Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

We consider classes of 2π-periodic functions that are represented in terms of convolutions with fixed kernels Ψ _β ⁻ whose Fourier coefficients tend to zero at exponential rate. We determine exact values of the best approximations of these classes in the uniform and integral metrics. In several cases, we determine the exact values of the Kolmogorov, Bernstein, and linear widths for these classes in the metrics of the spaces C and L. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 946–971, July, 2005.     

On Approximation Characteristics of Some Classes of Functions of Small Smoothness

We find the exact order of the entropy numbers and the Kolmogorov widths of a class of functions defined by a condition on the uniform norm of blocks of the Fourier series taken over a lacunar sequence of indices. This result generalizes a result by B. S. Kashin and V. N. Temlyakov.     

On Embeddings for Classes of Functions with Generalized Smoothness on Metric Spaces

Given a metric space with a Borel measure, we consider the classes of functions whose increment is controlled by the measure of a ball containing the corresponding points and a nonnegative function summable with some power. We prove embedding theorems for these spaces defined by two different measures satisfying the doubling condition.     

周期Besov函数类的N-宽度

本文研究了带有标准信息的各向同性Besov周期函数类的逼近问题,求得了带混淆范数的多维周期Besov函数类的Kolmogorov,Gel'fand和线性N-宽度的精确阶．     

N-Width for Some Classes of Periodic Functions

This paper considers the Kolmogorov width,the Gelfand width and the Linear width of the periodic smooth functions with boundary conditions in Orlicz space.Furth...     

Estimates of The Fourier Widths of the Classes Of Periodic Functions With Given Majorant of the Mixed Modulus of Smoothness

We obtain some order-sharp estimates for the Fourier widths of Nikol'skii–Besov and Lizorkin–Triebel function classes with given majorant of the mixed modulus of smoothness in the Lebesgue space for a few relations between the parameters of the class and the space. The upper bounds follow from estimates of the approximation of functions of these classes by special partial sums of their Fourier series with respect to the multiple system of periodized Meyer wavelets.     

Approximation of Classes of Periodic Functions in Several Variables

We study the approximation of the classes and of periodic functions of several variables by multiple Fourier sums of fixed order constructed with regard to individual properties of functions from these classes. In a number of cases, such approximations allow us to achieve a better degree of approximation of the classes indicated above as compared to their approximation by staircase hyperbolic Fourier sums.     

On Fractional Smoothness of Modulus of Functions

We consider the Nemytskii operators u→|u| and u→u~±in a bounded domain ? with C~2 boundary. We give elementary proofs of the boundedness in H~s(?) with 0 ≤ s 3/2.     

Inverse Theorems on the Approximation of Periodic Functions with High Generalized Smoothness

Mathematical Notes -     

关于某些调和函数

用H表示形如f(z)=h(z) g(z)的调和函数族,其中h和g是单位圆盘内的解析函数.本文考虑H的三类子族函数.其中的两族为PH(α)=f∶Ref(z)z≥α和NH(α)=f∶Ref(z)θz≥θα,其中0≤α<1和θ=argz.本文得到了函数f属于其中一族的一个充分必要条件,并且获得了一些系数不等式和偏差定理.     

On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

We obtain the exact asymptotics (as n ) of the best L ₁-approximations of classes of periodic functions by splines s S _{2n, r – 1} and s S _{2n, r + k – 1} (S _{2n, r} is the set of 2-periodic polynomial splines of order r and defect 1 with nodes at the points k/n, k Z) under certain restrictions on their derivatives.     

On the Order of Decrease of Uniform Moduli of Smoothness for the Classes of Functions E p,m [ɛ]

In this paper, we solve the problem of the exact order of decrease of uniform moduli of smoothness for the classes of 2π-periodic functions of several variables with a given majorant of the sequence of total best approximations in the metric of L _p , 1 ≤ p < ∞.     

Optimal Recovery of the Derivative of Periodic Analytic Functions from Hardy Classes

LetS_β{z : |Im z|<β}. For 2π-periodic functions which are analytic inS_βwithp-integrable boundary values, we construct an optimal method of recovery off′(ξ), ξS_β, using information about the valuesf(x₁), mldr;, f(x_n), x_j[0, 2π).     

On Smoothness Properties and Approximability of Optimal Control Functions

The theory of discretization methods to control problems and their convergence under strong stable optimality conditions in recent years has been thoroughly investigated by several authors. A particularly interesting question is to ask for a natural smoothness category for the optimal controls as functions of time.In several papers, Hager and Dontchev considered Riemann integrable controls. This smoothness class is characterized by global, averaged criteria. In contrast, we consider strictly local properties of the solution function. As a first step, we introduce tools for the analysis of L elements at a point. Using afterwards Robinson's strong regularity theory, under appropriate first and second order optimality conditions we obtain structural as well as certain pseudo-Lipschitz properties with respect to the time variable for the control.Consequences for the behavior of discrete solution approximations are discussed in the concluding section with respect to L as well as L ₂ topologies.