Approximation of almost periodic functions by periodic ones

It is not the purpose of this paper to construct approximations but to establish a class of almost periodic functions which can be approximated, with an arbitrarily prescribed accuracy, by continuous periodic functions uniformly on =(+).     

Approximation of weakly differentiable periodic functions

An asymptotic equality is found for the lower bounds of the best approximations of the classes C , under a condition of slow growth of (·).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 406–412, March, 1990.     

Approximation of measurable periodic functions in measure by step functions

For spaces defined by a function ? of the type of modulus of continuity, we prove direct and inverse Jackson theorems for the approximation by step functions with uniform partition.     

Approximation of almost periodic functions of two variables

In this paper we study the deviations of periodic functions of two variables from the integral mean values and their Fourier transforms. In the class of uniform almost periodic functions of two variables we obtain estimates for the deviation from sums of Marcinkiewicz-Zygmund type.     

Approximation of periodic functions by constants in the metric spaces ϕp(L)

By using the best approximations of functions by constants, we establish necessary conditions for the moduli of continuity of periodic functions in metric spaces with integral metric and find the Young constants of these spaces.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1095–1098, August, 1994.     

Approximation of semigroups and cosine functions in spaces of periodic functions

In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials.     

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 with small smoothness

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces C and L by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (n - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these classes.     

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 periodic functions by trigonometric polynomials in the mean

For arbitrary summation methods we obtain inequalities between upper bounds of deviations in the L metric and corresponding upper bounds in the C metric with respect to a certain class of functions. These inequalities constitute a generalization of known relationships due to S. M. Nikol'skii. We consider the cases wherein these inequalities become exact or asymptotic equalities.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 15–26, July, 1974     

Approximation of infinitely differentiable periodic functions by interpolation trigonometric polynomials

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the classes of periodic infinitely differentiable functions C C whose elements can be represented in the form of convolutions with fixed generating kernels. We obtain asymptotic equalities for upper bounds of approximations by interpolation trigonometric polynomials on the classes C _, and C H .Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 495–505, April, 2004.     

Approximation of periodic functions by Fejer-Zygmund means in various metrics

Translated from Matematicheskie Zametki, Vol. 48, No. 4, pp. 48–57, October, 1990.