Bounded and continuous random Fourier series on non- commutative groups

Abstract

The purpose of this note is to show that while I holds, when appropriately interpreted, for Fourier series on arbitrary compact groups, II does not hold in general if the group is noncommutative. The fact that a generalization of I holds for arbitrary groups follows easily from Billard's original proof and some results of Kahane on random series in Banach spaces [4, Theoreme 2] and [5, Chapter II, Theorem 1]. With the help of another result of Kahane [4, Theoreme 3] and [5, Chapter II, Theorem 5] one can easily show that II holds for Fourier series defined on an arbitrary commutative compact group. Our example in ?3 shows that II can fail at least for some noncommutative groups. One should notice that II is a stronger result than I, in fact an application of Fubini's theorem yields that if II is true, the conclusion also holds for random Fourier series of the type