Let 0 < rn < 1 and wn = e i2 n, n = 1, 2,.... For a function f holomorphic in the open unit disc U, we consider the linear functionals sn defined by the means sn(rn, f) = (1/n) k = 1n f(rnwnk). If 0 < rn {variant} < 1, we prove that f is uniquely determined by sn(rn, f), n = 1, 2,..., and in fact, f can be represented by a polynomial series whose coefficients involve sn(rn, f). The case 0 < rn 1 is also considered. In particular, if rn = 1 for all large n, there exist nontrivial functions f, holomorphic in U and continuous on the closure of U, such that sn(rn, f) = 0 for n = 1, 2,.... 1978.