Asymptotic Relations for Sharp Constants of Approximation Theory

In this talk we discuss asymptotic relations between sharp constants of approximation theory
in a general setting. We first present a general model that includes a circle of problems of
finding sharp or asymptotically sharp constants in some areas of univariate and multivariate
approximation theory, such as inequalities for approximating elements, approximation of individual
elements, and approximation on classes of elements. Next we discuss sufficient conditions that
imply limit inequalities and equalities between various sharp constants. Finally, we present
applications of these results to sharp constants in Bernstein-V. A. Markov type inequalities of
different metrics for univariate and multivariate trigonometric and algebraic polynomials and
entire functions of exponential type.