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 | References | Similar Articles | Additional Information
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.
- 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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Keywords:
Weighted <IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${L^2}$">-approximation,
complete,
dense
Article copyright:
© Copyright 1972
American Mathematical Society