Topological properties of analytically uniform spaces

Berenstein, C. A.; Dostál, M. A.

doi:10.1090/S0002-9947-1971-0397393-7

Topological properties of analytically uniform spaces
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by C. A. Berenstein and M. A. Dostál PDF

Trans. Amer. Math. Soc. 154 (1971), 493-513 Request permission

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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