadrature sums for the pth power (0< p<∞ is arbitrary) of trigonometric polynomials established by Lubinsky, Máté, and Nevai. For technical reasons we discuss only the trigonometric cases.

Some weighted polynomial inequalities

The author considers a weight function of sub-exponential type with zeros and singularities in [ -1,1]. Some inequalities of Remez, Bernstein-Markov and Nikolsky type are proved. The results hold for polynomials and generalized polynomials. Moreover an upper and a lower bound for the Christoffel functions are derived.

Some Weighted Polynomial Inequalities in L2-Norm

In this paper we give a new characterization of the classical orthogonal polynomials (Hermite, generalized Laguerre, Jacobi) by extremal properties in some weighted polynomial inequalities in L 2-norm.

Weighted polynomial inequalities

For the weights exp (−|x|λ), 0<λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0<λ<1. The proof is based on the solution of the problem of how fast a polynomialPn can decrease on [−1,1] ifPn (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedLp spaces.

Some weighted polynomial inequalities

We shall use the following notation: x will always denote a variable defined on (-co, co), y will always denote a variable defined on (0, co), ]/f(x)]], will stand for the norm off in I,,(--co, co) and ]]f(y)]], for the norm of f in L,(O, co); for p > 0, W,(x) = (1 + x*)O’* exp(-x2/2) and V,(y) = (1 + y)“‘* exp(--y/2); n will denote a strictly positive integer, and q, an arbitrary polynomial of degree at most n; by c we shall denote positive numbers depending at most on /3, and by c(s) positive numbers depending at most on j3 and on the variables enclosed by the parentheses, but not necessarily the same positive number if they appear more than once in the same formula. This paper is a sequel to [ 11, and like it has been deeply influenced by the ideas of G. Freud. The first five theorems below present polynomial inequalities on (-co, co) involving the weight W,(x); the case p = 0 of these results was proved in [ 11. The remaining five theorems present polynomial inequalities on [0, co) involving the weight V,(y). The functions W,(x) were introduced by Freud in [2]. Note that if Qo(x) = ln[ WD(x)] and p > 16[exp(1/16) l] > 1.04, then Q0[(p/16)“‘] 1, W,(x) is neither very strongly regular nor superregular in the sense of Mhaskar [4. 51. Hence the theorems in this paper are not contained in, nor can be trivially inferred from, the results of these authors. We start with: