Exact Values Of N-widths Research Articles

On the Best Linear Approximation of Holomorphic Functions

Let Ω be an open subset of the complex plane ℂ and let E be a compact subset of Ω. The present survey is concerned with linear n-widths for the class H ∞ (Ω) in the space C(E) and some problems on the best linear approximation of classes of Hardy–Sobolev-type in L p -spaces. It is known that the partial sums of the Faber series give the classical method for approximation of functions f ∈ H ∞ (Ω) in the metric of C(E) when E is a bounded continuum with simply connected complement and Ω is a canonical neighborhood of E. Generalizations of the Faber series are defined for the case where Ω is a multiply connected domain or a disjoint union of several such domains, while E can be split into a finite number of continua. The exact values of n-widths and asymptotic formulas for the e-entropy of classes of holomorphic functions with bounded fractional derivatives in domains of tube type are presented. Also, some results about Faber’s approximations in connection with their applications in numerical analysis are mentioned.

Exact Jackson–Stechkin-type inequalities for 2π-periodic functions in L 2 and widths of some classes of functions

We consider the problem of finding exact inequalities for the best approximations of periodic differentiable functions by trigonometric polynomials and the m-order moduli of continuity in the space L 2 and present their applications. For some classes of functions defined by the indicated moduli of continuity, we calculate the exact values of n-widths in the space L 2.

On the Best Polynomial Approximations of 2π-Periodic Functions and Exact Values of n-Widths of Functional Classes in the Space L2

To solve extremal problems of approximation theory in the space L2, we use τ-moduli introduced by Ivanov. We determine the exact values of constants in Jackson-type inequalities and the exact values of n-widths of functional classes determined by these moduli.

K-functionals and exact values ofn-widths for several classes fromL 2

We obtain exact values of differentn-widths for certain classes of 2π-periodic functions for which theK-functionals of theirrth-order derivatives are majorized by functions satisfying specified restrictions. We also consider some examples of majorants satisfying all the restrictions imposed in the present paper.