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Transactions of the American Mathematical Society

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

 
 

 

Two Hilbert spaces in which polynomials are not dense


Authors: D. J. Newman and D. K. Wohlgelernter
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
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Abstract: Let $ S$ be the Hilbert space of entire functions $ f(z)$ such that $ \vert\vert f(z)\vert{\vert^2} = \iint {\vert f(z){\vert^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 [Enhancements On Off] (What's this?)

  • [1] Juan Horváth, Approximation and quasi-analytic functions, Univ. Madrid. Publ. Sec. Mat. Fac. Ci. I. 1956 (1956), no. 1, 93 (Spanish). MR 0081359

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0294655-X
Keywords: Weighted $ {L^2}$-approximation, complete, dense
Article copyright: © Copyright 1972 American Mathematical Society

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