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Z] ZEUNER, H. [1992] Moment functions and laws of large numbers on hypergroups. Math. Zeitschrift 211, p. 369-407. Universite de Brest, Departement de Mathematiques, 6 Avenue Le Gorgeu BP 809 29285 BREST - FRANCE. From September 1st 1996: Universite de Tours, Departement de Mathematiques, Faculte des Sciences et Techniques, Parc de Grandmont 37200 TOURS - FRANCE About Some Random Fourier Series and Multipliers Theorems on Compact Commutative Hypergroups Marc-Olivier Gebuhrer Abstract The character theory of compact commutative hypergroups is yet far from being weIl understood; as this paper shows, the behaviour of the Plancherel measure is related to some deeper harmonie analysis involving notably Sidon sets.

Remark 4. We may replace 1 - Izl2 by 1 - Izl in the above result because the ratio of the two terms is bounded above and below on D. We are now ready to construct our example. Fix 0:, ß > 0 (to be determined later) and define the function w( ()) on ßD by ()a w(())= { if 0 < () < 1/2 (1_1()1 2ß)1/2 if-1/2<()<0; 1/ v'2 otherwise. Clearly w belongs to LOO and log w is integrable. We therefore may define b to be the out er function in Hoo with Ib(()) 1= w(()). It is easy to check that 10g(1 - IW) is integrable, so by the previously mentioned theorem of de Leeuw and Rudin, b is not an extreme point of the unit ball of Hoo.

Sarason When b is in the unit ball of Hoo but not an extreme point, a theorem of K. de Leeuw and W. 16, p. 343]. We take a to be out er and positive at the origin; this defines a uniquely. The space M(ä) is defined to be the range of Ta on H 2 ; as with H(b), we make M(ä) into a Hilbert space by giving it the range norm IITa11IM(a) = 111112. ) It turns out that every multiplier of H(b) belongs to M(ä) and that the space M(ä) n H OO is the auxiliary space analogous to the space KOO (p) mentioned above.

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