Some Weighted Polynomial Inequalities in L2-Norm

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.

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Markov type polynomial inequality for some generalized Hermite weight

Tatra Mountains Mathematical Publications ◽

10.2478/v10127-011-0030-4 ◽

2011 ◽

Vol 49 (1) ◽

pp. 111-118

Author(s):

Branislav Ftorek ◽

Mariana Marˇcokov´A

Keyword(s):

Weight Function ◽

Hermite Polynomials ◽

Polynomial Inequality ◽

Markov Type ◽

Polynomial Inequalities ◽

Weighted Polynomial Inequalities ◽

Special Case

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.

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Some weighted polynomial inequalities

Journal of Computational and Applied Mathematics ◽

10.1016/0377-0427(95)00116-6 ◽

1995 ◽

Vol 65 (1-3) ◽

pp. 279-292 ◽

Cited By ~ 4

Author(s):

G. Mastroianni

Keyword(s):

Polynomial Inequalities ◽

Weighted Polynomial Inequalities

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Weighted polynomial inequalities

Constructive Approximation ◽

10.1007/bf01893420 ◽

1986 ◽

Vol 2 (1) ◽

pp. 113-127 ◽

Cited By ~ 33

Author(s):

Paul Nevai ◽

Vilmos Totik

Keyword(s):

Polynomial Inequalities ◽

Weighted Polynomial Inequalities

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Singularities

10.1093/oso/9780198805021.003.0012 ◽

2018 ◽

Author(s):

S. G. Rajeev

Keyword(s):

Stokes Equations ◽

Negative Answer ◽

Three Dimensions ◽

Picard Iteration ◽

Navier Stokes ◽

Scale Invariant ◽

Navier Stokes Equations ◽

L2 Norm ◽

Two Dimensional Flow ◽

Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

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