Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] J. Korevaar and S. Hellerstein, Discrete sets of uniqueness for bounded holomorphic functions $ f(z,w)$, Proc. Sympos. Pure Math., vol. 11, Amer. Math. Soc., Providence, R. I., 1968, pp. 273-284. MR 38 #3462. MR 0235150 (38:3462)
  • [2] W. Rudin, Function theory in polydiscs, Benjamin, New York, 1969. MR 0255841 (41:501)
  • [3] L. Schwartz, Étude des sommes exponentielles, 2ième éd., Actualités Sci. Indust., no. 959, Hermann, Paris, 1959. MR 21 #5116. MR 0106383 (21:5116)
  • [4] W. Seidel, On the distribution of values of bounded analytic functions, Trans. Amer. Math. Soc. 36 (1934), 201-226. MR 1501738
  • [5] A. Zygmund, Trigonometrical series, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498. MR 0107776 (21:6498)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32.44

Retrieve articles in all journals with MSC: 32.44


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0285724-8
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

American Mathematical Society