Rate of pointwise convergence of a new kind of gamma operators for functions of bounded variation. (English) Zbl 1176.41022
Summary: We investigate the behavior of the operators \(L_n(f,x)\), defined as
\[ L_n(f;x)= \frac{(2n+3)!x^{n+3}}{n!(n+2)!} \int_0^\infty \frac{t^n}{(x+t)^{2n+4}} f(t)\,dt, \quad x>0, \]
and give an estimate of the rate of pointwise convergence of these operators on a Lebesgue point of bounded variation function \(f\) defined on the interval \((0,\infty)\). We use analysis instead of probability methods to obtain the rate of pointwise convergence. This type of study is different from the earlier studies on such a type of operator.
\[ L_n(f;x)= \frac{(2n+3)!x^{n+3}}{n!(n+2)!} \int_0^\infty \frac{t^n}{(x+t)^{2n+4}} f(t)\,dt, \quad x>0, \]
and give an estimate of the rate of pointwise convergence of these operators on a Lebesgue point of bounded variation function \(f\) defined on the interval \((0,\infty)\). We use analysis instead of probability methods to obtain the rate of pointwise convergence. This type of study is different from the earlier studies on such a type of operator.
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