The pointwise complete asymptotic expansion for the approximation of Lipschitz functions by Hermite-Fejer interpolation polynomials

The pointwise complete asymptotic expansion is derived for the approximation of Lipschitz functions by Hermite-Fejér interpolation polynomials based on the Chebyshev polynomials of the first kind.     

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.     

Order of mean approximation by mixed quasi Hermite-Fejer interpolation

In this paper we introduce a new kind of the mixed Hermite-Fejer interpolation with boundary conditions and obtain the mean approximation order. Our results include a new theorem of Varma and Prasad. Besides, we also get some other results about the mean approximation.     

On the approximation of entire functions by trigonometric polynomials

Let a set B have the following properties: if z ∈ B, then z ± 2π ∈ B and the intersection of B with the vertical strip 0 ≤ Re x ≤ π is a closed and bounded set. In this paper we study the approximation of a continuous on B and 2π-periodic function f(z) by trigonometric polynomials T _n (z). We establish the necessary and sufficient conditions for the function f(z) to be entire and specify a formula for calculating its order. In addition, we describe some metric properties of periodic sets in a plane.     

On the approximation of harmonic functions by harmonic polynomials

Siberian Mathematical Journal -     

On the approximation of measurable functions by continuous functions

We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization one can easily obtain results that have appeared recently in the literature, in a more general form. We also examine what happens when the range is a subset of an arbitrary Banach space, and show that this case often reduces to the separable case.     

On the approximation of differentiable functions by Bernstein polynomials and Kantorovich polynomials

E.V. Voronovskaya and S.N. Bernstein established an asymptotic representation for the deviation of functions from Bernstein polynomials under the condition that the function has an even-order derivative. In the present paper, a similar problem is solved in the case when the function has an odd-order derivative. In addition, analogous representations are obtained for the deviations of functions from Kantorovich polynomials.     

On approximation of functions on the half-line by algebraic polynomials

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On the rate of approximation of functions by the Bernstein polynomials

The estimates are refined for the deviations of the Bernstein polynomials from functions at a fixed point and for the remainders in the asymptotic formulas for these deviations for differentiable functions.     

On the nonclassical approximation method for periodic functions by trigonometric polynomials

We study the approximation of functions by linear polynomial means of their Fourier series with a function-multiplier φ that is equal to 1 not only at zero, in contrast with classical methods of summability. The exact order of convergence to zero of the sequence $$ \mathop{\max}\limits_{{x\in \left[ {-\pi, \pi } \right]}}\left| {f(x)-\sum\limits_{{\left| k \right|\leq n}} {\varphi \left( {\frac{{k\pi }}{n}} \right){{\hat{f}}_k}{e^{ikx }}} } \right| $$ ( $ {{\hat{f}}_k} $ Fourier coefficients) as n→∞ is obtained. The answer is given in terms of the values of difference operators of a continuous function f and a special K-functional (step of $ \frac{\pi }{n} $ ). In addition, we obtain not only the sufficient conditions for φ but the necessary ones as well.