Estimates for approximations by Fourier sums, best approximations and best orthogonal trigonometric approximations of the classes of (\psi, \beta)-differentiable functions

We obtain the exact-order estimates for approximations by Fourier sums, best approximations and best orthogonal trigonometric approximations in metrics of spaces L_s, 1\leq s<\infty, of classes of 2\pi-periodic functions, whose (\psi,\beta)-derivatives belong to unit ball of the space L_\infty.Comment: 8 page

Uniform approximations by Fourier sums on classes of generalized Poisson integrals

We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2\pi$-periodic functions, representable in the form of convolutions of functions $\varphi$, which belong to unit balls of spaces $L_{p}$, with generalized Poisson kernels. For obtained asymptotic equalities we introduce the estimates of remainder, which are expressed in the explicit form via the parameters of the problem.Comment: 31 page

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

We obtain the estimates for the best approximation in the uniform metric of the classes of $2\pi$-periodic functions whose $(\psi ,\beta)$-derivatives have a given majorant $\omega$ of the modulus of continuity. It is shown that the estimates obtained here are asymptotically exact under some natural conditions on the parameters $\psi ,$ $\omega$ and $\beta$ defining the classe

Exact values of Kolmogorov widths of classes of Poisson integrals

We prove that the Poisson kernel $P_{q,\beta}(t)=\sum\limits_{k=1}^{\infty}q^k\cos(kt-\dfrac{\beta\pi}{2})$, ${q\in(0,1)}$, $\beta\in\mathbb{R}$, satisfies Kushpel's condition $C_{y,2n}$ beginning with a number $n_q$ where $n_q$ is the smallest number $n\geq9$, for which the following inequality is satisfied: $$\dfrac{43}{10(1-q)}q^{\sqrt{n}}+\dfrac{160}{57(n-\sqrt{n})}\; \dfrac{q}{(1-q)^2}\leq (\dfrac{1}{2}+\dfrac{2q}{(1+q^2)(1-q)})(\dfrac{1-q}{1+q})^{\frac {4}{1-q^2}}.$$ As a consequence, for all $n\geq n_q$ we obtain lower bounds for Kolmogorov widths in the space $C$ of classes $C_{\beta,\infty}^q$ of Poisson integrals of functions that belong to the unit ball in the space $L_\infty$. The obtained estimates coincide with the best uniform approximations by trigonometric polynomials for these classes. As a result, we obtain exact values for widths of classes $C_{\beta,\infty}^q$ and show that subspaces of trigonometric polynomials of order $n-1$ are optimal for widths of dimension $2n$

Approximation of classes of analytic functions by de la Vallee Poussin sums in uniform metric

In this paper asymptotic equalities are found for the least upper bounds of deviations in the uniform metric of de la Vallee Poussin sums on classes of 2\pi-periodic (\psi,\beta)-differentiable functions admitting an analytic continuation into the given strip of the complex plane. As a consequence, asymptotic equalities are obtained on classes of convolutions of periodic functions generated by the Neumann kernel and the polyharmonic Poisson kernel.Comment: Supported in part by the Ukrainian Foundation for Basic Research (project no. 035/001

Order estimations of the best approximations and approximations of the Fourier sums on the classes of infinitely differentiable functions

We obtained order estimations for the best uniform approximations by trigonometric polynomials and approximations by Fourier sums of classes of $2\pi$-periodic continuous functions, which $(\psi,\beta)$-derivatives $f_{\beta}^{\psi}$ belong to unit balls of spaces $L_{p}, 1\leq p<\infty$ in case at consequences $\psi(k)$ decrease to nought faster than any power function. We also established the analogical estimations in $L_{s}$-metric, $1<s\leq\infty$, for classes of the summable $(\psi,\beta)$-differentiable functions, such that $\parallel f_{\beta}^{\psi}\parallel_{1}\leq1$.Comment: 22 pages, in Ukrainia

Order estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in uniform metric

We obtain exact for order estimates of best uniform approximations and uniform approximations by Fourier sums of classes of convolutions the periodic functions belong to unit balls of spaces $L_{p}, \ {1\leq p<\infty}$, with generating kernel $\Psi_{\beta}$, whose absolute values of Fourier coefficients $\psi(k)$ are such that $\sum\limits_{k=1}^{\infty}\psi^{p'}(k)k^{p'-2}<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$, and product $\psi(n)n^{\frac{1}{p}}$ can't tend to nought faster than power functions.Comment: 20 pages, in Ukrainia

Order estimates of the best orthogonal trigonometric approximations of classes of convolutions of periodic functions of not high smoothness

We obtain order estimates for the best uniform orthogonal trigonometric approximations of $2\pi$-periodic functions, whose $(\psi,\beta)$-derivatives belong to unit balls of spaces $L_{p}, \ 1\leq p<\infty$, in case at consequences $\psi(k)$ are that product $\psi(n)n^{\frac{1}{p}}$ can tend to zero slower than any power function and $\sum\limits_{k=1}^{\infty}\psi^{p'}(k)k^{p'-2}<\infty$ when $1<p<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$ and $\sum\limits_{k=1}^{\infty}\psi(k)<\infty$ when $p=1$. We also establish the analogical estimates in $L_{s}$-metric, $1< s\leq \infty$, for classes of the summable $(\psi,\beta)$-differentiable functions, such that $\parallel f_{\beta}^{\psi}\parallel_{1}\leq1$.Comment: 21 pages, in Ukrainia

Estimates of best mm-term trigonometric approximation of classes of analytic functions

In metric of spaces $L_{s}, \ 1\leq s\leq\infty$, we obtain exact in order estimates of best $m$-term trigonometric approximations of classes of convolutions of periodic functions, that belong to unit all of space $L_{p}, \ 1\leq p\leq\infty$, with generated kernel$\Psi_{\beta}(t)=\sum\limits_{k=1}^{\infty}\psi(k)\cos(kt-\frac{\beta\pi}{2})$,$\beta\in \mathbb{R}$, whose coefficients$\psi(k)$tend to zero not slower than geometric progression. Obtained estimates coincide in order with approximation by Fourier sums of the given classes of functions in$L_{s}$-metric. This fact allows to write down exact order estimates of best orthogonal trigonometric approximation and trigonometric widths of given classes.Comment: 7 pages, in Ukrainia

Uniform approximations by Fourier sums on classes of convolutions of periodic functions

We establish asymptotic estimates for exact upper bounds of uniform approximations by Fourier sums on the classes of $2\pi$-periodic functions, which are represented by convolutions of functions $\varphi (\varphi\bot 1)$ from unit ball of the space $L_{1}$ with fixed kernels $\Psi_{\beta}$ of the form $\Psi_{\beta}(t)=\sum\limits_{k=1}^{\infty}\psi(k) \cos\left(kt-\frac{\beta\pi}{2}\right)$, $\sum\limits_{k=1}^{\infty}k\psi(k)<\infty$, $\psi(k)\geq 0$, $\beta\in\mathbb{R}$