Approximation of functions of several variables by trigonometric polynomials with harmonics from hyperbolic crosses

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Vallee-Poussin operators, Ces`aro operators, Abel operators, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the Lq space for 1 ⩽ q < ∞, and the Fourier partial summation operators and the Vallee-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the Lq space for 1 ⩽ q < ∞.

We obtain exact-order estimates for the best approximation of periodic functions of several variables from the classes $ MB_{{p,\theta }}^{\omega } $ by trigonometric polynomials with the “numbers” of harmonics from graded hyperbolic crosses in the metric of the space L q :

Asymptotic expansions in powers of δ as δ → +∞ of the series $\sum\limits_{k = 0}^\infty {( - 1)^{(\beta + 1)k} \frac{{Q((\delta ^\alpha - (ak + b)^\alpha ) + )}} {{(ak + b)^{r + 1} }}} , $ where β ∈ ℤ, α, a, b > 0, and r ∈ ℂ, while Q is an algebraic polynomial satisfying the condition Q(0) = 0, are obtained. In special cases, these series arise from the approximation of periodic differentiable functions by the Riesz and Cesàro means.

We consider the approximation of periodic functions by trigonometric polynomials in metric (not normed) spaces that are generalizations of the spaces L p , 0 < p < 1, and L 0. In particular, we prove the multidimensional Jackson theorem in L p (T m ), 0 < p < 1.

Denote by C A the set of functions that are analytic in the disk |z| < 1 and continuous on its closure |z| ≤ 1; let ℜ n , n = 0, 1, 2, ..., be the set of rational functions of degree at most n. Denote by R n (f) (R n (f) A ) the best uniform approximation of a function f ∈ C A on the circle |z| = 1 (in the disk |z| ≤ 1) by the set ℜ n . The following equality is proved for any n ≥ 1: sup{R n (f) A /R n (f): f ∈ C A ∖ ℜ n } = 2. We also consider a similar problem of comparing the best approximations of functions in C A by polynomials and trigonometric polynomials.

CONTENTS Introduction Chapter I. The analytic capacity of sets § 1. Definition and some properties of analytic capacity § 2. The connection between the capacity of a set and measures § 3. On removable singularities of analytic functions § 4. The analytic C-capacity of sets § 5. Estimates of the coefficients in the Laurent series § 6. The change in the capacity under a conformal transformation of a set Chapter II. The separation of singularities of functions § 1. The construction of a special system of partitions of unity § 2. Integral representations of continuous functions § 3. Separation of singularities § 4. The approximation of functions in parts § 5. Approximation of functions on sets with empty inner boundary § 6. The additivity of capacity for some special partitions of a set Chapter III. Estimation of the Cauchy integral § 1. Statement of the result § 2. Estimate of the Cauchy integral § 3. Estimation of the Cauchy integral along a smooth Lyapunov curve § 4. Proof of the principal theorem § 5. Some consequences § 6. A refinement of the Maximum Principle and the capacity analogue to the theorem on density points Chapter IV. Classification of functions admitting an approximation by rational fractions § 1. Examples of functions that cannot be approximated by rational fractions § 2. A criterion for the approximability of a function § 3. Properties of the second coefficient in the Laurent series § 4. Proof of the principal lemma § 5. Proof of the theorems of § 2 Chapter V. The approximation problem for classes of functions § 1. Removal of the poles of approximating functions from the domain of analyticity of the function being approximated § 2. Necessary conditions for the algebras to coincide § 3. A criterion for the equality of the algebras § 4. Geometrical examples § 5. Some problems in the theory of approximation Chapter VI. The approximation of functions on nowhere dense sets § 1. The instability of capacity § 2. A capacity criterion for the approximability of functions on nowhere dense sets § 3. Theorems on the approximation of continuous functions in terms of Banach algebras § 4. A capacity characterization of the Mergelyan function and of peak points References

Let B be a set of integers with certain arithmetic properties. We obtain estimates of the best approximation of functions in the space L p , 0 < p <1, by trigonometric polynomials that are constructed by the system $$ \{e^{ikx}\}_{k\in \mathbb{Z}\backslash B} $$ . Bibliography: 13 titles.

We obtain new inverse theorems on the approximation of periodic functions f(·) that establish conditions for the existence of their (ψ, β)-derivatives. These theorems also guarantee a certain smoothness of these derivatives.

The purpose of this paper is to get the degree of pointwise approximation for continuously differentiable functions of order r (r∈Z+) on the interval [−1, 1] by the Vallee-Poussin sums of Tchebycheff-Fourier series.

In the present paper we shall consider certain problems of best mean approximation for functions of several variables. It is wellknown that there is an essential difficulty in the extension of the theory of best Chebyshev approximation to the multivariate case. The basic problem here is the absence of Haar subspaces of functions of more than one variable. However, this fact seems not to be an insuperable obstacle in developing the theory of best Lt-approximation of functions of several variables. Some first steps in this direction have been made in [1]. In particular, it was shown in [1] that the L~-approximation of continuous functions of two variables by the tensor product of a two-dimensional Haar subspaces and an arbitrary Haar subspace is unique. This result gives some hope that the general problem of unicity of best Ll-approximation of continuous functions of several variables by tensor products ofHaar subspaces can be solved positively. Nevertheless, this problem is still unsolved even in the case of ordinary polynomials. In the present paper we continue the study of uniqueness of multivariate L~-approximation presenting another approach to this problem. We shall consider approximation by arbitrary multivariate polynomials but restrict ourselves only to approximating rational functions. The main advantage of this approach is that it includes the study of the unicity of polynomials of several variables with fixed leading coefficients which have minimal Ll-deviation from zero. In case of Chebyshev approximation this problem was considered in [2], [6] and [7]. The results of our paper imply the unicity of the polynomials of least Ll-deviation having fixed leading coefficients. Thus we show that the solution of the Korkin--Zolotarjov type extremal problem is unique in the multivariate case, too. For rectangular regions this unique solution is given by the product of Chebyshev polynomials of second kind.

i. A. N. Guz', "Construction of a theory of the stability of unidirectional fiber composites," Prikl. Mekh., 5, No. 2, 62-70 (1969). 2. A. N. Guz', Stability of Three-Dimensional Deformable Solids [in Russian], Naukova Dumka, Kiev (1971). 3. A. N. Guz', Stability of Elastic Bodies with Finite Strains [in Russian], Naukova Dumka, Kiev (1973). 4. A. N. Guz', Principles of a Three-Dimensional Theory of the Stability of Deformable Bodies [in Russian], Vishch. Shkola, Kiev (1986). 5. A. N. Guz', Fracture Mechanics of Composites in Compression [in Russian], Naukova Dumka, Kiev (1989). 6. A. N. Guz' and I. A. Guz', "Local instability of composite laminates," Dokl. Akad. Nauk SSSR, 312, No. 3, 210-212 (1990).

The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

In the first section the best approximations of periodic functions of one real variable by trigonometric polynomials are studied. Estimates of these approximations in terms of averaged differences are obtained. A multidimensional generalization of these estimates is presented in the second section. As a consequence. The multidimensional Jackson's theorem is proved.

The rate of ϕ-strong approximation of periodic functions by trigonometric polynomials constructed on the basis of interpolating polynomials with equidistant nodes is considered.

, Manfred Sommer, Hans Strauss, On Strong Uniqueness in One-Sided $L^1$-Approximation of Differentiable Functions, Proceedings of the American Mathematical Society, Vol. 106, No. 4 (Aug., 1989), pp. 1011-1016