On the generalized Favard operators (Q1280026)

Given a positive null sequence \(\{\gamma_n\}\), the generalized Favard operators are formally defined for any function \(f\) which is defined on \((-\infty,\infty)\), by \[ G_nf(x):=\sum_{j=-\infty}^\infty f\bigl({j\over n}\bigr)p_{n,j}(x), \] where \[ p_{n,j}(x):={1\over n\gamma_n\sqrt{2\pi}}\exp\bigl(-{(j-nx)^2\over 2n^2\gamma_n^2}\bigr). \] For \(m\geq 0\), the author lets \(w_m(x):=(1+x^{2m})^{-1}\), introduces the weighted norm \(\| f\| _m:=\| fw_m\| \), where the latter is the ordinary sup-norm on \((-\infty,\infty)\), and denotes by \(C_m(R)\) the space of all continuous functions \(f\) such that \(\| f\| _m<\infty\). Now as usual, for \(f\in C_m(R)\), the author sets the second weighted modulus of smoothness of \(f\) to be \(\omega_2(f,t)_m:=\sup_{0<h\leq t} \| \Delta^2_nf\| _m\). However, not as usual, there are functions in \(C_m(R)\) for which \(\omega_2(f,t)_m\not\to 0\), as \(t\to 0\), and this makes it unclear what smoothness it exactly measures. Still for this modulus of smoothness the main result of the paper is that \[ \omega_2(G_nf,t)_m\leq M\{(1+t_0^2)\omega_2(f,t)_m+t^2\| f\| _m\},\quad 0<t\leq t_0. \] There is also an application to estimating the degree of approximation of the ordinary Favard operators to \(f\), measured in a certain weighted Hölder type norm.