zbMATH — the first resource for mathematics

Approximation by some operators of the Szasz-Mirakjan type in exponential weight spaces. (English) Zbl 0880.41017
Let \(C(R_+,R)\) be the space of real-valued functions continuous on \(R_+ =[0, \infty)\), \(W_p(x)= e^{-px}\) with \(x\in R_+\) and \(p>0\), \(C_p= \{f\in C(R_+,R)\): \(W_pf\) is uniformly continuous and bounded on \(R_+\}\), \(|f|_p =\sup\{W_p(x) |f(x) |: x\in R_+\}\), \(\Delta (f;x,h) =f(x+h)- f(x)\) for \(f\in C_p\) and \(h\geq 0\), \(\omega_p (f;\delta) =\sup \{|\Delta (f;h) |_p: h\in [0,\delta]\}\), for \(\delta\geq 0\). The linear positive operators \(L_n\) and \(U_n\) of the Szász-Mirakjan type in the space \(C_p\) are \(L_n(f;x) =\sum^\infty_{k= 1} a_{n,k} (x)f(2kn^{-1})\), \(U_n(f;x)= \sum^\infty_{k=0} a_{n,k} (x)2^{-1} n \int_{I_{n,k}} f(t) d(t)\), where \(x\in R_+\), \(n\in\mathbb{N} =\{1,2, \dots\}\), \(k\in N_0 =N \cup \{0\}\), \(a_{n,k} (x) =\text{(chnx)}^{-1} (nx)^{2k} ((2k)!)^{-1}\), \(I_{n,k} (x)= \{x\in R_+ :2kn^{-1} \leq x \leq (2k+2) n^{-1}\}\). The aim of Section 1 of this paper is the following theorem: Suppose that \(p>0\), \(r>p\), \(n_0\) is a fixed natural number satisfying \(n_0> p(\ln rp^{-1})^{-1}\), \(f\in C_p\), \(g\in C^1_p =\{f \in C_p: f' \in C_p\}\). There exists a positive constant \(M_{p,r}\) such that I. \(w_r(x) |T_n (f,x) -f(x) |\leq (3+ M_{p,r}) \omega_p (f,({x+1 \over n})^{{1 \over 2}})\), II. \(w_r(x) |T_n(g,x) -g(x) |\leq M_{p, r}|g' |_p ({x+1 \over n})^{1/2}\), where \(T_n(h, \cdot) \in \{L_n (h, \cdot)\), \(U_n (h, \cdot) \}\), \(x\geq 0\), \(n\geq n_0\). In Section 2 of this paper the authors give direct approximation theorems for functions of two variables.
Reviewer: I.Badea (Craiova)

41A36 Approximation by positive operators