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