On rational Abel – Poisson means on a segment and approximations of Markov functions

Approximations on the segment [−1, 1] of Markov functions by Abel – Poisson sums of a rational integral operator of Fourier type associated with the Chebyshev – Markov system of algebraic fractions in the case of a fixed number of geometrically different poles are investigated. An integral representation of approximations and an estimate of uniform approximations are found. Approximations of Markov functions in the case when the measure µ satisfies the conditions suppµ = [1, a], a > 1, dµ(t) = φ(t)dt and φ(t) ≍ (t − 1)α on [1, a], a are studied and estimates of pointwise and uniform approximations and the asymptotic expression of the majorant of uniform approximations are obtained. The optimal values of the parameters at which the majorant has the highest rate of decrease are found. As a corollary, asymptotic estimates of approximations on the segment [−1, 1] are given by the method of rational approximation of some elementary Markov functions under study.

Uniform approximations of the first symmetric elliptic integral in terms of elementary functions

We consider the standard symmetric elliptic integral $$R_F(x,y,z)$$ R F ( x , y , z ) for complex x, y, z. We derive convergent expansions of $$R_F(x,y,z)$$ R F ( x , y , z ) in terms of elementary functions that hold uniformly for one of the three variables x, y or z in closed subsets (possibly unbounded) of $$\mathbb {C}{\setminus }(-\infty ,0]$$ C \ ( - ∞ , 0 ] . The expansions are accompanied by error bounds. The accuracy of the expansions and their uniform features are illustrated by means of some numerical examples.

Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

The Zygmund sums of a function $f\in L_{1}$ are trigonometric polynomials of the form $$Z^{s}_{n-1}(f;t):=\frac{a_{0}}{2}+\sum_{k=1}^{n-1}\Big(1-\big(\frac{k}{n}\big)^{s}\Big) \big(a_{k}(f)\cos kt+b_{k}(f)\sin kt\big), s>0,$$ where $a_{k}(f)$ and $b_{k}(f)$ are the Fourier coefficients of $f$. We establish the exact-order estimates of uniform approximations by the Zygmund sums $Z^{s}_{n-1}$ of $2\pi$-periodic continuous functions from the classes $C^{\psi}_{\beta,p}$. These classes are defined by the convolutions of functions from the unit ball in the space $L_{p}$, $1\leq p<\infty$, with generating fixed kernels $$\Psi_{\beta}(t)\sim\sum_{k=1}^{\infty}\psi(k)\cos\left(kt+\frac{\beta\pi}{2}\right), \Psi_{\beta}\in L_{p'}, \beta\in \mathbb{R}, \frac1p+\frac{1}{p'}=1.$$ We additionally assume that the product $\psi(k)k^{s+1/p}$ is generally monotonically increasing with the rate of some power function, and, besides, for $1< p<\infty$ it holds that $\sum_{k=n}^{\infty}\psi^{p'}(k)k^{p'-2}<\infty$, and for $p=1$ the following condition $\sum_{k=n}^{\infty}\psi(k)<\infty$ is true. It is shown, that under these conditions Zygmund sums $Z^{s}_{n-1}$ and Fejér sums $\sigma_{n-1}=Z^{1}_{n-1}$ realize the order of the best uniform approximations by trigonometric polynomials of these classes, namely for $1<p<\infty$ $${E}_{n}(C^{\psi}_{\beta,p})_{C}\asymp{\cal E}\left(C^{\psi}_{\beta,p}; Z_{n-1}^{s}\right)_{C}\asymp\Big(\sum_{k=n}^{\infty}\psi^{p'}(k)k^{p'-2}\Big)^{1/p'}, \ \frac{1}{p}+\frac{1}{p'}=1,$$ and for $p=1$ $$ {E}_{n}(C^{\psi}_{\beta,1})_{C}\asymp{\cal E}\left(C^{\psi}_{\beta,1}; Z_{n-1}^{s}\right)_{C}\asymp {\left\{{\begin{array}{l l} \sum\limits_{k=n}^{\infty}\psi(k), & \cos \frac{\beta\pi}{2}\neq 0,\\ \psi(n)n, &\cos \frac{\beta\pi}{2}= 0, \end{array}} \right.} $$ where $${E}_{n}(C^{\psi}_{\beta,p})_{C}:=\sup_{f\in C^{\psi}_{\beta,p}}\inf\limits_{t_{n-1}\in\mathcal{T}_{2n-1}}\|f(\cdot)-t_{n-1}(\cdot)\|_{C}, $$ and $\mathcal{T}_{2n-1}$ is the subspace of trigonometric polynomials $t_{n-1}$ of order $n-1$ with real coefficients, $${\cal E}\left(C^{\psi}_{\beta,p}; Z_{n-1}^{s}\right)_{C}:=\mathop{\sup}\limits_{f\in C^{\psi}_{\beta,p}}\|f(\cdot)-Z^{s}_{n-1}(f;\cdot)\|_{C}.$$

Asymptotic estimates for the best uniform approximations of classes of convolution of periodic functions of high smoothness

We find two-sided estimates for the best uniform approximations of classes of convolutions of $2\pi$-periodic functions from a unit ball of the space $L_p, 1 \le p <\infty,$ with fixed kernels such that the moduli of their Fourier coefficients satisfy the condition $\sum\limits_{k=n+1}^\infty\psi(k)<\psi(n).$ In the case of $\sum\limits_{k=n+1}^\infty\psi(k)=o(1)\psi(n),$ the obtained estimates become the asymptotic equalities.

Rational interpolation of the function |x|α by an extended system of Chebyshev – Markov nodes

In this paper, we study the approximations of a function |x|α, α > 0 by interpolation rational Lagrange functions on a segment [–1,1]. The zeros of the even Chebyshev – Markov rational functions and a point x = 0 are chosen as the interpolation nodes. An integral representation of an interpolation remainder and an upper bound for the considered uniform approximations are obtained. Based on them, a detailed study is made:a) the polynomial case. Here, the authors come to the famous asymptotic equality of M. N. Hanzburg;b) at a fixed number of geometrically different poles, the upper estimate is obtained for the corresponding uniform approximations, which improves the well-known result of K. N. Lungu;c) when approximating by general Lagrange rational interpolation functions, the estimate of uniform approximations is found and it is shown that at the ends of the segment [–1,1] it can be improved.The results can be applied in theoretical research and numerical methods.