On the convergence of multidimensional regular C-fractions with independent variables

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

The criterion of the best approximant for the multivariable functions in the space $$$L_{p_1,...,p_{i-1},1,p_{i+1},...,p_n}$$$

The questions of the characterization of the best approximant for the multivariable functions in the space with mixed integral metric were considered in this article. The criterion of the best approximant in space $$$L_{p_1,...,p_{i-1},1,p_{i+1},...,p_n}$$$ is obtained.

The best polynomial approximations of some classes of analytic functions in the Hardy spaces

Problems of the best polynomial approximation of classes of analytic functions $$$H^m_{p,R}$$$, $$$m\in \mathbb{Z}_+$$$, $$$R \geqslant 1$$$, $$$1 \leqslant p \leqslant \infty$$$, have been investigated in the Hardy spaces $$$H_p$$$. The best linear methods of approximation were constructed on the indicated classes.

Bojanov-Naidenov problem for positive (negative) parts of differentiable functions on the real domain

We solve the extremal problem $$$\| x^{(k)}_{\pm} \|_{L_p[a,b]} \rightarrow \sup$$$, $$$k = 0, 1, ..., r-1$$$, over the set of pairs $$$(x, I)$$$ of functions $$$x\in W^r_{\infty} (\mathbb{R})$$$ and intervals $$$I = [a,b]$$$ with restrictions on the local norm of function $$$x$$$ and the measure of support $$$\mu \{ \mathrm{supp}_{[a,b]} x^{(k)}_{\pm} \}$$$.

On mean approximation of function and its derivatives

Some properties of the functions being integrable on the segment were considered in this article. Estimates for approximation are obtained.

The order of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions

We show that the error of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions comprising $$$N$$$ elements has the order $$$N^{-2}$$$ as $$$N \rightarrow \infty$$$.

On one property of zeros of polynomials of the least $$$(\alpha; \beta)$$$-deviation from zero in weighted space $$$L_p$$$

We prove the property of monotonity of zeros of polynomials of the least $$$(\alpha; \beta)$$$-deviation from zero in the space with integral metrics with weight.

Inequalities of various metrics for the norms $$$\|x\|_{p,\delta} = \sup \bigl\{ \| x \|_{L_p[a,b]} \colon a,b\in \mathbb{R}, b-a\leqslant \delta \bigr\}$$$ of differentiable functions on the real domain

We prove sharp inequalities of various metrics for the norms $$$\| x \|_{p, \delta}$$$ of differentiable functions defined on the real line, trigonometric polynomials and periodic splines.

Lower estimates on the saturation order of approximation of twice continuously differentiable functions by piecewise constants on convex partitions

We consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants. We show that the saturation order of piecewise constant approximation in $$$L_p$$$ norm on convex partitions with $$$N$$$ cells is $$$N^{-2/(d+1)}$$$, where $$$d$$$ is the number of variables.

Pointwise estimates of the best one-sided approximations of classes $$$W^r_{\infty}$$$ for $$$0 < r < 1$$$

The asymptotic pointwise estimation of the best one-sided approximations to the classes $$$W^r_{\infty}$$$, $$$0 < r < 1$$$, has been established.