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Random Fourier series on compact Abelian hypergroups. (English) Zbl 0576.43004

Problems concerning random Fourier series are studied in the generality of a compact metrizable abelian hypergroup K whose dual \(\hat K\) is also a hypergroup. The space of almost surely continuous functions on K is investigated. In particular, it is shown that Dudley’s entropy condition is sufficient for almost sure continuity. Many results on almost sure membership in \(L^ p\), \(p<\infty\) are generalized to the setting of compact hypergroups.
A condition on \(\hat K=\{\psi_ 1,\psi_ 2,...\}\) is given that implies the de Leeuw-\(Kahane\)-\(Katznelson\) phenomenon: whenever \(\sum^{\infty}_{n=1}| b_ n|^ 2 \hat m(\psi_ n)<\infty\) \((\hat m\) the invariant measure on \(\hat K)\) there exists \(f\in C(K)\) such that \(| \hat f(\psi_ n)| \geq b_ n\) for all n. Further multiplier interpretations of the foregoing material are offered. Finally, two examples are given illustrating the general theory and its limitations.
Reviewer: R.Lasser

MSC:

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
60G17 Sample path properties
60G50 Sums of independent random variables; random walks
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