On Voronovskaja formula for linear combinations of Mellin-Gauss-Weierstrass operators (Q449463)

For every \(w\geq 1\), the Mellin-Gauss-Weierstrass kernel is defined by the kernel NEWLINE\[NEWLINE K_w(t)=\frac{w}{\sqrt{4\pi}}\exp \left(-\left(\frac{w}{2}\log t\right)^2 \right), \quad t\in \mathbb{R}^+. NEWLINE\]NEWLINE Let NEWLINE\[NEWLINE K_{w,r}(t)= \sum_{j=1}^{r} \alpha_j K_{jw}(t), \quad t\in \mathbb{R}^+, NEWLINE\]NEWLINE where \(\alpha_j\) are nonzero real numbers such that \(\sum_{j=1}^{r} \alpha_j=1\). The authors study the pointwise approximation properties of the operators NEWLINE\[NEWLINE (G_{w,r}f)(s)=\int_{0}^{\infty}K_{w,r}(t)f(st)\frac{dt}{t}. NEWLINE\]NEWLINE For \(f\in C^2\) locally at \(s\), NEWLINE\[NEWLINE\lim_{w\to +\infty} w^2 ((G_{w,1}f)(s)-f(s))=f'(s)s+f''(s)s^2. NEWLINE\]NEWLINE The operators \(G_{w,r}\) provide a better order of pointwise approximation and lead to asymptotic formulae of the type NEWLINE\[NEWLINE\lim_{w\to +\infty} w^\nu((G_{w,r}f)(s)-f(s))=A(f,\nu), NEWLINE\]NEWLINE where \(\nu\in \mathbb{N}\) and \(A(f,\nu)\) is a differential operator containing the derivatives of \(f\) up to order \(\nu\).