DOI: 10.2478/s12175-010-0011-0 Math. Slovaca 60 (2010), No. 2, 265–278 THEOREMS FOR GENERALIZED FAVARD-KANTOROVICH AND FAVARD-DURRMEYEROPERATORS IN EXPONENTIAL FUNCTION SPACES Grzegorz Nowak* — Aneta Sikorska-Nowak** (Communicated by Michal Zajac ) ABSTRACT. We consider the Kantorovich and the Durrmeyer type modifica- tions of the generalized Favard operators and we prove some direct approxima- tion theorems for functions f such that w σ f ∈ L p (R), where 1 ≤ p ≤∞ and w σ (x) = exp(-σx 2 ), σ> 0. c 2010 Mathematical Institute Slovak Academy of Sciences 1. Introduction Let γ =(γ n ) ∞ n=1 be a positive sequence convergent to zero. For functions f : R → R the generalized Favard operatos are defined formally by F n f (x)= ∞  k=-∞ f (k/n)p n,k (x; γ ), (x ∈ R, n ∈ N), where p n,k (x; γ )= 1 nγ n √ 2π exp  - 1 2γ 2 n  k n - x  2  (see [5]). In the case where γ 2 n = ϑ/(2n) with a positive constant ϑ, F n become the known Favard operators introduced by J. Favard [4] as discrete analogs of the singular Weierstrass integral. Some approximation properties of the clasical Favard operators for continuous functions f on R are presented in [1, 2], and for 2000 Mathematics Subject Classification: Primary 41A25. K e y w o r d s: Favard-Kantorovich operator, Favard-Durrmeyer operator, direct approximation theorem, exponential weight space, weighted modulus of smoothness.