On the Beatson convolution operators in the unit disk (Q1409819)

For functions \(f\) analytic on the open and continuous on the closed unit disc the convolution-type operators \[ (L_{n,r} f):= {1\over \pi} \int^{2\pi}_0 f(ze^{it)} B_{m,r}(t)\,dt,\quad r,\,n\in\mathbb N,\;m= \Biggl[{n\over r}\Biggr]+ 1 \] are defined with Beatson kernels \[ B_{m,r}(t):= {m\over 2\pi} \int^{t+{\pi\over m}}_{t- {\pi\over m}} K_{m,r}(s)\,ds,\quad m\in \mathbb N \] (proved to be bell-shaped), where \(K_{m,r}(s)\) is a classical Jackson kernel known to be not bell-shaped. The author proves the global smoothness preservation of the operators \(L_{n,r}\), their degree of approximation and some further interesting geometric properties.