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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.