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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
DOI: https://doi.org/10.1090/S0002-9947-1971-0285724-8
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 <!– MATH ${L^p}({I^n})$ –> <IMG WIDTH="65" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img12.gif" ALT="${L^p}({I^n})$">, space of continuous functions, monomials, spanning set, Blaschke product, sets of uniqueness in <!– MATH ${H^\infty }({U^n})$ –> <IMG WIDTH="83" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img11.gif" ALT="${H^\infty }({U^n})$">, polydisc algebra
Article copyright: © Copyright 1971 American Mathematical Society