Direct and inverse theorems of approximation theory

Approximation in Smirnov spaces: Direct and inverse theorems

We give direct and inverse approximation theorems for Dirichlet series in Smirnov spaces over convex polygons. We estimate the degree of convergence and the regularity of the functions with moduli of arbitrary order k. Moreover, we consider the influence of differentiability conditions on the rate of approximation and vice versa. This work extends results by Yu. I. Mel’nik and gives an example ...

Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method

For an arbitrary self-adjoint operator B in a Hilbert space H, we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector x ∈ H with respect to the operator B, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator B, and the k-modulus of continuity of the vector x with respect to the op...

Inverse Approximation Theorems for . . .

Direct and Inverse Approximation Theorems forthe p - Version of the Finite Element

This is the third of a series devoted to the direct and inverse approximation theorems of the p-version of the nite element method in the framework of the weighted Besov spaces. In this paper we will prove the inverse theorems in terms of the weighted Besov spaces based upon the convergence rate of the the p-version nite element solutions. Key words. Weighted Besov spaces, modi ed weighted Beso...

Negative Theorems in Approximation Theory

1. INTRODUCTION. Negative theorems have a rich tradition in mathematics. In fact mathematics seems to be unique among the sciences in that negative results are very much a part of the mathematical edifice. Most mathematical theories try to explain what is possible and also what is not. To understand the structure of a mathematical theory is also to understand its limitations. There are many dif...