Acta Mathematica Academiae Scientiarum Hungaricae Tomits 20 ( 3 ~ ) , (1969), pp, 451--46]. ON A TRIGONOMETRIC CONVOLUTION OPERATOR WITH KERNEL HAVING TWO ZEROS OF SIMPLE MULTIPLICITY By P. L. BUTZER and E. L. STARK (Aachen) Dedicated to Professor G. ALExrrs on the occasion of his 70th birthday, January 5, 1969 1. Introduction If T,(f; x) is a positive (more exactly: non-negative) bounded linear trigono- metric polynomial operator of degree n, then, as P. P. KOROVKIN [8] has shown, the optimal order of approximation of functions fEC2~ by such operators is O(n-2), n~; this order of approximation cannot be improved by supposing f to be arbitrarily smooth. In particular, let the operator 7", be a singular i~tegrat ~t of convolutiorl type defined by T~(f; x~ = (f*p~)(x) = (t/~)ff(x-u)p,,(u)du having an even kernel p,(x)~ C2~. The question then arises whethe r the order of approximation by f ~p, can be improved if the kernel is allowed to be negative for some x. In this respect, P. P. KOROVK1N [10], cf. [9], (investigating the analogous algebraic case, only remarking that all results are valid in the periodic case as well) and inde- pendently P.L. BUTZER--R. J. NESS~L--K. SCHERER [5] have shown that if the (even) kernel p~(x) has 2m changes of sign in the interval (-re, rc] -- the number of changes being independent of// --, then the optimal order of approximation cannot exceed O(n -2~-2) provided such a kernel exists. A. I. KOVALENKO [11] has actually stated a scheme in how to construct such operators f .p, which approxi- mate with order O(//-2m-2) if fCC(2~m+2); (for an algebraic analogon, see G, N. VINOGRADOVA [15]). The purpose of this note is to give an explicit example of a convolution operator approximating with order O(11-4) provided e.g. JcEC (4)2=. Thus, we shall not be content with a scheme but will in fact construct the associated kernel N,_t.r cf. Theorem 1. As the general case in [11] is very complicated, we prefer a direct approach in whmh the choice of the function ~o*(t) = sin a r~t, t C[0, 1], is essential; this function as a special example satisfies the conditions given in [11]. The authors are indebted to KARL SCHERER for various valuable suggestions and discussions. 2. Basic results Let C2~, [{~ (I _<-p< o~) be the space of all 2~t-per~odir ffmctions which are continuous on tlae whole real line or Lebesgue integrabIe to the pth power over (-~r, ~z), respectively. X~ denotes one of these spaces endowed with the usual norm. Cg~ denotes the space of functions fE C2= whose derivatives of order s are continuous. C(a, b) means those functions continuous on (a, b). n, m are natural numbers. Acta Mathematica Ac~demiae Sclentiarum Hungar ~o, r969