Trigonometric polynomial approximation,K-functionals and generalized moduli of smoothness

2017 ◽  
Vol 208 (2) ◽  
pp. 237-254 ◽  
Author(s):  
K V Runovskii
Keyword(s):  
Trigonometric Polynomial ◽  
Polynomial Approximation ◽  
Moduli Of Smoothness ◽  
Trigonometric Polynomial Approximation

scholarly journals Polynomial approximation on spheres - generalizing de la Vallée-Poussin

2011 ◽  
Vol 11 (4) ◽  
pp. 540-552 ◽  
Author(s):  
Ian H. Sloan
Keyword(s):  
Trigonometric Polynomial ◽  
Polynomial Approximation ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
Fast Convergence ◽  
Piecewise Polynomial ◽  
Smooth Functions ◽  
Higher Dimensional ◽  
Trigonometric Polynomial Approximation ◽  
De La Vallée Poussin

AbstractFor trigonometric polynomial approximation on a circle, the century-old de la Vallée-Poussin construction has attractive features: it exhibits uniform convergence for all continuous functions as the degree of the trigonometric polynomial goes to infinity, yet it also has arbitrarily fast convergence for sufficiently smooth functions. This paper presents an explicit generalization of the de la Vallée-Poussin construction to higher dimensional spheres S^d ≤ R^{d+1}. The generalization replaces the C^∞ filter introduced by Rustamov by a piecewise polynomial of minimal degree. For the case of the circle the filter is piecewise linear, and recovers the de la Vallée-Poussin construction, while for the general sphere S^d the filter is a piecewise polynomial of degree d and smoothness C^{d−1}. In all cases the approximation converges uniformly for all continuous functions, and has arbitrarily fast convergence for smooth functions.

A SURVEY OF POLYNOMIAL APPROXIMATION ON THE SPHERE

2009 ◽  
Vol 07 (06) ◽  
pp. 749-771 ◽  
Author(s):  
FENG DAI ◽  
KUNYANG WANG
Keyword(s):  
Polynomial Approximation ◽  
Moduli Of Smoothness ◽  
Spherical Approximation ◽  
Jackson Inequality ◽  
Cesàro Means ◽  
Bernstein Type ◽  
Cubature Formulas ◽  
Polynomial Inequalities ◽  
Spherical Polynomials ◽  
Weighted Polynomial Inequalities

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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