Rational Approximation with Varying Weights, II

Rational approximation with varying weights in the complex plane

Given an open bounded set G in the complex plane and a weight function W (z) which is analytic and di erent from zero in G, we consider the problem of locally uniform rational approximation of any function f(z), which is analytic in G, by particular ray sequences of weighted rational functions of the form Wm+n(z)Rm;n(z), where Rm;n(z) = Pm(z)=Qn(z); with deg Pm m and degQn n: The main result of...

Rational Approximation, II

Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, II. General weights.∗

We study compact embeddings for weighted spaces of Besov and TriebelLizorkin type where the weights belong to Muckenhoupt Ap classes. We focus our attention on the influence of singular points of the weights on the compactness of the embeddings as well as on the asymptotic behaviour of their entropy and approximation numbers.

Constrained Approximation with Jacobi Weights

In this paper, we prove that for l = 1 or 2 the rate of best l-monotone polynomial approximation in the Lp norm (1 ≤ p ≤ ∞) weighted by the Jacobi weight wα ,β(x) ∶= (1 + x)α(1 − x)β with α, β > −1/p if p <∞, or α, β ≥ 0 if p =∞, is bounded by an appropriate (l + 1)-st modulus of smoothness with the same weight, and that this rate cannot be bounded by the (l+2)-ndmodulus. Related results on con...

Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

and the coefficients in (1.3), which depend on the parameter N , {an,N : bn,N}n=0, bn,N > 0. ∗The authors acknowledge support from NATO Collaborative linkage grant PST.CLG.979738 and INTAS Research Network 03-51-6637. The first author was supported by grants RFBR-05-01-00522, NSh1551.2003.1, Program No. 1 DMS RAS. The second author was supported by an NSF grant. The third author was supported b...