×

Rajchman measures on compact groups. (English) Zbl 0673.43005

On a compact group Rajchman measures are Radon measures whose Fourier- Stieltjes transforms have norms \(>\epsilon\) only for finitely many elements of the dual object. The norms on the dual object considered are \(d_{\sigma}^{- 1/p}\| {\hat \mu}\|_{\phi_ p}\), \(1\leq p\leq \infty\). \(\| \cdot \|_{\phi_ p}\) denotes the von Neumann norm. There are essentially only two different classes of Rajchman measures \({\mathcal R}_ p\), one corresponding to the norms for \(p<\infty\) and one to the operator norm \(\| \cdot \|_{\phi_{\infty}}\). It is shown that Rajchman measures are intermediate between continuous and absolutely continuous measures and \(\nu \ll \mu \in {\mathcal R}_ p\Rightarrow \nu \in {\mathcal R}_ p\), which extends theorems of C. F. Dunkl and D. E. Ramirez [Mich. Math. J. 17, 311-319 (1970; Zbl 0188.206), ibid. 19, 65-69 (1972; Zbl 0213.136)], for \({\mathcal R}_{\infty}\). The main theorem characterizes \({\mathcal R}_ p\) by vanishing on a certain class of Borel sets and answers a question of R. Lyons [Ann. Math., II. Ser. 122, 155-170 (1985; Zbl 0583.43006)] how his result on Rajchman measures on l.c. Abelian groups extends to l.c. non Abelian groups for compact groups. A similar characterization holds for Rajchman measures on homogeneous spaces. (Author)
A. Rajchman conjectured (1922) that there exists some class of sets in T (denoted by W) which could be used to characterize those Radon measures \(\mu\) on T of which the Fourier transform \({\hat \mu}\) vanishes at infinity (denote such class by R) in such a way: \(\mu\in R\) iff \(\mu (E)=0\), \(\forall E\in W\). R. Lyons was the first to establish such result for locally compact Abelian groups [Ann. Math., II. Ser. 122, 155- 170 (1985; Zbl 0583.43006)]. This paper establishes such result for compact non-Abelian groups. The author uses \((d_{\sigma}^{-1/p}\| A\|_{\phi_ p)\sigma \in \hat G}\in c_ 0(\hat G)\) to define the class \(R_ p\) of Radon measures \(\mu\) on G, where A is \(\mu\)’s Fourier coefficient operator, and \(\| A\|_{\phi_ p}\) is von Neumann norm of A, \(1\leq p\leq \infty\), and defines the similar classes \(W_ p\) and \(\bar W_ p\) of sets in G, and obtains the corresponding characterization: Theorem 3. \(\mu \in R_ p\) iff \(\mu (E)=0\), \(\forall E\in W_ p\), or equivalently \(\mu (E)=0\), \(\forall E\in \bar W_ p\), \(1\leq p\leq \infty\).
Reviewer: R.Long

MSC:

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
43A05 Measures on groups and semigroups, etc.
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Dunkl, C., Ramirez, D.: Translation in measure algebras and the correspondence to Fourier transforms vanishing at infinity. Mich. Math. J.17, 311-319 (1970) · Zbl 0188.20601 · doi:10.1307/mmj/1029000517
[2] Dunkl, C., Ramirez, D.: Helson sets in compact and locally compact groups. Mich. Math. J.19, 65-69 (1971) · Zbl 0213.13602
[3] Hewitt, E., Ross, K.: Abstract harmonic analysis. II. Berlin Heidelberg New York: Springer 1970 · Zbl 0213.40103
[4] Hewitt, E., Stromberg, K.: Real and abstract analysis. Berlin Heidelberg New York: Springer 1965 · Zbl 0137.03202
[5] Lyons, R.: Characterizations of measures whose Fourier-Stieltjes transforms vanish at infinity. Bull. Am. Math. Soc., New Ser.10, 93-96 (1984) · Zbl 0525.43004 · doi:10.1090/S0273-0979-1984-15198-X
[6] Lyons, R.: Fourier-Stieltjes coefficients and asymptotic distribution modulo 1. Ann Math.122, 155-170 (1985) · Zbl 0583.43006 · doi:10.2307/1971372
[7] Rajchman, A.: Sur l’unicité du développement trigonométrique. Fundam. Math.3, 287-302 (1922) · JFM 48.0304.03
[8] Révész, P.: On a problem of Stainhaus. Acta Math. Acad. Sci. Hung.16, 310-318 (1965)
[9] ?reîder, J.: On the ring of functions with bounded variation (Russian). Usp. Mat. Nauk.24, 224-225 (1948)
[10] Vilenkin, N.: Special functions and the theory of group representations. Vol. 22. Providence: Am. Math. Soc. 1968 · Zbl 0172.18404
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.