Saturation theorems for diseretized linear operators

There is a well developed theory of saturation for positive convolution operators on C*, the space of 2n-periodic and continuous functions. Up until recently, this was suitable to handle almost all the important approximation processes on C*. Now, however, some new and interesting sequences of operators have been obtained for C* by discretizing convolution operators [1], [5]. Such a discretized operator L has a simple form because the value of L ( f ) depends only on a finite number of values of f . Since these operators are not given by convolution, the existing saturation theorems do not apply to determine their saturation properties. Our interest in this note is to prove a saturation theorem for positive operators that map C* to C*, with no additional structure assumptions on the operators (such as convolution). Thus, this saturation theorem will determine the saturation properties of the discretized convolution operators, provided the other hypotheses of the saturation theorem are satisfied. Our theorem may be considered either as an extension of TURECKI]['S saturation theorem for convolution operators [2, p. 69] or as the trigonometric analogue of MOHLBACH'S theorem [4] for C [ 1 , 1]. In fact, Tureckii's theorem is a special case of our theorem when one assumes that the operators are given by convolution. Our proof will use a modification of the parabola technique (see [2], Ch. 5), which is used in the proof of Miihlbach's theorem. If {Ln} is a sequence of positive linear operators, we define ( . ~ ( t x ] ] (1) #n(X)=4Ln[sln [ ~ l , x . ,