Vol. 43, No. 2, 1972

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Applications of random Fourier series over compact groups to Fourier multipliers

Alessandro Figà-Talamanca and John Frederick Price

Vol. 43 (1972), No. 2, 531–541

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.

Mathematical Subject Classification 2000
Primary: 43A22
Secondary: 42A36
Received: 3 August 1971
Published: 1 November 1972
Alessandro Figà-Talamanca
John Frederick Price