tailieunhanh - On some properties of Szasz-Mirakyan operators in Holder spaces

We study some properties of modified Szasz-Mirakyan operators in Holder exponential weighted spaces. We give theorems on the degree of approximation of functions by these operators. | Turk J Math 27 (2003) , 435 – 446. ¨ ITAK ˙ c TUB On Some Properties of Szasz-Mirakyan Operators in H¨ older Spaces L. Rempulska, Z. Walczak Abstract We study some properties of modified Szasz-Mirakyan operators in H¨ older exponential weighted spaces. We give theorems on the degree of approximation of functions by these operators. Key Words: Szasz-Mirakyan operator, H¨ older space, modulus of smoothness, degree of approximation. 1. Introduction . Paper [1] examined approximation properties of Szasz-Mirakyan operators Sn (f; x) := ∞ X k=0 k , pk (nx)f n (1) x ∈ R0 = [0, +∞), n ∈ N := {1, 2, · · ·}, where pk (t) := e−t tk k! for t ∈ R0 , k ∈ N0 := N ∪ {0}, (2) in exponential weighted spaces Cq . The space Cq , with a fixed q > 0, is related with the weighted function vq (x) := e−qx , x ∈ R0 , and Cq is the set of all real-valued functions f . Subject Classification: 41A36 435 REMPULSKA, WALCZAK continuous on R0 for which vq f is uniformly continuous and bounded on R0 . The norm in Cq is defined by kfkq ≡ kf (·) kq := sup vq (x)|f(x)|. (3) x∈R0 It is obvious that Cq ⊂ Cp if 0 q > 0 and n > q/ ln(p/q). Recently in many papers were introduced various modifications of operators Sn (see . [3, 4, 5, 8, 10]). In the paper [8] were introduced for f ∈ Cq , q > 0, the following modified SzaszMirakyan operators Sn,q (f; x) := ∞ X pk (nx)f k=0 k n+q x ∈ R0 , n ∈ N, , (4) where pk (·) is defined by (2). Also in [8] was proved that, for every n ∈ N and q > 0, Sn,q is positive linear operator from the space Cq into Cq and kSn,q (f; ·)kq ≤ kfkq , n ∈ N, (5) for every f ∈ Cq . Moreover, in [8] were given approximation theorems for f ∈ Cq and Sn,q (f). . The purpose of this paper is the examination of approximation properties of older spaces related with exponential weighted space Cq , q > 0. operators Sn,q in H¨ Approximation of 2π-periodic functions in the H¨ older spaces first was considered by S. Pr¨ ossdorf and J. Prestin in