On the best linear approximation methods and the widths of certain classes of analytic functions

We discuss the best linear approximation methods in the Hardy spaceH _q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained. Translated fromMatermaticheskie Zametki, Vol. 65, No. 2, pp. 186–193, February, 1999.     

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

To solve extremal problems of approximation theory in the space L ₂, we use -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.     

Ismagilov Type Theorems for Linear, Gel′fand and Bernstein n-Widths

Using a variational principle for s-numbers, we obtain estimates for the linear, Gel′fand. and Bernstein n-widths. A simple proof of some results concerned with the exact values of n-widths of diagonal operators is given. We also calculate the exact values at the Bernstein n-widths for the Hardy-Sobolev classes.     

Approximation of smooth functions on compact two-point homogeneous spaces

Estimates of Kolmogorov n-widths and linear n-widths , (1q∞) of Sobolev's classes , (r>0, 1p∞) on compact two-point homogeneous spaces (CTPHS) are established. For part of (p,q)[1,∞]×[1,∞], sharp orders of or were obtained by Bordin et al. (J. Funct. Anal. 202(2) (2003) 307). In this paper, we obtain the sharp orders of and for all the remaining (p,q). Our proof is based on positive cubature formulas and Marcinkiewicz–Zygmund-type inequalities on CTPHS.     

On the best mean square approximations by entire functions of exponential type in L 2(ℝ) and mean ν-widths of some functional classes

We consider some extremal problems of approximation theory of functions on the whole real axis ? by entire functions of the exponential type. In particular, we find the exact values of the mean ν-widths of classes of functions, defined by the modules of continuity of the mth order ω _m and majorants ψ satisfying the special type of restriction.     

Exact Values of Widths of Classes of Analytic Functions on the Disk and Best Linear Approximation Methods

In the Hardy space H _p, (p1, 0< 1, H _p,1 H _p) we develop best linear approximation methods (previously studied by Taikov and Ainulloev) for the classes W(r,,) of analytic functions on the unit disk and calculate the exact values of linear, Gelfand, and informational n-widths of these classes.     

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. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 494–499, October, 1999.     

Continuous Algorithms in n-Term Approximation and Non-Linear Widths

In the present paper we investigate optimal continuous algorithms in n-term approximation based on various non-linear n-widths, and n-term approximation by the dictionary V formed from the integer translates of the mixed dyadic scales of the tensor product multivariate de la Vallée Poussin kernel, for the unit ball of Sobolev and Besov spaces of functions with common mixed smoothness. The asymptotic orders of these quantities are given. For each space the asymptotic orders of non-linear n-widths and n-term approximation coincide. Moreover, these asymptotic orders are achieved by a continuous algorithm of n-term approximation by V, which is explicitly constructed.     

Asymptotic Results and a Markovian Approximation for the M(n)/M(n)/s+GI System

In this paper for the M(n)/M(n)/s+GI system, i.e. for a s-server queueing system where the calls in the queue may leave the system due to impatience, we present new asymptotic results for the intensities of calls leaving the system due to impatience and a Markovian system approximation where these results are applied. Furthermore, we present a new proof for the formulae of the conditional density of the virtual waiting time distributions, recently given by Movaghar for the less general M(n)/M/s+GI system. Also we obtain new explicit expressions for refined virtual waiting time characteristics as a byproduct.     

On one extremal problem for positive series

The approximation properties of the spaces S _ϕ ^p introduced by Stepanets’ were studied in a series of works of Stepanets’ and his disciples. In these works, problems related to the determination of exact values of n-term approximations of q-ellipsoids in these spaces were reduced to some extremal problems for series with terms that are products of elements of two nonnegative sequences one of which is fixed and the other varies on a certain set. Since solutions of these extremal problems may be of independent interest, in the present work we develop a new method for finding these solutions that enables one to obtain the required result in a substantially shorter and more transparent way. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1677–1683, December, 2005.     

On Some Extremal Properties of Lagrange Interpolatory Polynomials

In this paper, we will show that Lagrange interpolatory polynomials are optimal for solving some approximation theory problems concerning the finding of linear widths.In particular, we will show that

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, where _n is a set of the linear operators with finite rank n+1 defined on −1,1], and where _n+1 denotes the set of polynomials p=∑_i=0ⁿ⁺¹a_ixⁱ of degreen+1 such that a_n+11. The infimum is achieved for Lagrange interpolatory polynomial for nodes .     

Construction of the extremal function for a functional on the classH ω (n)

For a functional on the classH _ω ⁽ⁿ⁾ ,n≥3, we construct the extremal function on which the upper bound obtained by A. I. Stepanets is attained. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 519–529, April, 1997. Translated by N. K. Kulman     

An extrapolation theorem for nonlinear approximation and its applications

We prove an extrapolation theorem for the nonlinear m-term approximation with respect to a system of functions satisfying very mild conditions. This theorem allows us to prove endpoint L^p-L^q estimates in nonlinear approximation. As a consequence, some known endpoint estimates can be deduced directly and some new estimates are also obtained. Finally, applications of these new estimates are given to spherical m-widths and m-term approximation of the weighted Besov classes.     

Minimal Non—Commutative n—Insertive Rings

Let n be an integer with |n| > 1. If p is the smallest prime factor of |n|, we prove that a minimal non-commutative n-insertive ring contains n ⁴ elements and these rings have five (2p+4) isomorphic classes for p = 2 (p ≠ 2). This research is supported by the National Natural Science Foundation of China, and the Scientific Research Foundation for “Bai-Qian-Wan” Project, Fujian Province of China     

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i _X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i _X −1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i _X ≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρ_X> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.     

On the complexity of some subgraph problems

We study the complexity of the problem of deciding the existence of a spanning subgraph of a given graph, and of that of finding a maximum (weight) such subgraph. We establish some general relations between these problems, and we use these relations to obtain new NP-completeness results for maximum (weight) spanning subgraph problems from analogous results for existence problems and from results in extremal graph theory. On the positive side, we provide a decomposition method for the maximum (weight) spanning chordal subgraph problem that can be used, e.g., to obtain a linear (or O(nlogn)) time algorithm for such problems in graphs with vertex degree bounded by 3.     

Optimal Subspaces for n -Widths of Bounded Subsets in Hilbert Spaces

In this article, we consider the problem of proving the optimality of several approximation spaces by means of n-widths. Specifically, they are optimal subspaces for approximating bounded subsets in some Hilbert spaces with mesh-dependent norms. We prove that finite element spaces and newly developed generalized L-spline spaces are optimal subspaces for n-widths.     

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.     

Approximations for the delay probability in the M/G/s queue

This paper develops approximations for the delay probability in an M/G/s queue. For M/G/s queues, it has been well known that the delay probability in the M/M/s queue, i.e., the Erlang delay formula, is usually a good approximation for other service-time distributions. By using an excellent approximation for the mean waiting time in the M/G/s queue, we provide more accurate approximations of the delay probability for small values of s. To test the quality of our approximations, we compare them with the exact value and the Erlang delay formula for some particular cases.     

On some extremal problems of approximation theory of functions on the real axis II

The work is devoted to the solution of a number of extremal problems of approximation theory of functions on the real axis $ \mathbb{R} $ . In the space L ₂( $ \mathbb{R} $ ), the exact constants in Jackson-type inequalities are calculated. The exact values of average ν-widths are obtained for the classes of functions from L ₂( $ \mathbb{R} $ ) that are defined by averaged k-order moduli of continuity and for the classes of functions defined by K-functionals. In the chronological order, the sufficiently complete analysis of the final results related to the solution of extremal problems of approximation theory in the periodic case and on the whole real axis is carried out.