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An investigation of pointwise convergence of sequences {2. _a'f(T-x) :k= 1, 2,* } wheref lies in the space L1([O, 1]) of Lebesgue integrable functions on the unit interval, T is an invertible measure preserving transformation on [0, 1], and the sequence of polynomials { kz-j=1, 2, } is uniformly bounded and pointwise convergent for all z such that JzJ = 1. Spectral properties. An invertible measure preserving transformation T on the unit interval I is known to induce a unitary operator on the space L2(I) of square integrable functions on 1 [6, p. 13]. By the spectral theorem [5, p. 71] there exists a spectral measure E on the Borel subsets of the unit circle C in the complex plane such that for any integer k, Uk=f zkE(dz) in the sense of strong convergence. Let the resolution of the identity Et, t in [0, 2w), be given by E({exp(is):0Os vexp(ijt) I t E({1}) E({exp(it)}) Et= U + j O 27rij 2,r 2 where, for each z in C, E({z})=lim(>ijn__ ziU-j)/(2n+ 1) and the symbol j10 denotes the limit as n tends to infinity of the sum _ Substituting the Fourier series 7 _ eP(i = s, 0 <s < 2s = T, s = 0, on the right-hand side of the identity [1, p. 100] = + sin(1(2j + l)s) (-1)jcos(1(2j + l)s) s = v + 32 2 ---v (24+l (O _ s < 21), and theb thsideswit(2j + 1)3 and then integrating both sides with respect to the spectral measure Received by the editors June 7, 1972 and, in revised form, June 25, 1973. AMS (MOS) subject classifications (1970). Primary 28A65; Secondary 47A35. 1 This research was supported by a grant from the National Research Council of Canada ? American Mathematical Society 1974

Invertible measure preserving transformations and pointwise convergence