Fourier Coefficients of Continuous Functions on Compact Groups
نویسنده
چکیده
Let G be an infinite compact group with dual object 2 Letting 3C0 be the representation space for a G 2, S2(2) is the set [A = (A°) EÏÏ *('3C„): Mil, = 2„dalr(A°A°') < oo}. For A G S2(2), we show that there is a function/in C(G) such that l|/l|œ « CM!12 and Tr(/(o)/(o)') » Tt(A°A°') for every o G 2. In a 1977 paper [3], K. de Leeuw, Y. Katznelson and J.-P. Kahane proved that every square summable sequence is dominated by the sequence of Fourier coefficients of a continuous function on the circle group, T. As the authors mentioned, this result is true, with the same proof, for any compact abelian group in the role of T and its dual group in place of the integers, Z. This paper answers the same question for a compact nonabelian group. Using appropriate tools, our proof parallels that of [3]. All notation and terminology used here without explicit definition is as in [2]. Let G be an infinite compact group with dual object 2. For each a G 2, let U" be a representation in a and let %a, its representation space, have dimension dc. If <$>(%„) is the space of operators on %a, define || ||2 on £H9C„) by \\A°\\2 = Tt(A°A°')]/2. LetS(E) = no6z6$(3î,,)andletS2(2:)bethesetof^ =(Aa) 6 £(2) satisfying \\Ah=(2da\\A°\\l)]/~<TMFinally, T will designate the compact group Il^j^it/,,), where %(</„) is the group of all unitary operators on Xa. We make use of the following results. (l)Let/(K) = Iad0TT(B°V°)(VE T) be a finite sum. Then /|exp/(K)|^K^exp(||5||2). This statement and its proof are similar to [4, Lemma 2]. (2) Suppose A E S2(2). Then, for almost all V E Y, 2dnTr(A°V°U°(x)) a converges for almost every x E G [4, Lemma 8]. Received by the editors May 24, 1982. 1980 Mathematics Subject Classification. Primary 43A30. 43A77. 60BI5. ' I9K3 American Mathematical Society (XX)2-9939/82/(KMK)-0936/$02.(K) 685 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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