PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 40(54), pp. 7383 (1986) 

ON THE APPROXIMATION OF CONTINUOUS FUNCTIONSAlexandru LupasFacultatea de mecanica, 2400 Sibiu, RomaniaAbstract: We construct a sequence $(J_n)$ of linear positive operators defined on the space $C(K)$, $K=[a,b]$, with the properties: a) $J_nf$ ($f\in C(K)$) is a polynomial of degree $\leq n$; b) if $f\in C(K)$ then there exists a positive constant $C_0$ such that $\fJ_nf\\leq C_0\cdot\omega(f;1/n)$, $n=1,2,\ldots$, where $\\cdot\$ is the uniform norm and $\omega(f;\cdot)$ is the modulus of continuity; c) for $f\in C(K)$ there exists a $C_1>0$ such that $$  f(x)  (J_n f)(x)  \leq C_1 \cdot \omega \enskip (f; \Delta_n (x)), \quad x \in K $$ where $$ \Delta_n (x) = \sqrt {(x  a) (b  x)/n} + n^{2}, \quad n = 1, 2, \ldots; $$ d) if $ \Delta_n^{\ast} (x) = \sqrt {(x  a) (b  x)/n} $ and $$ (J^{\ast}_n f) (x) = (J_n f) (x) + {b  x \over b  a} [f(a)  (J_n f)(a)] + {x  a \over b  a} [f(b)  (J_n f)(b)], $$ then for every continuous function $f:[a,b]\to R$ there exists a positive constant $C_2$ such that $$  f(x)  (J^{\ast}_n f)(x)  \leq C_2 \cdot \omega (f; \Delta^{\ast}_n (x)), \quad x \in [a, b], \quad n = 1, 2, \ldots. $$ In this manner are presented constructive proofs of the wellknown theorems of Jackson [8], Timan [14] and Teljakovskii [13]. Likewise, some other approximation properties of the operators $ (J_n) $ are investigated. Classification (MSC2000): 41A35; 41A36 Full text of the article:
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