Abstract English |
Abstract<br>
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Let μ be a finite positive<br>
measure on the closed disk D¯<br>
in the complex plane, let 1 ≤ t < ∞,<br>
and let Pt(μ)<br>
denote the closure of the analytic polynomials in<br>
Lt(μ). We suppose<br>
that D<br>
is the set of analytic bounded point evaluations for<br>
Pt(μ), and<br>
that Pt(μ)<br>
contains no nontrivial characteristic functions. It is then known that the restriction of<br>
μ to<br>
∂D must be of the form<br>
h|dz|. We prove that every<br>
function f ∈ Pt(μ) has nontangential<br>
limits at h|dz|-almost<br>
every point of ∂D,<br>
and the resulting boundary function agrees with<br>
f as an<br>
element of Lt(h|dz|).<br>
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Our proof combines methods from James E. Thomson’s proof of the existence of bounded point<br>
evaluations for Pt(μ)<br>
whenever Pt(μ)≠Lt(μ)<br>
with Xavier Tolsa’s remarkable recent results on analytic capacity. These methods allow<br>
us to refine Thomson’s results somewhat. In fact, for a general compactly supported<br>
measure ν<br>
in the complex plane we are able to describe locations of bounded point evaluations<br>
for Pt(ν) in<br>
terms of the Cauchy transform of an annihilating measure.<br>
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As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We<br>
show that for 1 < t < ∞ dim<br>
ℳ∕zℳ = 1 for every nonzero<br>
invariant subspace ℳ<br>
of Pt(μ) if and<br>
only if h≠0.<br>
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We also investigate the boundary behaviour of the functions in<br>
Pt(μ) near the<br>
points z ∈ ∂D<br>
where h(z) = 0. In<br>
particular, for 1 < t < ∞<br>
we show that there are interpolating sequences for<br>
Pt(μ)<br>
that accumulate nontangentially almost everywhere on<br>
{z : h(z) = 0}. |