Rate of approximation by modified Gamma-Taylor operators

A. Izgi

Eurasian mathematical journal, Tome 5 (2014) no. 3, pp. 46-57

Résumé

In this paper we consider the following modification of the Gamma operators which were first introduced in [8] (see [17], [18] and [8] respectively) $$ A_n(f; x)=\int_0^\infty K_n(x, t)f(t)dt $$ where $$ K_n(x, t)=\frac{(2n+3)!}{n!(n+2)!}\frac{t^nx^{n+3}}{(x+t)^{2n+4}}, \quad x, t\in(0, \infty), $$ and the following modified Gamma-Taylor operators $$ A_{n,r}(f;x)=\int_0^\infty K_n(x, t)\left(\sum_{i=0}^r\frac{f^{(i)}(t)}{i!}(x-t)^i\right)dt. $$ We establish some approximation properties of these operators. At the end of the paper we also present some graphs allowing to compare the rate of approximation of $f$ by $A_n(f; x)$ and $A_{n,r}(f; x)$ for certain $n$, $r$ and $x$.