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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 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Functional calculus and positive-definite functions
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by Colin C. Graham
Trans. Amer. Math. Soc. 231 (1977), 215-231
DOI: https://doi.org/10.1090/S0002-9947-1977-0487285-2

Abstract:

For a LCA group G with dual group Ĝ, let $D(G) = D(\hat G)$ denote the convex (not closed) hull of $\{ \langle x,\gamma \rangle :x \in G,\gamma \in \hat G\}$. The set $D(G)$ is the natural domain for functions that operate by composition from the class, $P{D_1}(\hat G)$, of Fourier-Stieltjes transforms of probability measures on G to $B(\hat G)$, the class of all Fourier-Stieltjes transforms on Ĝ. Little is known about the behavior of F on the boundary of $D(G)$. In §1, we show (1) if F operators from $P{D_1}(G)$ to $B(G)$ and G is compact, then $K(z) = {\lim _{t \to {1^ - }}}F(tz)$ exists for all $z \in D(G)$ and K operates from $P{D_1}(\hat G)$ to $B(\hat G)$; (2) if F operates from $P{D_1}(\hat G)$ to $PD(\hat G) = { \cup _{r > 0}}rP{D_1}(\hat G)$ and G is compact, then K operates from $P{D_1}(\hat G)$ to $PD(\hat G)$, and so also does $F - K$; (3) if $G = {{\mathbf {D}}_q},q \geqslant 2$, and F operates from $P{D_1}(\hat G)$ to $B(\hat G)$, then $F = K$ on $D(G) \cap \{ z:|z| < 1\}$. This third result is shown to be sharp for compact groups of bounded order. In §2, an example is given that fills a gap in the theory of functions operating from $P{D_1}(\hat G)$ to $B(\hat G)$. In §3 we show that most Riesz products and all continuous measures on K-sets have a property that is very useful in proving symbolic calculus theorems. Applications of this are indicated. Some open questions are given in §4.
References
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Bibliographic Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 231 (1977), 215-231
  • MSC: Primary 43A25
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0487285-2
  • MathSciNet review: 0487285