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 functions of several variables by nuclear operators

Translated from Matematicheskii Zametki, Vol. 47, No. 3, pp. 32–41, March, 1990.     

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 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 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 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 sobolev classes of functions by sums of products of functions of fewer variables

Translated from Matematicheskie Zametki, Vol. 48, No. 6, pp. 10–21, December, 1990.     

Approximation by Fourier sums of classes of functions with several bounded derivatives

An ordered estimate is obtained for the approximation by Fourier sums, in the metric ofd=(d ₁ , ...,d _n ), 1<dj<,j=1, ...,_n of classes of periodic functions of several variables with zero means with respect to all their arguments, having m mixed derivatives of order a¹..., a^m., aⁱ rⁿ. which are bounded in the metrics ofp ⁱ =p ₁ ⁱ , ..., p _n ⁱ , i

j ⁱ <,i=i, ...,n, j=1, ...,n by the constants ₁, ., _m, respectively.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 197–212, February, 1978.     

Best trigonometric and bilinear approximations of classes of functions of several variables

Order-sharp estimates of the best orthogonal trigonometric approximations of the Nikol’skii-Besov classes B _p,θ ^r of periodic functions of several variables in the space L _q are obtained. Also the orders of the best approximations of functions of 2d variables of the form g(x, y) = f(x?y), x, y ∈ $\mathbb{T}$ ^d = Π _j=1 ^d [?π, π], f(x) ∈ B _p,θ ^r , by linear combinations of products of functions of d variables are established.     

Approximation of classes of differentiable functions

Conditions are derived under which is finite or infinite. The value of F is calculated for certain special cases.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 105–112, February, 1971.