Weighted Polynomial Approximation for Weights with Slowly Varying Extremal Density

A Survey of Weighted Polynomial Approximation with Exponential Weights

Let W : R → (0, 1] be continuous. Bernstein’s approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm f → ‖fW‖L∞(R). The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950’s. Quantitative forms of the problem were actively investigated starting from the 1960’s. We survey old and recent aspects of this t...

Weighted Extremal Domains and Best Rational Approximation

Let f be holomorphically continuable over the complex plane except for finitely many branch points contained in the unit disk. We prove that best rational approximants to f of degree n, in the L2-sense on the unit circle, have poles that asymptotically distribute according to the equilibrium measure on the compact set outside of which f is single-valued and which has minimal Green capacity in t...

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...

Extremal graphs for weights

Strong limit theorems for random sets and fuzzy random sets with slowly varying weights

Theories of random sets and fuzzy random sets are useful concepts which are frequently applied in scientific areas including information science, probability and statistics. In this paper strong limit theorems are derived for random sets and fuzzy random sets with slowly varying weights in separable Banach spaces. Both independent and dependent cases are covered to provide a wide range of appli...