Nonlinear approximations of classes of periodic functions of many variables

Order-sharp estimates are established for the best N-term approximations of functions in the classes $B_{pq}^{sm} (\mathbb{T}^k )$ and $L_{pq}^{sm} (\mathbb{T}^k )$ of Nikol’skii-Besov and Lizorkin-Triebel types with respect to the multiple system of Meyer wavelets in the metric of $L_r (\mathbb{T}^k )$ for various relations between the parameters s, p, q, r, and m (s = (s ₁, ..., s _n ) ∈ ? ₊ ⁿ , 1 ≤ p, q, r ≤ ∞, m = (m ₁, ..., m _n ) ∈ ? ⁿ , and k = m ₁ + ... + m _n ). The proof of upper estimates is based on variants of the so-called greedy algorithms.     

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 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.     

On estimates of approximation characteristics of the Besov classes of periodic functions of many variables

We obtain order estimates for some approximate characteristics of the Besov classes B _p,ϑ ^r of periodic functions of many variables. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 9, pp. 1250–1261, September, 1997.     

Trigonometric and orthoprojection widths of classes of periodic functions of many variables

We obtain exact order estimates for trigonometric and orthoprojection widths of the Besov classes B ^r _p,θ and Nikol’skii classes Hr p of periodic functions of many variables in the space L _q for certain relations between the parameters p and q.     

On the best approximations and Kolmogorov widths of besov classes of periodic functions of many variables

Order estimates are obtained for the best approximations of the classesB _1, ^r in the spaceL _q with 1<q< and classesB _, ^r in a uniform metric. The behavior of Kolmogorov widths of the classesB _p, ^r ,1<p, in the metric of L is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 79–92, January, 1995.     

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.     

Best approximations of the classes B p,θ r of periodic functions of many variables in uniform metric

We obtain estimates exact in order for the best approximations of the classes B _∞,θ ^r of periodic functions of two variables in the metric of L _∞ by trigonometric polynomials whose spectrum belongs to a hyperbolic cross. We also investigate the best approximations of the classes B _p,θ ^r , 1 ≤ p < ∞, of periodic functions of many variables in the metric of L _∞ by trigonometric polynomials whose spectrum belongs to a graded hyperbolic cross. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1395–1406, October, 2006.     

Linear widths of the classes B p,θ Ω of periodic functions of many variables in the space L q

We obtain exact order estimates for the linear widths of the classes B _p,θ ^Ω of periodic functions of many variables in the space L _q for certain values of the parameters p and q. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 93–104, January, 2006.     

Best cubature formulas for some classes of functions of many variables

The problem of minimizing the error in the cubature formula for a given class of functions is considered. For cubature formulas with a lattice arrangement of points this problems is solved exactly for a wide class of functions of m variables.Basic contents of this paper presented with proofs at the Seminar on Theory of Functions at Dnepropetrovsk State University, December, 1965.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 565–576, May, 1968.     

Approximation of continuous periodic functions of many variables by spherical Riesz means

We find in succession exact upper bounds for the magnitudes of the least upper bounds of the deviations of spherical Riesz means on classes of continuous periodic functions of many variables and, in a number of cases, we prove the asymptotic exactness of these estimates.Translated from Matematicheskie Zametki, Vol. 15, No. 5, pp. 821–832, May, 1974.The author thanks S. B. Stechkin for suggesting the topic of this paper.     

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.     

Best trigonometric and bilinear approximations for the Besov classes of functions of many variables

We obtain order estimates for the best trigonometric and bilinear approximations for the classesB _p, ^r of functions of many variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1097–1111, August, 1995.     

On approximate characteristics of functions of many variables occurring in saturation classes of continuity moduli of different order

For the spaces L_p and C of periodic functions of many variables we establish general theorems connecting the saturation classes generated by approximation methods with similar classes for continuity moduli. __________ Translated from Problemy Matematicheskogo Analiza, No. 33, 2006, pp. 79–90.     

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.