Approximation of classes of periodic functions of several variables

We study classes of periodic functions of several variables with bounded generalized derivative in the metric of the space L_p. We obtain order estimates of deviations of Fourier sums, which are constructed depending on the behavior of functions that define the operator of generalized differentiation. We find estimates of the Kolmogorov widths, which are realized by the Fourier sums that are constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 662–672, May, 1992.     

Approximation of classes of periodic multivariable functions by linear positive operators

In an N-dimensional space, we consider the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is positive. We establish that the least upper bound of this approximation does not exceed the sum of properly chosen least upper bounds in m-and ((N − m))-dimensional spaces. We also consider the cases where the inequality obtained turns into the equality. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 12–19, January, 2006.     

Approximation of classes of smooth functions of several variables

The paper deals with imbedding theorems and harmonic approximation for classes of periodic functions of several variables. Necessary and sufficient conditions for imbedding are found for various classes of smooth functions. The paper contains asymptotic estimates for fractional order derivatives of multidimensional Dirichlet and Favard kernels, for Fourier best approximation and for the Kolmogorov diameter of smooth function classes.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 207–226, 1984.     

Approximation of the Besov classes of periodic functions of several variables in a space Lq

Order estimates are obtained for best approximations by polynomials constructed according to hyperbolic crosses on the classes B _p ^r , of periodic functions of several variables. The order is found of the Kolmogorov width on these classes in the spaces L_q for 1 <p q 2.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1398–1408, October, 1991.     

Approximation of some classes of periodic functions of many variables

We obtain the exact order of deviations of Fejér sums on the class of continuous functions. This order is determined by a given majorant of the best approximations.     

Approximation of classes B_{p,\theta }^r of periodic functions of one and several variables

We obtain order-sharp estimates of best approximations to the classes $B_{p,\theta }^r$ of periodic functions of several variables in the space L _q , 1 ≤ p, q ≤ ∞ by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes $ B_{1,\theta }^{r_1 } $ in the space L ₁.     

Approximation of certain classes of smooth periodic functions of several variables by means of interpolation splines defined over a uniform net

Translated from Matematicheskie Zametki, Vol. 57, No. 6, pp. 917–919, June, 1995.     

Approximation characteristics of the classes B p,θ Ω of periodic functions of many variables

We obtain exact order estimates for the approximation of the classes B _p,θ ^Ω of periodic functions of many variables in the space L _q by using operators of orthogonal projection and linear operators satisfying certain conditions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 692–704, May, 2006.     

Approximation of functions of several variables by spherical Riesz means

For even N ≥ 2 and δ 2N-3 (for N-2 or 4 we assume that δ > (N-1)/2) we find asymptotic approximations for the quantity $$E_R^\delta (H_{\rm N}^\omega ) = \mathop {sup}\limits_{f \in H_{\rm N}^\omega } \parallel f(x) - S_R^\omega (x,f)\parallel _ \in (R \to \infty ),$$ , where S _R ^δ (x,f) is the spherical Riesz mean of order δ of the Fourier kernel of the functionf(x), and H _N ^ω is the class of periodic functions of N variables whose moduli of continuity do not exceed a given convex modulus of continuity ω(δ). For N 2 and δ > 1/2 the result is known.     

Approximation of almost periodic functions of two variables

In this paper we study the deviations of periodic functions of two variables from the integral mean values and their Fourier transforms. In the class of uniform almost periodic functions of two variables we obtain estimates for the deviation from sums of Marcinkiewicz-Zygmund type.