Smooth equilibrium measures and approximation

A necessary and sufficient condition is given for approximation with weighted expressions of the form wnPn, where w is a given continuous weight function and Pn are polynomials of degree . The condition is that the extremal measure that solves an associated equilibrium problem is smooth (asymptotically optimal doubling). As corollaries we get all previous (positive and negative) results for approximation, as well as the solution of a problem of T. Bloom and M. Branker. A connection to level curves of homogeneous polynomials of two variables is also explored.     

Weighted Polynomial Approximation for Convex External Fields

It is proven that if Q is convex and w(x)= exp(-Q(x)) is the corresponding weight, then every continuous function that vanishes outside the support of the extremal measure associated with w can be uniformly approximated by weighted polynomials of the form w ⁿ P _n . This solves a problem of P. Borwein and E. B. Saff. Actually, a similar result is true locally for any parts of the extremal support where Q is convex. February 10, 1998. Date revised: July 23, 1998. Date accepted: August 17, 1998.     

Uniform approximation by Baskakov and Meyer-König and Zeller operators

Necessary and sufficient conditions are given for a function to have uniform approximation or a Lipschitz order of approximation by Baskakov and Meyer-König and Zeller operators.     

Pointwise convergence of multiple Fourier series using different convergence notions

, . , L ^p (²) p>1 . , C(T³), - , . - .

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This work was completed while the first named author was a visiting professor at Indiana University, Bloomington, Indiana; and the second named author was a visiting professor at Ohio State University, Columbus, Ohio, U.S.A.     

The distribution of extreme points in best complex polynomial approximation

LetK be a compact point set in the complex plane having positive logarithmic capacity and connected complement. For anyf continuous onK and analytic in the interior ofK we investigate the distribution of the extreme points for the error in best uniform approximation tof onK by polynomials. More precisely, if $$A_n (f): = \{ z \in K:|f(z) - p_n^* (f;z)| = \parallel f - p_n^* (f)\parallel _K \} ,$$ wherep _n ^* (f) is the polynomial of degree≤n of best uniform approximation tof onK, we show that there is a subsequencen _k with the property that the sequence of (n _k +2)-point Fekete subsets of \(A_{n_k }\) has limiting distribution (ask→∞) equal to the equilibrium distribution forK. Analogues for weighted approximation are also given.     

Trigonometric series with a generalized monotonicity condition

In this paper,we consider numerical and trigonometric series with a very general monotonicity condition.First,a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting.In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.