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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some analytic varieties in the polydisc and the Müntz-Szasz problem in several variables

Author: Simon Hellerstein
Journal: Trans. Amer. Math. Soc. 158 (1971), 285-292
MSC: Primary 32.44
MathSciNet review: 0285724
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Abstract: For $ 1 \leqq {p_1} < {p_2} < \infty $ and $ n \geqq 2$ it is shown that there exists a sequence of monomials $ \{ \prod _{j = 1}^nS_j^\lambda mj\} $ with $ {\lambda _{mj}} \sim m$ for each $ j = 1, \ldots ,n$ whose linear span is dense in $ {L^{{p_1}}}({I^n})$ but not in $ {L^{{p_2}}}({I^n})$ ($ {I^n}$ is the Cartesian product of $ n$ copies of the closed unit interval $ [0, 1]$). Construction of the examples is via duality, making use of suitable analytic varieties in the polydisc.

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Keywords: Analytic varieties, polydisc, Banach spaces $ {L^p}({I^n})$, space of continuous functions, monomials, spanning set, Blaschke product, sets of uniqueness in $ {H^\infty }({U^n})$, polydisc algebra
Article copyright: © Copyright 1971 American Mathematical Society