Degree of approximation of functions in the Hölder metric by (e,c) means

Degree of approximation of functions by the (e, c) means of its Fourier series in the Hölder metric is studied.     

On integral convolution operators

We study integral convolutions defined on functions ofn variables in symmetric spaces and obtain new additive estimates for the mean value of the nonincreasing permutation(K ^* f) ^** (t) of the absolute value of the integral convolution on the interval [0,t] for anyt>0. As an example of application of the estimates obtained, we prove the boundedness of the integral convolution operator acting from the intersection of symmetric spaces into the Marcinkiewicz space. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 551–555, October, 1999.     

Approximation of functions on the real axis by Féjer-type operators in the generalized Hölder metric

In this paper, we consider the orders of approximation of functions on the whole real axis by operators of Fejér type in the Banach space with the so-called generalized Hölder metric.     

On subspaces of C[0, 1] consisting of nonsmooth functions

For each Hölder spa?e H ^ω, we construct an infinite-dimensional closed subspace G of C[0, 1], isomorphic to l ¹ and such that, for each function x ∈ G not identically zero, its restriction to the set of positive measure does not belong to the Hölder space H ^ω.     

Degree of approximation of functions by their fourier series in the generalized Hölder metric

The paper studies the degree of approximation of functions by matrix means of their Fourier series in the generalized Hölder metric, generalizing many known results in the literature     

Degree of approximation of functions associated with Hardy-Littlewood series in the generalized Hölder metric

The paper studies the degree of approximation of functions associated with Hardy Littlewood series in the generalized Hölder metric.     

A difference between continuous and absolutely continuous norms in Banach function spaces

On examples we show a difference between a continuous and absolutely continuous norm in Banach function spaces.     

加权(C)eby(s)ev-Ostrowski型不等式

关于著名的Cebysev不等式，已有众多的研究成果．通过建立积分不等式，来建立全新的加权Cebysev型积分不等式．给予了独立的证明，并给出了此类不等式的新评价．     

Stability of time-dependent diffusion semigroups and kernels

The main result of this paper is that when the coefficients of the time-dependent divergence form operators are Hölder continuous in time with order not too much smaller than 1/2, the distance of the semigroups of two operators is bounded by the L ₂ distance of the coefficients of their corresponding operators.     

Continuous Approximations of Multivalued Mappings and Fixed Points

In the present paper, we prove a fixed-point theorem for completely continuous multivalued mappings defined on a bounded convex closed subset X of the Hilbert space H which satisfies the tangential condition , where T _X (x) is the cone tangent to the set X at a point x. The proof of this theorem is based on the method of single-valued approximations to multivalued mappings. In this paper, we consider a simple approach for constructing single-valued approximations to multivalued mappings. This approach allows us not only to simplify the proofs of already-known theorems, but also to obtain new statements needed to prove the main theorem in this paper.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 212–222.Original Russian Text Copyright © 2005 by B. D. Gel’man.     

Harmonic maps from Riemannian polyhedra to geodesic spaces with curvature bounded from above

The hypothesis of local compactness of the target is removed from an earlier result about interior Hölder continuity of locally energy minimizing maps ? from a Riemannian polyhedron (X, g) to a suitable ball B of radius R <  π/2 (best possible) in a geodesic space with curvature ≤ 1. Furthermore, the variational Dirichlet problem for harmonic maps from an open set \(\Omega \Subset X\) to B is shown to be uniquely solvable, and the solution is continuous up to the boundary ?Ω at any regular point of ?Ω at which the prescribed boundary map is continuous.     

广义Hilbert空间中几何凸函数的几个定理及其应用

研究广义Hilbert空间中几何凸函数的性质,给出一些重要定理,并运用几何凸函数的Jensen不等式建立了三重的双参数Hlder不等式和三重的多参数Minkowski不等式.     

Two-weight inequalities for convolution operators in Lebesgue space

In this paper, we prove a theorem on the boundedness of a convolution operator in a weighted Lebesgue space with kernel satisfying a certain version of Hörmander’s condition.     

Kernel based approximation in Sobolev spaces with radial basis functions

In this paper, we study several radial basis function approximation schemes in Sobolev spaces. We obtain an optional error estimate by using a class of smoothing operators. We also discussed sufficient conditions for the smoothing operators to attain the desired approximation order. We then construct the smoothing operators by some compactly supported radial kernels, and use them to approximate Sobolev space functions with optimal convergence order. These kernels can be simply constructed and readily applied to practical problems. The results show that the approximation power depends on the precision of the sampling instrument and the density of the available data.     

Some remarks on James-Schreier spaces

The James-Schreier spaces Vp, where 1?p<∞, were recently introduced by Bird and Laustsen (in press) [5] as an amalgamation of James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach-Saks property on the other. The purpose of this note is to answer some questions left open in Bird and Laustsen (in press) [5]. Specifically, we prove that (i) the standard Schauder basis for the first James-Schreier space V₁ is shrinking, and (ii) any two Schreier or James-Schreier spaces with distinct indices are non-isomorphic. The former of these results implies that V₁ does not have Pe?czyński's property (u) and hence does not embed in any Banach space with an unconditional Schauder basis.     

Geometry and Topology of Continuous Best and Near Best Approximations

The existence of a continuous best approximation or of near best approximations of a strictly convex space by a subset is shown to imply uniqueness of the best approximation under various assumptions on the approximating subset. For more general spaces, when continuous best or near best approximations exist, the set of best approximants to any given element is shown to satisfy connectivity and radius constraints.     

一个推广的Hilbert型不等式及其应用

引入单参数λ及Beta函数,给一个Hilbert型不等式作具有最佳常数因子的推广.作为应用,建立它的等价式及逆向形式.     

A new trigonometric method of summation and its application to the degree of approximation

The object of the present investigation is to introduce a new trigonometric method of summation which is both regular and Fourier effective and determine its status with reference to other methods of summation (see §2-§4) and also give an application of this method to determine the degree of approximation in a new Banach space of functions conceived as a generalized Holder metric (see §5).