Reproducing Kernels and Extremal Functions in Dirichlet-Type Spaces

Some identities are established for reproducing kernels and extremal functions in Dirichlet-type spaces. In certain cases, these identities are characteristic. A new proof of Aleman's theorem on extremal functions as contractive multipliers is obtained. In contrast to the original one, this proof can be extended to vector-valued spaces. Bibliography: 11 titles.     

Pseudo-uniform Convexity in Hp and Some Extremal Problems on Sobolev Spaces

We extend Newman and Keldysh theorems to the behavior of sequences of functions in H^p (μ) which explain geometric properties of discs in these spaces. Through Keldysh's theorem we obtain asymptotic results for extremal polynomials in Sobolev spaces.     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

Generalized Interpolation in Bergman Spaces and Extremal Functions

In a recent paper A. Schuster and K. Seip [SchS] have characterized interpolating sequences for Bergman spaces in terms of extremal functions (or canonical divisors). As these are natural analogues in Bergman spaces of Blaschke products, this yields a Carleson type condition for interpolation. We intend to generalize this idea to generalized free interpolation in weighted Bergman spaces B^{p, α} as was done by V. Vasyunin [Va1] and N. Nikolski [Ni1] (cf.also [Ha2]) in the case of Hardy spaces. In particular we get a strong necessary condition for free interpolation in B^{p, α} on zero–sets of B^{p, α}–functions that in the special case of finite unions of B^{p, α}–interpolating sequences turns out to be also sufficient.     

Wiener criterion on metric spaces: Boundary regularity in axiomatic and Poincaré-Sobolev spaces

We obtain a Wiener-type criterion for the Hölder continuity of extremal functions on general metric spaces in an abstract setting. Then we use this result to establish the boundary regularity of quasi-minimizers of the p-energy integral in the axiomatic framework of Gol’dshtein-Troyanov and also for extremal functions from the class of Poincaré-Sobolev functions.     

Orthogonal polynomials and quadratic extremal problems

The purpose of this paper is to analyse a class of quadratic extremal problems defined on various Hilbert spaces of analytic functions, thereby generalizing an extremal problem on the Dirichlet space which was solved by S.D. Fisher. Each extremal problem considered here is shown to be connected with a system of orthogonal polynomials. The orthogonal polynomials then determine properties of the extremal function, and provide information about the existence of extremals.

     

A rigidity theorem in Alexandrov spaces with lower curvature bound

Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang–Schroeder–Sturm. The purpose of this paper is to study the extremal cases of these inequalities and to prove rigidity results. The spaces which we shall deal with here are Alexandrov spaces which possibly have infinite dimension and are not supposed to be locally compact.     

Extremal functions as divisors for kernels of Toeplitz operators

In this paper, an extremal function of a Banach space of analytic functions in the unit disk (not all functions vanishing at 0) is a function solving the extremal problem for functions f of norm 1. We study extremal functions of kernels of Toeplitz operators on Hardy spaces H^p, 1<p<∞. Such kernels are special cases of so-called nearly invariant subspaces with respect to the backward shift, for which Hitt proved that when p=2, extremal functions act as isometric divisors. We show that the extremal function is still a contractive divisor when p<2 and an expansive divisor when p>2 (modulo p-dependent multiplicative constants). We give examples showing that the extremal function may fail to be a contractive divisor when p>2 and also fail to be an expansive divisor when p<2. We discuss to what extent these results characterize the Toeplitz operators via invariant subspaces for the backward shift.     

BEST APPROXIMATION FOR WEIERSTRASS TRANSFORM CONNECTED WITH SPHERICAL MEAN OPERATOR

Using reproducing kernels for Hilbert spaces,we give best approximation for Weierstrass transform associated with spherical mean operator.Also,estimates of extremal functions are checked.     

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.     

Best Approximation and Cyclic Variation Diminishing Kernels

We study best uniform approximation of periodic functions from

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where the kernelK(x, y) is strictly cyclic variation diminishing, and related problems including periodic generalized perfect splines. For various approximation problems of this type, we show the uniqueness of the best approximation and characterize the best approximation by extremal properties of the error function. The results are proved by using a characterization of best approximants from quasi-Chebyshev spaces and certain perturbation results.     

On some properties of bitopological QHC spaces

Some important questions connected with bitopological QHC spaces are investigated. New conditions are found, under which such spaces are compact with respect to one component of the topology. It is shown that a pairwise extremal disconnected bitopological QHC space is S-closed in the sense of [4]. Theorems on the second category of a base set and on the almost Baire property of bitopological QHC spaces are proved. Also, several properties of QHC bitopological spaces are found under some known bitopological mappings. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 186–192, April–June, 2006.     

Asymptotically optimal approximation in fractional Sobolev spaces and the numerical solution of differential equations

We study two extremal problems in fractional Sobolev spaces which were motivated by the consideration of optimal finite difference methods for approximating the solution of a well posed Cauchy problem for linear partial differential equations. The limiting forms as the mesh increment tends to zero of the solutions of the extremal problem are shown to be generalized spline functions. Computations illustrating the effectiveness of these generalized spline functions are included.This research was partially supported by ONR Contract N 0014-69-C-0023.     

Exposed and denting points in duals of operator spaces

We study the extremal structure of the dual unit balls of various operator spaces. Mainly, we show that the classes of [w*-] strongly exposed, [w*-] exposed, and denting points in the dual unit balls of spaces of compact operators between Banach spacesX andY are completely — and in a canonical way — determined by the corresponding classes of points in the unit balls of the (bi-)duals of the factor spacesX andY. Applications to the duality of operator spaces and differentiability properties of the norm in operator spaces are given.     

Extremal functions of cubature formulas on a multidimensional sphere and spherical splines

We establish the general form of extremal cubature formulas on multidimensional spheres. The domains of definition for the cubature formulas under consideration are Sobolev-type spaces on the sphere. The smoothness of the class function under study may be fractional. We prove that, for a given set of nodes, there exists a one-to-one correspondence between the set of extremal functions of cubature formulas on the sphere and the set of natural spherical splines with zero spherical mean.     

A Class of Extremal Functions and Trigonometric Polynomials

In this paper we present two classes of extremal approximating functions. These functions have the property that they are entire, have finite exponential type, and provide excellent approximations along the real line for a specific set of functions. One class of functions provides majorants and minorants, while the other class minimizes theL¹-norm on the real line. As applications we construct extremal trigonometric polynomials and obtain an inequality involving almost periodic trigonometric polynomials.     

On Multivariate Adaptive Approximation

Recently, A. Cohen, R. A. DeVore, P. Petrushev, and H. Xu investigated nonlinear approximation in the space BV (R ² ). They modified the classical adaptive algorithm to solve related extremal problems. In this paper, we further study the modified adaptive approximation and obtain results on some extremal problems related to the spaces V _σ,p ^r (R ^d ) of functions of ``Bounded Variation" and Besov spaces B ^α (R ^d ). November 23, 1998. Date revised: June 25, 1999. Date accepted: September 13, 1999.     

On upper bounds for code distance and covering radius of designs in polynomial metric spaces

The purpose of this paper is to present new upper bounds for code distance and covering radius of designs in arbitrary polynomial metric spaces. These bounds and the necessary and sufficient conditions of their attainability were obtained as the solution of an extremal problem for systems of orthogonal polynomials. For antipodal spaces the behaviour of the bounds in different asymptotical processes is determined and it is proved that this bound is attained for all tight 2k-design.     

A sharp estimate for extremal functions

We prove a sharp pointwise estimate for extremal functions of invariant subspaces of some weighted Bergman spaces on the unit disk. The allowed weights include standard radial weights and logarithmically subharmonic weights.

     

An extension of the Uchiyama theorem and its application to composition operators which preserve BMO

Uchiyama constructed certain extremal BMO functions on Euclidean spaces which show that the John-Nirenberg estimate is in some sense remarkably precise. We extend his result to the case of general subdomains of Euclidean spaces. Also, as an application, we characterize Lebesgue measurable maps between general subdomains of Euclidean spaces which preserve BMO. To the memory of Professor Nobuyuki Suita