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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Two Hilbert spaces in which polynomials are not dense
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by D. J. Newman and D. K. Wohlgelernter PDF
Trans. Amer. Math. Soc. 168 (1972), 67-72 Request permission

Abstract:

Let $S$ be the Hilbert space of entire functions $f(z)$ such that $||f(z)|{|^2} = \iint {|f(z){|^2}}dm(z)$, where $m$ is a positive measure defined on the Borel sets of the complex plane. Two Hilbert spaces are constructed in which polynomials are not dense. In the second example, our space is one which contains all exponentials and yet in which the exponentials are not complete. This is a somewhat surprising result since the exponentials are always complete on the real line.
References
  • Juan Horv├íth, Approximation and quasi-analytic functions, Univ. Madrid. Publ. Sec. Mat. Fac. Ci. I. 1956 (1956), no.┬á1, 93 (Spanish). MR 0081359
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 168 (1972), 67-72
  • MSC: Primary 30A82
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0294655-X
  • MathSciNet review: 0294655