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 interpolation polynomials in L1

Asymptotically precise estimates are obtained for the deviation, in the L₁-norm, of interpolation polynomials with equally-spaced nodes from certain classes of functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 781–786, June, 1990.     

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 properties of a new type Bernstein-Stancu polynomials of one and two variables

A new generalization of Bernstein-Stancu type polynomials for one and two variables are constructed and the theorems on convergence and the degree of convergence are established. In addition some numerical examples, corresponding to obtaining results are given.     

Concerning the order of approximation of periodic continuous functions by trigonometric interpolation polynomials

Letx _kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR _n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R _n(x_kn)=f(x_kn),R _n ^(j)(x_kn)=0,j=1,2,4,k=0,1…,n−1. We setw ₂(t,f) as the second modulus of continuity off(x). Then we prove that |R _n(x)-f(x)|=0(nw₂(1/nf)). We also examine the question of lower estimate of ‖R _n-f‖. This generalizes an earlier work of the author.     

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 trigonometric functions by trigonometric polynomials with interpolation

The paper studies the approximation order of periodic functions by trigonometric polynomials with interpolation in arbitrary set of nodes. A method of construction of Hermite interpolation polynomials is pointed out.     

Approximation of continuous functions by trigonometric polynomials almost everywhere

We consider the problem of the rate of approximation of continuous 2π-periodic functions of class W^rH[ω]C by trigonometric polynomials of order n on sets of total measure. We prove that when r≥0,ω(δ)δ ^?1 → ∞ (δ → 0) there exists a function f ε W^rH[ω]C such thatf ε W^rH[ω]C and for any sequence {t_{n n=1} ^∞ we have almost everywhere on [0, 2π] $\begin{array}{l} \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0, \\ \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {\tilde f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0. \\ \end{array}$      

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