Integral approximation of the characteristic function of a spherical cap by algebraic polynomials

A problem of optimal boundary control of thermal sources for a stationary model of natural thermal convection of a high-viscosity inhomogeneous incompressible fluid in the Boussinesq approximation is investigated. Conditions for the solvability of the problem, as well as necessary and sufficient optimality conditions, are specified. Optimality conditions and the corresponding adjoint problems defining the gradient of the quality functional are written for several special cases of the functional. Computational procedures for finding an optimal control based on gradient methods are described. The results of numerical experiments are given.     

On Bellman function for extremal problems in BMO

In this Note we describe our results on construction of the Bellman function solving an extremal problem for a large class of integral functionals on BMO.     

On the approximation order of extremal point methods for hyperbolic minimal energy problems

Summary. Let be an analytic Jordan curve in the unit disk We regard the hyperbolic minimal energy problem where () denotes the set of all probability measures on . There exist several extremal point discretizations of *, among others introduced by M. Tsuji (Tsuji points) or by K. Menke (hyperbolic Menke points). In the present article, it is proven that hyperbolic Menke points approach the images of roots of unity under a conformal map from onto geometrically fast if the number of points tends to infinity. This establishes a conjecture of K. Menke. In particular, explicit bounds for the approximation error are given. Finally, an effective method for the numerical determination of * providing a geometrically shrinking error bound is presented.Mathematics Subject Classification (1991): 30C85, 30E10, 31C20The notation Menke points has been introduced by D. Gaier.     

On some extremal problems of approximation theory in the complex plane

In the Hardy Banach spaces H _q , Bergman Banach spaces H_q, and Banach spaces (p, q, ), we determine the exact values of the Kolmogorov, Bernstein, Gelfand, linear, and trigonometric n-widths of classes of functions analytic in the disk |z| < 1 and such that the averaged moduli of continuity of their r-derivatives are majorized by a certain function. For these classes, we also consider the problems of optimal recovery and coding of functions.__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1155–1171, September, 2004.     

Some extremal problems of theory of quadrature and approximation of functions

This is the author-review of the dissertation presented for the degree of the Doctor of Physicomathematical Sciences. The dissertation was defended on March 13, 1975 in a meeting of the scientists of the V. A. Steklov Mathematics Institute of the Academy of Sciences of the USSR. The official opponents were Doctor of Physicomathematical Sciences Prof. N. S. Bakhvalov, Doctor of Physicomathematical Sciences Prof. P. K.Suetin, and Doctor of Physicomathematical Sciences Prof. S. B. Stechkin.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 299–311, February, 1976.     

Massiveness of the sets of extremal functions in some problems in approximation theory

It is proved that the sets of extremal functions are massive in some problems in approximation theory.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1356–1361, October, 1993.     

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.     

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

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

On a class of extremal problems

In the paper we introduce a class of trigonometrical polynomial extremal problems depending on a continuous parameter 0≤r≤1. It turns out that the two border cases r=0 and r=1 are known problems investigated earlier by Kamae, Mendes-France, Ruzsa and the present author. We also introduce another set of extremal problems for measures with similar parametrization, and prove a duality relationship between the two type of extremal quantities. The proof relies on a minimax theorem proved earlier by the author. The known duality results are proved as corollaries. 1980 MS Classification. Primary 42A05; Secondary 46B25, 46N05.     

Development of studies on the exact solution of extremal problems of the theory of best approximation

We give a survey of studies on the exact solution of problems of best approximation of function classes by concrete approximating sets, the calculation of widths, and also some similar problems. We pay primary attention to the fundamental results of N. P. Korneichuk with whose name the development of these directions of the theory of approximations is closely associated. The influence of his ideas and the powerful methods he created on the studies of other authors is traced.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 4–17, January, 1990.     

Uniqueness of the extremal function for thep-capacity of a condenser

The uniqueness of the extremal function for the p-capacity of a condenser is established. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 228–234.     

The strong approximation of extremal processes

Summary If X ₁, X ₂, ..., are i.i.d. random variables and Y _n =Max(X ₁, ..., X _n ); if for some sequences A _n , B_n, n=1, 2, ..., E _n (t)=A_nY_[nt]+B_n is such that E _n (1) weakly converges to a non degenerate limit distribution, then we prove that it is possible to construct a sequence of replicates of extremal processes E ⁽ⁿ⁾(t) on the same probability space, such that d(E _n (.), E ⁽ⁿ⁾(.))0 a.s., with the Levy metric. We give the rates of consistency of the approximations.     

An extremal problem for the derivative of a rational function

Erd?s’ well-known problem on the maximum absolute value of the derivative of a polynomial on a connected lemniscate is extended to the case of a rational function. Moreover, under the assumption that certain lemniscates are connected, a sharp upper bound for the absolute value of the derivative of a rational function at any point in the plane different from the poles is found. The role of the extremal function is played by an appropriate Zolotarev fraction.     

Some extremal problems for circular polygons

The main result of this paper is the solution of the following problem posed by J. Hersch: find the maximal conformal radius on the family of all hyperbolic polygons with n sides (n ≥ 3). It is proved that the maximum is attained on a regular polygon. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 127–136. Translated by A. Yu. Solynin.     

Some extremal problems for trigonometric polynomials

Three extremal problems for trigonometric polynomials are studied in this paper. The first was initiated by Maiorov. It relates to the trigonometric polynomials with n nonzero harmonics. The L_p-norm of the Weyl derivative is compared with the L_q-norm of the polynomial. The other two problems have appeared in the recent paper by Oswald. They deal with the polynomials of degree n. The minimum of L_p-norm with respect to the choice of phases is compared with l_q-norm of its coefficients.