Sharp estimates for the convergence rate of double Fourier series in terms of orthogonal polynomials in the space L2((a, b) × (c, d); p(x)q(y))

Sharp estimates are obtained for the convergence rate of double Fourier series in terms of general orthogonal polynomials in some classes of functions and for the Kolmogorov N-widths of these classes. These results find applications in numerical analysis. Original Russian Text ? V.A. Abilov, M.K. Kerimov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 8, pp. 1364–1368.     

Certain Classes of Analytic Functions Related to the Crescent-Shaped Regions

In this paper, we study certain classes of analytic functions which satisfy a subordination condition and are associated with the crescent-shaped regions. We first give certain integral representations for the functions belonging to these classes and also present a relevant example. Making use of some known lemmas, we derive sufficient conditions for the functions to be in these classes. Some results on coefficient estimates are also obtained.     

MEROMORPHIC FUNCTIONS WITH BOUNDED BOUNDARY ROTATION

In this article, we generalize the class of meromorphic functions with bounded boundary rotation and related classes. Characterizations and some properties of these classes of functions are given.     

Generalized Affine Functions and Generalized Differentials

We study some classes of generalized affine functions, using a generalized differential. We study some properties and characterizations of these classes and we devise some characterizations of solution sets of optimization problems involving such functions or functions of related classes.     

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.     

Meromorphic functions with bounded boundary rotation

In this article, we generalize the class of meromorphic functions with bounded boundary rotation and related classes. Characterizations and some properties of these classes of functions are given.     

A sufficient condition for self-concordance,with application to some classes of structured convex programming problems

Recently a number of papers were written that present low-complexity interior-point methods for different classes of convex programs. The goal of this article is to show that the logarithmic barrier function associated with these programs is self-concordant. Hence the polynomial complexity results for these convex programs can be derived from the theory of Nesterov and Nemirovsky on self-concordant barrier functions. We also show that the approach can be applied to some other known classes of convex programs.This author's research was supported by a research grant from SHELL.On leave from the Eötvös University, Budapest, Hungary. This author's research was partially supported by OTKA No. 2116.     

Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator

In the present paper, we investigate some subordination- and superordination-preserving properties for certain classes of analytic and multivalent functions in the open unit disk, which are associated with such multiplier transformations as the Srivastava-Attiya operator. Various sandwich-type theorems for functions belonging to these classes are also obtained.     

On the linear combination of <Emphasis Type="Italic">q</Emphasis>-additive functions at prime places

Necessary and sufficient conditions are given for linear combinations of q-ary additive functions to belong to some function classes when the summation is extended to the set of primes. Supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA T46993.     

Large Deviation Probabilities for Square-Gaussian Stochastic Processes

We investigate properties of square-Gaussian stochastic processes. These processes are formed by quadratic forms of Gaussian processes or by limits in the mean square of quadratic forms of Gaussian processes. Special classes of these processes are determined and investigated. For processes from these classes estimates of large deviation probability are obtained. These estimates we use to estimate the probability that Gaussian vector-valued process leave some region on some interval of time. We construct asymptotic confidence regions for estimates of covariance functions of vector-valued Gaussian processes. Criterion of hypothesis testing on covariance functions of these processes is constructed.     

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.     

Some classes of analytic and multivalent functions involving a linear operator

By using a linear operator, which is defined here by means of the Hadamard product (or convolution), the authors introduce some new classes of analytic and multivalent functions in the open unit disk and investigate their inclusion relationships and convolution properties. Integral transforms of functions in these classes are also discussed.     

Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings

In this paper, some gap functions for three classes of a system of generalized vector quasi-equilibrium problems with set-valued mappings (for short, SGVQEP) are investigated by virtue of the nonlinear scalarization function of Chen, Yang and Yu. Three examples are then provided to demonstrate these gap functions. Also, some gap functions for three classes of generalized finite dimensional vector equilibrium problems (GFVEP) are derived without using the nonlinear scalarization function method. Furthermore, a set-valued function is obtained as a gap function for one of (GFVEP) under certain assumptions.      

Submultiplicative and subadditive functions and integral inequalities of Bellman-Bihari type

We define two classes of functions which contain the classes of nondecreasing submultiplicative and subadditive functions. Then we discuss the properties of these classes and use them to give generalizations of some well-known integral inequalities like the Bellman-Bihari inequality.     

Enumeration of algebras close to absolutely free algebras and binary trees

We study three classes of algebras: absolutely free algebras, free commutative non-associative, and free anti-commutative non-associative algebras. We study asymptotics of the growth for free algebras of these classes and for their subvarieties as well. Mainly, we study finitely generated algebras, also the codimension growth for varieties in theses classes is studied. For these purposes we use ordinary generating functions as well as exponential generating functions. The following subvarieties are studied in these classes: solvable, completely solvable, right-nilpotent, and completely right-nilpotent subvarieties. The obtained results are equivalent to an enumeration of binary labeled and unlabeled rooted trees that do not contain some forbidden subtrees. We enumerate these trees using generating functions. For solvable and right-nilpotent algebras the generating functions are algebraic. For completely solvable and completely right-nilpotent algebras the respective functions are rational. It is known that these three varieties of algebras satisfy Schreier's property, i.e., subalgebras of free algebras are free. For free groups, there is Schreier's formula for the rank of a subgroup of a free group. We find analogues of this formula for these varieties. They are written in terms of series. As an application, we study invariants of finite groups acting on absolutely free algebras.     

Lagrangian Duality and Cone Convexlike Functions

In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called image space approach initiated by Giannessi.     

α-Well-posedness for Nash Equilibria and For Optimization Problems with Nash Equilibrium Constraints

We present the concepts of α-well-posedness for parametric noncooperative games and for optimization problems with constraints defined by parametric Nash equilibria. We investigate some classes of functions that ensure these types of well-posedness and the connections with α-well-posedness for variational inequalities and optimization problems with variational inequality constraints.     

Elementary properties of arcwise connected sets and functions

The purpose of this paper is to investigate some elementary, basic properties of arcwise connected sets and functions. Since these concepts are generalizations of convexity, it is natural to ask if any of the basic properties of convex sets and functions are carried over to these new generalized classes. All the functions involved are considered to be not necessarily differentiable.     

由从属关系刻画的多叶调和函数新子类

本文引入了由广义Dziok-Srivastava算子定义的多叶调和函数新子类, 给出了该类中函数的系数不等式, 偏差估计和极值点等性质.     

Generalized arcwise-connected functions and characterizations of local-global minimum properties

In this paper, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexity, and their associate subclasses. Functions belonging to these classes satisfy certain local-global minimum properties. Conversely, it is shown that, under some mild regularity conditions, functions for which the local-global minimum properties hold must belong to one of the classes of functions introduced.Dedicated to R. BellmanThe authors are indebted to I. Kozma, N. Megiddo, and A. Tamir for valuable discussions and to S. Schaible for valuable remarks. This research was partially supported by the Fund for the Encouragement of Research at the Technion.