In this paper we obtain various approximation theorems by means of k-positive linear operators defined on the space of all analytic functions on a bounded domain of the complex plane.

In this paper we have studied the Chebyshev and interpolation errors for functions in C(E), the normed algebra of analytic functions on a compact set E of positive transfinite diameter. The (p,q)-order and generalized (p,q)-type have been characterized in terms of these approximation errors. Finally, we have obtained a saturation theorem for f ∈ C(E) which can be extended to an entire function of (p,q)-order 0 or 1 and for entire functions of minimal generalized (p,q)-type.

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.

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.