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, thenLEP(G), the closure inLp (G) of theE-trigonometric polynomials, does not have enough continuous linear functionals to separate points. Ifμ is actually singular, thenLEp(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 spanLP (T), where T is the unit circle; and they show that theHp spaces of the “big disc algebra” have one-dimensional dual.
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