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Topological properties of analytically uniform spaces


Authors: C. A. Berenstein and M. A. Dostál
Journal: Trans. Amer. Math. Soc. 154 (1971), 493-513
MSC: Primary 46F05; Secondary 32A30
DOI: https://doi.org/10.1090/S0002-9947-1971-0397393-7
MathSciNet review: 0397393
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Abstract: In the first part of the article we study certain topological properties of analytically uniform spaces (AU-spaces, cf. L. Ehrenpreis, Fourier transforms in several complex variables, Interscience, New York, 1970). In particular we prove that AU-spaces and their duals are always nuclear. From here one can easily obtain some important properties of these spaces, such as the Fourier type representation of elements of a given AU-space, etc.

The second part is devoted to one important example of AU-space which was not investigated in the aforementioned monograph: the scale of Beurling spaces $ {\mathcal{D}_\omega }$ and $ {\mathcal{D}'_\omega }$. We find a simple family of majorants which define the topology of the space $ {\hat{\mathcal{D}}_\omega }$. This shows that the spaces of Beurling distributions are AU-spaces. Moreover, it leads to some interesting consequences and new problems.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0397393-7
Keywords: Analytically uniform spaces, Beurling spaces of distributions
Article copyright: © Copyright 1971 American Mathematical Society

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