By using the properties of convergence and global smoothness preservation of multivariate Weierstrass singular integrals, we establish multivariate complex Carleman type approximation results with rates. Here the approximants fulfill the global smoothness preservation property. Furthermore Mergelyan's theorem for the unit disc is strengthened by proving the global smoothness preservation property.

We consider a certain analog of Cauchy type integral taking values in a three-dimensional harmonic algebra with two-dimensional radical. We establish sufficient conditions for an existence of limiting values of this integral on the curve of integration.

We consider linear difference equations whose coefficients are meromorphic at $\infty$. We characterize the meromorphic equivalence classes of such equations by means of a system of meromorphic invariants. Using an approach inspired by the work of G. D. Birkhoff we show that this system is free.

We show that if a Cantor set E as considered by Garnett in [G2] has positive Hausdorff h-measure for a non-decreasing function h satisfying ∫01 r−3 h(r)2 dr < ∞, then the analytic capacity of E is positive. Our tool will be the Menger three-point curvature and Melnikov’s identity relating it to the Cauchy kernel. We shall also prove some related more general results.

Let $D$ be a Carathéodory domain. For $1\le p\le \infty$, let $L^p\left(D\right)$ be the class of all functions $f$ holomorphic in $D$ such that ${\parallel f\parallel }_{D,p}=[\frac{1}{A}\int {\int }_D^{}{\left|f\left(z\right)\right|}^p{dxdy]}^{1/p}<\infty$, where $A$ is the area of $D$. For $f\in L^p\left(D\right)$, set $$E_n^p\left(f\right)=\underset{t\in {\pi }_n}{inf}{\parallel f-t\parallel }_{D,p};$$${\pi }_n$ consists of all polynomials of degree at most $n$. In this paper we study the growth of an entire function in terms of approximation...

We continue studying the estimation of Bernstein-Walsh type for algebraic polynomials in regions with piecewise smooth boundary.

Let $E={\bigcup }_{n=1}^{\infty }{I}_n$ be the union of infinitely many disjoint closed intervals where $I_n=[a_n$, $b_n]$, $0<a_1<b_1<a_2<b_2<\cdots <b_n<\cdots$, ${lim}_{n\to \infty }{b}_n=\infty .$ Let $\alpha \left(t\right)$ be a nonnegative function and ${\left\{{\lambda }_n\right\}}_{n=1}^{\infty }$ a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system $\left\{t^{{\lambda }_n}{log}^{m_n}t\right\}$ in $C_0\left(E\right)$ is obtained where $C_0\left(E\right)$ is the weighted Banach space consists of complex functions continuous on $E$ with $f\left(t\right){\mathrm{e}}^{-\alpha \left(t\right)}$ vanishing at infinity.