Rate of approximation by modified Gamma-Taylor operators

Abstract

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

\( K_n(x, t) = \frac{(2n + 3)!}{n!(n+2)!} \cdot \frac{t^n x^{n+3}}{(x + t)^{2n + 4}} \),   \( x, t \in (0, \infty) \).

and the following modified Gamma-Taylor operators

\( A_{n,r}(f ; x) = \int_0^\infty K_n(x, t) \sum_{i=0}^r \frac{f^{(i)}(t)}{i!} (x - t)^i \, dt \).

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 \( x \).