DEMONSTRATIO MATHEMATICA Vol. XXXVI No 4 2003 Grzegorz Nowak DIRECT RESULTS FOR GENERALIZED FAVARD-KANTOROVICH AND FAVARD-DURRMEYER OPERATORS IN WEIGHTED FUNCTION SPACES Abstract. We consider the Kantorovich and the Durrmeyer type modifications of the generalized Favard operators and we prove some direct approximation theorems for functions / such that W2 m f 6 L P (R), where 1 < p < oo and u>2 m (x) = (1 + x 2m ) -1 , m G No- 1. Introduction Let 7 = (7n)£Li be a positive sequence convergent to 0. For functions / : i? — R the generalized Favard operators are defined formally by OO £ Fnf{x)= Y1 /(-)pn ,fc(s; 7), ( xeR,neN), fc=—oo where Pn,k{x-, 7 ) = !_exp(--^(£-a:) ) n-y n V2n \ 2% \n J J (see [6]). In the case where 7^ = p/(2n) with a positive constant p, F n be- come the known Favard operators introduced by J. Favard [5] as discrete analogs of the singular Weierstrass integral. Some approximation properties of the classical Favard operators for continuous functions f on R are pre- sented in [1], [2], and for the generalized operators F n f are given e.g. in [6], [8]. For measurable functions / on R we introduce, also formally, the 1991 Mathematics Subject Classification: 41A25. Key words and phrases: Favard-Kantorovich operator, Favard-Durrmeyer operator, direct approximation theorem, weighted function space, weighted modulus of smoothness.