Approximation in the mean by polynomials (Q5905392)

Let \(\mu\) be a positive measure with compact support in the complex plane and let \(t\in[1,\infty)\). Denote by \(P^ t(\mu)\) the closure in \(L^ t(\mu)\) of the polynomials in one complex variable. The paper deals with the description of \(P^ t(\mu)\). The main results are the following: There exists a Borel partition \(\{\Delta_ i\}^ \infty_{i=0}\) of \(\hbox{supp}\mu\) such that for each \(i\leq 1\), \(P^ t(\mu\mid\Delta_ i)\) contains no nontrivial characteristic functions and \(P^ t(\mu)=L^ t(\mu\mid\Delta_ 0)\oplus\left(\bigoplus^ \infty_{i=1}P^ t(\mu\mid\Delta_ i)\right)\). If \(W_ i\) is the set of analytic bounded point evaluations for \(P^ t(\mu\mid\Delta_ i)\), \(i\geq 1\), then \(W_ i\) is a simply connected region and \(\Delta_ i\subset\overline{W}_ i\).