The Fourier series of a
function on a compact group can be “randomized” by operating on each of the
Fourier coefficients by independent random unitary operators. In this paper the
theory of random Fourier series is used to prove several new results for a
type of Rudin-Shapiro sequence and for Fourier multipliers. Thus in §2 it is
shown in effect that M(Lp,Lq) ⊆ M(L2,L2) for all p,q ∈ [1,∞] except for
the pair (p,q) = (∞,1), while in §3 the theory of random Fourier series
is used to construct a type of Rudin-Shapiro sequence. This sequence is
then used in §4 to obtain, for compact groups in one case, and compact
Lie groups in another, slightly more restricted versions of several known
families of strict inclusions for Fourier multipliers over compact Abelian
groups.
