Best Approximation Of Classes Research Articles

Наилучшее приближение аналитических в единичном круге функций в весовом пространстве Бергмана B_(2,μ)

The paper studies the issues of the best approximation of analytical functions in the Bergman weight space B_(2,μ). In this space, for best approximations of functions analytic in the circle by algebraic complex polynomials we obtain the exact inequalities by means of generalized modules of continuity of higher order derivatives Ω_m (z^r f^((r) ),t),m∈N,r∈Z. For classes of functions analytic in the unit circle defined by the characteristic Ω_m (z^r f^((r) ),t), and the majorant Φ, the exact values of some n-widths are calculated. When proving the main results of this work, we use methods for solving extremal problems in normalized spaces of functions analytic in the circle, N.P. Korneichuk’s method for estimating upper bounds for the best approximations of classes of functions by a subspace of fixed dimension, and a method for estimating from below the n-widths of function classes in various Banach spaces.

Exact Constants of the Best One Sided Approximations of the Sum Analytic Functions from Different Classes

The task of obtaining the exact values of the best approxima-tions by trigonometric polynomials of continuous or summable functions originates from the works of P. L. Chebyshev, who, back in the 1950s, posed the problem of finding the polynomial, that de-viates the least from a given continuous function. Subsequently this direction in the theory of approximation got further development thanks to the works of K. Weierstrass, D. Jackson, S. N. Bernstein, Valle-Poussin and others. On this time there is an increase attention to problems of one-sided approximation of individual functions and their classes in the metric space L. Problems of this content arise up in number theory, coding theory, and other areas of math-ematics. The first results of this direction were obtained in the 1880th by A.A. Markov and T.Y. Stieltijes. In the future, these studies were continued in the works of J. Karamata (1930), G.Freud and T. Hanelius (mid-20th century). General issues related to the problem of the best approximation of classes of functions by trigonometric polynomials: the existence of the best approximation polynomial, its characteristic properties, one sided approximations are detailed in many works, in particular, for example, in the book of M. P. Korneychuk [1], the works of T.Ganelius [4] , V.G. Doronin, A.A. Ligun [5].The exact constants of the best one sided approximation of the sum ofthe majorant functions of the classes that allow analytical extension into a strip of fixed width and of functions harmonic in a circle of radius 1 have been found in this work.

The best approximation of classes, defined by powers of self-adjoint operators acting in Hilbert space, by other classes

The best approximation of class of elements such that $\| A^k x \| \leqslant 1$ by classes of elements such that $\| A^r x \| \leqslant N$, $N > 0$ for powers $k < r$ of self-adjoint operator $A$ in Hilbert space $H$ is found.

Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol’skii–Besov spaces

We have obtained the exact-order estimates for some approximative characteristics of the Sobolev classes $$ {\mathbbm{W}}_{p,\alpha}^r $$ and Nikоl’skii–Besov classes $$ {\mathbbm{B}}_{p,\theta}^r $$ of periodic functions of one and several variables in the norm of the space B∞,1. Properties of the linear operators realizing the orders of the best approximation for the classes $$ {\mathbbm{B}}_{\infty, \theta}^r $$ in this space by trigonometric polynomials generated by a set of harmonics with “numbers” from step hyperbolic crosses are investigated.

Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol'skii-Besov spaces

We have obtained the exact-order estimates for some approximative characteristics of the Sobolev classes $\mathbb{W}^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}$ and Nikоl'skii--Besov classes $\mathbb{B}^{\boldsymbol{r}}_{p,\theta}\ $ of periodic functions of one and several variables in the norm of the space $B_{\infty, 1}$. Properties of the linear operators realizing the orders of the best approximation for the classes $\mathbb{B}^{\boldsymbol{r}}_{\infty, \theta}$ in this space by trigonometric polynomials generated by a set of harmonics with $``$numbers$"$ from step hyperbolic crosses are investigated.

On exact values of relative approximations of classes of periodic functions by splines

We obtain the conditions under which the exact values of the best relative approximation of classes of periodic functions by splines coincide with the exact values of the best approximations of these classes by splines without constraints.

Approximation properties of generalized Fup-functions

Generalized Fup-functions are considered. Almost-trigonometric basis theorem is proved. Spaces of linear combinations of shifts of the generalized Fup-functions are constructed and an upper estimate of the best approximation of classes of periodic differentiable functions by these spaces in the norm of $L_2[-\pi,\pi]$ is obtained.

Best approximation by analogues of “own” and “alien” hyperbolic crosses

The author studies the best approximation of classes of periodic functions of several real variables with a given majorant of the mixed moduli of continuity by analogues of “own” and “alien” hyperbolic crosses. The majorant has the power factors in different degrees and logarithmic factors in different degrees.

Asymptotic behavior of best approximations of classes of Poisson integrals of functions from [formula omitted

We find asymptotic formulas for the least upper bounds of approximation in the metric of the space C by using a linear method U n ∗ for classes of Poisson integrals of continuous 2 π -periodic functions in the case where their moduli of continuity do not exceed fixed convex majorants. On the basis of these results, we obtain asymptotic formulas for the best uniform approximations by trigonometric polynomials and, under some restrictions, asymptotic formulas for Kolmogorov widths of these classes in the uniform metric.

On the exact values of the best approximations of classes of differentiable periodic functions by splines

We obtain the exact values of the best L1-approximations of classes WrF, r ∈ ℕ, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F, as well as of classes WrHω of periodic functions whose rth derivative has a given convex (upward) majorant ω(t) of the modulus of continuity, by subspaces of polynomial splines of order m ≥ r + 1 and of deficiency 1 with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ℕ, k ∈ ℤ, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

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.

On best approximation of classes by radial functions

We investigate the radial manifolds R n generated by a linear combination of n radial functions on R d . We consider the best approximation of function classes by the manifold R n . In particular, we prove that the deviation of the manifold R n from the Sobolev class W 2 r, d in the Hilbert space L 2 behaves asymptotically as n − r d−1 . We show the connection between the manifold R n and the space of algebraic polynomials P d,s of degree s. Namely, we prove there exist constants c 1 and c 2 such that the space P d,s is either contained or not in R n as n⩾ c 1 s d−1 or n< c 2 s d−1 , respectively.

The best approximation of classes $W^r H^{\omega}$ by algebraic polynomials in $L_1$ space

We consider asymptotic behavior of the best approximation of classes $W^r H^{\omega}$ by algebraic polynomials in $L_1$ space.

On the best approximation, by polynomials, of some classes of functions

We obtain asymptotically exact estimates of the best approximations of classes of conjugate functions by algebraic polynomials in the spaces $C$ and $L_1$.

On the best approximation of classes of convolutions of periodic functions by trigonometric polynomials

We present sufficient conditions for kernels to belong to the classN n * . In certain cases, this enables us to find exact values of the best approximations of classes of convolutions by trigonometric polynomials in the metrics ofC andL.

Best approximations of classes of functions defined by means of the modulus of continuity

This paper is devoted to an exact solution of problems of best approximation in the uniform and integral metrics of classes of periodic functions representable as a convolution of a kernel not increasing the oscillation with functions having a given convex upwards majorant of the modulus of continuity. The approximating sets are taken to be the trigonometric polynomials in the case of the uniform and integral metrics, and convolutions of the kernel defining the class with polynomial splines in the case of the integral metric.

Best approximation of classes of functions with bounded r-th derivative on a segment by finite-dimensional subspaces

Best approximation of classes of functions with bounded r-th derivative on a segment by finite-dimensional subspaces

The best approximation of classes w 1 r (Is) in the space L?(Is)

A method is proposed which enables us to obtain the best upper bounds for the n-diameters of multidmensional Sobolev classes W 1 r (Is) in the metric of L∞ (Is).