Approximation in the Mean by Polynomials
Copyright policy )Approximation in the Mean by Polynomials
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 \(\Delta_ i\subset\overline{W}_ i\).
subnormal operator, Approximation by polynomials, analytic functions, Spaces of bounded analytic functions of one complex variable, Subnormal operators, hyponormal operators, etc., Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
subnormal operator, Approximation by polynomials, analytic functions, Spaces of bounded analytic functions of one complex variable, Subnormal operators, hyponormal operators, etc., Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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