Let C,, denote the Banach space of all 2r-periodic continuous functions on the real line with the supremum norm II . Jo . Let {L,} be a sequence of bounded linear operators of C,, into itself. Suppose that there exists a sequence {&} of positive numbers converging to zero such that every f in C,, for which (1 L,(f) -fij = ~(4~) is a constant function, and there exists a nonconstant function f0 in C,, such that 11 L,(f,) f0 11 = U(+,). Then the sequence {L,} is said to be saturated with the order { I#+J and the class Y(L:,), consisting of all fin C,, for which jj L,(f) fi’ = 0(J an excellent source for references and a systematic treatment of theorems of saturation can be found in Butzer and Nessel [2] and DeVore [5]. Saturation theory in an arbitrary Banach space setting is treated by Butzer, Nessel, and Trebels [3,4]. Here we are concerned with trigonometric polynomial operators which can be defined as follows. Let (h(n; I?)),,~>~ b e a lower triangular matrix, that is, an infinite real matrix satisfying h(n; k) = 0 whenever k > n. Let f~ C,, , let its Fourier series be