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Cyclicity in Dirichlet Spaces
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 62 / Issue 2 / June 2019
- Published online by Cambridge University Press:
- 11 January 2019, pp. 247-257
- Print publication:
- June 2019
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- Article
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Let
$\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and 
${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset 
$E$ of the unit circle with zero 
$c_{\unicode[STIX]{x1D707}}$-capacity, there exists a function 
$f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that 
$f$ is cyclic (i.e., 
$\{pf:p\text{ is a polynomial}\}$ is dense in 
${\mathcal{D}}(\unicode[STIX]{x1D707})$), 
$f$ vanishes on 
$E$, and 
$f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in 
${\mathcal{D}}(\unicode[STIX]{x1D707})$.
