On the best approximation of classes of convolutions of periodic functions by trigonometric polynomials

We present sufficient conditions for kernels to belong to the classN _n ^* . In certain cases, this enables us to find exact values of the best approximations of classes of convolutions by trigonometric polynomials in the metrics ofC andL.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1261–1265, September, 1995.     

On the best approximation by trigonometric polynomials on convolution classes of analytic periodic functions

For a continuous 2π-periodic real-valued function K(t), whose amplitudes decrease as a geometric progression with a denominator q ∈ (0, 1) starting from a given number n ∈ ?, we find sharp upper bounds for q ensuring that K(t) satisfies the Nagy condition N* _n .     

On asymptotically exact estimates for the approximation of certain classes of functions by algebraic polynomials

We present a survey of results obtained for the last decade in the field of approximation of specific functions and classes of functions by algebraic polynomials in the spaces C and L ₁ and approximation with regard for the location of a point on an interval.     

On the best linear approximation methods and the widths of certain classes of analytic functions

We discuss the best linear approximation methods in the Hardy spaceH _q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained. Translated fromMatermaticheskie Zametki, Vol. 65, No. 2, pp. 186–193, February, 1999.     

Best approximation by splines on classes of periodic functions in the metric of L

We have obtained the exact value of the upper bound on the best approximations in the metric of L on the classes W^rH of functionsf C ₂ ^r for which ¦f ^(r) (x)-f ^(r) (x)) ¦ <(¦ x-xf) [ (t) is the upwards-convex modulus of continuity] by subspaces of r-th order polynomial splines of defect 1 with respect to the partitioning k/n.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 655–664, November, 1976.     

On uniqueness of the polynomial of best approximation of the function cos kx by trigonometric polynomials in the L metric

In this paper we clarify a problem concerning uniqueness of the polynomial which best approximates cos kx in the L metric with respect to a trigonometric system of order n in which cos kx is absent. We prove uniqueness in the case n=(2l +l)k. In the remaining cases there is no uniqueness. An analogous problem in the C metric is solved and the relationship between n and k in the case of uniqueness ia distinguished from the conditions in the L metric.     

On polynomials of best approximation

In the space of continuous functions defined on the ball from ?ⁿ, n ≥ 2, we study the set of algebraic polynomials of least deviation from zero. Its width and dimension are estimated.     

Best approximation by holomorphic functions. Application to the best polynomial approximation of classes of holomorphic functions

We establish necessary and sufficient conditions under which a real-valued function from , 1 ≤ p < ∞, is badly approximable by the Hardy subspace H _p ⁰ := {ƒ ∈ H _p : ƒ(0) = 0}. In a number of cases, we obtain the exact values of the best approximations in the mean of functions holomorphic in the unit disk by functions holomorphic outside this disk. We use the obtained results for finding the exact values of the best polynomial approximations and n-widths of some classes of holomorphic functions. We establish necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1047–1067, August, 2007.     

On the approximation of functions from Weyl-Nagy classes by Zygmund sums in theL q metric

In the integral metric, estimates exact by order are found for deviations of the Zygmund linear means from functions, which belong to Weyl-Nagy classes. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 268–270, February, 1999.     

On the best approximation of vector valued functions by polynomials with coefficients in vector spaces

In this paper, we prove some basic results concerning the best approximation of vector-valued functions by algebraic and trigonometric polynomials with coefficients in normed spaces, called generalized polynomials. Thus we obtain direct and inverse theorems for the best approximation by generalized polynomials and results concerning the existence (and uniqueness) of best approximation generalized polynomials. This paper was written during the 2005 Spring Semester when the second author (S.G. Gal) was a Visiting Professor at the Department of Mathematical Sciences, The University of Memphis, TN, USA. Mathematics Subject Classification (2000) 41A65, 41A17, 41A27