On the Approximation of Continuous Functions

Alexandru Lupas

Publications de l'Institut Mathématique, _N_S_40 (1986) no. 54, p. 73

Résumé

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 $\|f-J_nf\|\leq C_0\cdot\omega(f;1/n)$, $n=1,2,\ldots$, 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) | łeq 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) | łeq 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 well-known theorems of Jackson [8], Timan [14] and Teljakovskii [13]. Likewise, some other approximation properties of the operators $ (J_n) $ are investigated.

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Classification : 41A35 41A36