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.     

Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables

We obtain exact order estimates for the best M-term trigonometric approximations of the Besov classes B _∞,θ ^r in the space L _q . We also determine the exact orders of the best bilinear approximations of the classes of functions of 2d variables generated by functions of d variables from the classes B _p,θ ^r with the use of translation of arguments.     

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.     

Best approximations of functions of several variables by sums of functions of fewer variables

The minimum of the mean-square deviation of approximations of functions of several independent variables by sums of functions of fewer variables in a multidimensional parallelepiped is investigated. The approximating function yielding the minimum mean-square deviation is obtained and this minimum deviation is calculated.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 217–226, February, 1969.The author wishes to thank M. D. Millionshchikov and V. A. Khodakov for their useful suggestions concerning this work.     

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 .     

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.     

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.     

Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes B_pq^sm(T^k) with respect to the multiple trigonometric system T^(k) in the metric of L_r(T^k) for a number of relations between the parameters s, p, q, r, and m (s = (s₁,..., s_n) ∈ R₊ⁿ, 1 ≤ p, q, r ≤ ∞, m = (m₁,..., m_n) ∈ Nⁿ, k = m₁ +... + m_n). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates.     

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 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.     

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.     

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 cubature formulas for some classes of continuous functions of two variables

The exact value of the error of a cubature formula is determined for some classes of continuous functions of two variables defined by strictly monotone moduli of continuity.