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 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.     

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.     

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.     

Approximative characteristics of the isotropic classes of periodic functions of many variables

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B _p,θ ^r ) and Nukol’skii (H _p ^r ) classes of periodic functions of many variables in the metric of L _q , 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L ₁ and L _∞ by trigonometric polynomials with the corresponding spectrum.     

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.     

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.     

Best trigonometric approximations for some classes of periodic functions of several variables in the uniform metric

We study best M-term trigonometric approximations and best orthogonal trigonometric approximations for the classes B _pθ ^r and W _pα ^r of periodic functions of several variables in the uniform metric.     

The best trigonometric approximations and the Kolmogorov diameters of the Besov classes of functions of many variables

The order estimates for the best trigonometric approximations and the Kolmogorov diameters of the classesB _p, ^r of functions of many variables in the spaceL _q are obtained for certain values of the parametersp andq.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 663–675, May, 1993.     

Best M-Term trigonometric approximations of the classes $$ B_{p,\theta }^\Omega $$ of periodic functions of many variables

We obtain exact order estimates for the best M -term trigonometric approximations of the classes B_p,q^W B_{p,\theta }^\Omega of periodic functions of many variables in the space L _q .     

Nonlinear interpolation of functions of very many variables

The difficulty is first shown in the nonlinear interpolation of functions defined in a space of very many dimensions. There is a method using a sampling technique [J. ACM, Vol. 17, pp. 420–425, July 1970] that works fairly well in the regime ofk~10?20 (k=number of dimensions). The sampling errors, however, increase exponentially with increasingk, so that fork greater than the above values the computation is no more feasible. This is due to the subtraction between two large sums of about the same magnitude, each of which suffers stochastic fluctuations accompanying the samplings. To avoid this difficulty, a “pairwise” sampling method is devised where one draws two samples at a time when required, one from each of these two sums of terms. With the use of this technique, standard errors are reduced by orders of magnitude. Some details of algorithm are given together with typical computed examples.     

Best bilinear approximations of the classes $ S_{p,\theta }^\Omega B $ of periodic functions of many variables

We obtain exact-order estimates for the best bilinear approximations of the classes S_p,q^W B S_{p,\theta }^\Omega B of periodic functions of many variables in the space L _q under certain restrictions on the parameters p, q, and θ.     

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.     

Widths and best approximations for classes of convolutions of periodic functions

We establish exact lower bounds for the Kolmogorov widths in the metrics ofC andL for classes of functions with high smoothness; the elements of these classes are sourcewise-representable as convolutions with generating kernels that can increase oscillations. We calculate the exact values of the best approximations of such classes by trigonometric polynomials. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 674–687, May, 1999.     

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.     

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.