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Approximation in the mean by polynomials. (English) Zbl 0736.41008

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 \(\Delta_ i\subset\overline{W}_ i\).

MSC:

41A10 Approximation by polynomials
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
30H05 Spaces of bounded analytic functions of one complex variable
47B20 Subnormal operators, hyponormal operators, etc.
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