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Transactions of the American Mathematical Society

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

 

 

Functional calculus and positive-definite functions


Author: Colin C. Graham
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
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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:\vert z\vert < 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.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0487285-2
Keywords: Symbolic calculus for measures, Riesz product measures, Fourier-Stieltjes transforms with locally almost singular powers, functional calculus for positive-definite functions, positive-definite functions on LCA groups
Article copyright: © Copyright 1977 American Mathematical Society