Polynomial approximation in the mean with respect to harmonic measure on crescents

Abstract

For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-slanted-equals s greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> ⩽ </mml:mo> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leqslant s &gt; \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and "nice" crescents <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , this paper gives a necessary condition (Theorem 2.6) and a sufficient condition (Theorem 2.5) for density of the polynomials in the generalized Hardy space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript s Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^s}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These conditions are easily tested and almost equivalent.