Related papers: Approximation in the mean by rational functions

For $1\le t < \infty$, a compact subset $K\subset\mathbb C$, and a finite positive measure $\mu$ supported on $K$, $R^t(K, \mu)$ denotes the closure in $L^t(\mu)$ of rational functions with poles off $K$. Conway and Yang (2019) introduced…

For $1 \le t < \infty ,$ a compact subset $K$ of the complex plane $\mathbb C,$ and a finite positive measure $\mu$ supported on $K,$ $R^t(K, \mu)$ denotes the closure in $L^t (\mu )$ of rational functions with poles off $K.$ The paper…

In 1991, J. Thomson obtained celebrated structural results for $P^t(\mu).$ Later, J. Brennan (2008) generalized Thomson's theorem to $R^t(K,\mu)$ when the diameters of the components of $\mathbb C\setminus K$ are bounded below. The results…

For a compact set $K\subset \mathbb C,$ a finite positive Borel measure $\mu$ on $K,$ and $1 \le t < \i,$ let $\text{Rat}(K)$ be the set of rational functions with poles off $K$ and let $R^t(K, \mu)$ be the closure of $\text{Rat}(K)$ in…

For a compact subset $K\subset \mathbb C$ and a positive finite Borel measure $\mu$ supported on $K,$ let $\text{Rat}(K)$ denote the space of rational functions with poles off $K,$ let $R^\infty (K,\mu)$ be the weak-star closure of…

Let $K$ be a compact subset in the complex plane and let $A(K)$ be the uniform closure of the functions continuous on $K$ and analytic on $K^{\circ}$. Let $\mu$ be a positive finite measure with its support contained in $K$. For $1 \leq q <…

Let $K$ be a compact subset of the complex plane $\mathbb C.$ Let $P(K)$ and $R(K)$ be the closures in $C(K)$ of analytic polynomials and rational functions with poles off $K,$ respectively. Let $A(K) \subset C(K)$ be the algebra of…

For $1 \le t < \infty ,$ a compact subset $K$ of the complex plane $\mathbb C,$ and a finite positive measure $\mu$ supported on $K,$ $R^t(K, \mu)$ denotes the closure in $L^t (\mu )$ of rational functions with poles off $K$. Let $\Omega$…

In 1991, J. Thomson obtained a celebrated decomposition theorem for $P^t(\mu),$ the closed subspace of $L^t(\mu)$ spanned by the analytic polynomials, when $1 \le t < \i.$ In 2008, J. Brennan \cite{b08} generalized Thomson's theorem to…

For $K\subset \mathbb C$ a compact subset and $\mu$ a positive finite Bore1 measure supported on $K,$ let $R^\infty (K,\mu)$ be the weak-star closure in $L^\infty (\mu)$ of rational functions with poles off $K.$ We show that if $R^\infty…

Let $A(K)$ be the algebra of continuous functions on a compact set $K\subset\mathbb C$ which are analytic on the interior of $K$, and $R(K)$ the closure (with the uniform convergence on $K$) of the functions that are analytic on a…

For a compact subset $K$ of the complex plane $\mathbb C,$ let $C(K)$ denote the algebra of continuous functions on $K$. For an open subset $U \subset K,$ let $A(K,U) \subset C(K)$ be the algebra of functions that are analytic in $U.$ We…

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform…

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue $L^2$-space to the existence of analytic bounded point evaluations. Analogues to the complex plane…

For a positive finite Borel measure $\mu$ compactly supported in the complex plane, the space $\mathcal{P}^2(\mu)$ is the closure of the analytic polynomials in the Lebesgue space $L^2(\mu)$. According to Thomson's famous result, any space…

We consider self-similar measures $\mu $ with support in the interval $0\leq x\leq 1$ which have the analytic functions $\left\{e^{i2\pi nx}:n=0,1,2,... \right\} $ span a dense subspace in $L^{2}(\mu) $. Depending on the fractal dimension…

Let $K$ be a non-polar compact subset of $\mathbb{R}$ and $\mu_K$ denote the equilibrium measure of $K$. Furthermore, let $P_n\left(\cdot, \mu_K\right)$ be the $n$-th monic orthogonal polynomial for $\mu_K$. It is shown that…

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$…

We study the classical problem of identifying the structure of $P^2(\mu)$, the closure of analytic polynomials in the Lebesgue space $L^2(\mu)$ of a compactly supported Borel measure $\mu$ living in the complex plane. In his influential…

We use weighted polynomial approximation to prove the existence of a compact set K with non-empty interior and a function f is dense in the space A(K) of all continuous functions on K that are holomorphic in the interior of K, endowed with…