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 Local Smoothness Classes of Periodic Functions

We obtain a characterization of local Besov spaces of periodic functions in terms of trigonometric polynomial operators. We construct a sequence of operators whose values are (global) trigonometric polynomials, and yet their behavior at different points reflects the behavior of the target function near each of these points. In addition to being localized, our operators preserve trigonometric polynomials of degree commensurate with the degree of polynomials given by the operators. Our constructions are “universal;” i.e., they do not require an a priori knowledge about the smoothness of the target functions. Several numerical examples are discussed, including applications to the problem of direction finding in phased array antennas and finding the location of jump discontinuities of derivatives of different order.     

The Average Widths and Non-linear Widths of the Classes of Multivariate Functions with Bounded Moduli of Smoothness

The classes of the multivariate functions with bounded moduli on Rd and Td are given and their average σ-widths and non-linear n-widths are discussed. The weak asymptotic behaviors are established for the corresponding quantities.     

具有混合光滑性的Sobolev-Wiener类在L_q(R~d)中的平均宽度

本文研究了具有混合光滑性的Sobolev-Wiener类SpqrW在Lq(Rd)中的平均σ-K宽度(1相似文献   

Linear Widths of the Besov Classes of Periodic Functions of Many Variables. II

We obtain order estimates for linear widths of the Besov classes of periodic functions of many variables in the space L _q for certain values of parameters p and q different from those considered in the first part of the work.     

Linear Widths of the Besov Classes of Periodic Functions of Many Variables. I

We obtain order estimates for linear widths of the Besov classes of periodic functions of many variables in the space L _q for certain values of the parameters p and q.     

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. I

In the Banach space of functions analytic in a Jordan domain , we establish order estimates for the Kolmogorov widths of certain classes of functions that can be represented in by Cauchy-type integrals along the rectifiable curve = and can be analytically continued to or to .     

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. II

In normed spaces of functions analytic in the Jordan domain , we establish exact order estimates for the Kolmogorov widths of classes of functions that can be represented in by Cauchy-type integrals along = with densities f(·) such that . Here, is a conformal mapping of onto {w: |w| > 1}, and is a certain subset of infinitely differentiable functions on T = {w: |w| = 1}.     

Optimal Recovery of Periodic Functions from Fourier Coefficients Given with an Error

We construct optimal methods of recovery of 2π-periodic functions analytic in a strip and its derivatives at a pointt∈ [0, 2π), using information about the Fourier coefficients given with an error in the uniform norm. The same problem is solved for the Sobolev spaceW^r₂.     

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.     

具有混合光滑性的Holder-Nikolskii-Wiener类在Lq(Rd)中的平均宽度（英文）

本文研究了具有混合光滑性的Holder-Nikolskii-Wiener类SrpqH在Lq(Rd)中的平均σ-K宽度(2≤q≤p<∞),得到了该量的弱渐近估计.     

On the K-Functional for the Mixed Generalized Modulus of Smoothness

Mathematical Notes -     

Conjugate Functions and the Modulus of Smoothness of Fractional Order

In the present paper, estimates of the partial moduli of smoothness of fractional order of the conjugate functions of several variables are obtained in the space C(Tⁿ). The accuracy of the obtained estimates is established by appropriate examples.     

Sharp Estimates for Mean Square Approximations of Classes of Differentiable Periodic Functions by Shift Spaces

Let L₂ be the space of 2π-periodic square-summable functions and E(f, X)₂ be the best approximation of f by the space X in L₂. For n ∈ ? and B ∈ L₂, let \({{\Bbb S}_{B,n}}\) be the space of functions s of the form \(s\left( x \right) = \sum\limits_{j = 0}^{2n - 1} {{\beta _j}B\left( {x - \frac{{j\pi }}{n}} \right)} \). This paper describes all spaces \({{\Bbb S}_{B,n}}\) that satisfy the exact inequality \(E{\left( {f,{S_{B,n}}} \right)_2} \leqslant \frac{1}{{^{{n^r}}}}\parallel {f^{\left( r \right)}}{\parallel _2}\). (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.