Integral-type operators on continuous function spaces on the real line (Q935066)

Let \( \alpha , \beta \in C (\mathbb R) \) be two fixed functions with \( \alpha (x) \geq 0 \) for all \( x \in \mathbb R \). Further, let \( E(\mathbb R)\) be the space of all functions \( f \in C (\mathbb R) \) such that \[ \int^\infty_{-\infty} | f(ay +b) | e^{-y^2/2}\,dy < \infty \] for \( a \geq 0 \) and \(b \in \mathbb R\). In the paper, linear positive operators \( G_n\) of the form \[ G_n (f) (x) := {1\over \sqrt{2\pi}}\int^\infty_{-\infty} f\Bigg(\sqrt{{2\alpha (x)}\over n} y + x + {\beta(x)\over n}\Bigg) e^{-y^2/2}\,dy \] are studied for \( f \in E (\mathbb R)\) and \( x \in \mathbb R\). In the special case \( \alpha \equiv 1\) and \( \beta \equiv 0\), \(G_n \) ist the \(n\)th Gauß-Weierstraß convolution operator. Approximation properties of \( G_n (f) \) are studied, and an asymptotic formula is proved that relates \( G_n ((1+{\gamma \over n})f) \) (with a bounded continuous function \(\gamma \)) to the differential operator \( Lu := \alpha u'' + \beta u' + \gamma u.\) Shape-preserving and regularity properties are also investigated.