Subspaces ofL
p (G) spanned by characters: 0<p<1 |
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Authors: | Joel H Shapiro |
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Institution: | (1) Department of Mathematics, Michigan State University, 48824 East Lansing, MI, USA |
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Abstract: | LetG be an infinite compact abelian group,μ a Borel measure onG with spectrumE, and 0<p<1. We show that ifμ is not absolutely continuous with respect to Haar measure, thenL
E
P
(G), the closure inL
p (G) of theE-trigonometric polynomials, does not have enough continuous linear functionals to separate points. Ifμ is actually singular, thenL
E
p
(G) does not have any nontrivial continuous linear functionals at all. Our methods recover the classical F. and M. Riesz theorem,
and a related several variable result of Bochner; they reveal the existence of small sets of characters that spanL
P (T), where T is the unit circle; and they show that theH
p spaces of the “big disc algebra” have one-dimensional dual. |
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Keywords: | |
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