Balázs-Shepard operators on infinite intervals. II (Q1362107)

The authors study uniform convergence of the so-called Balász-Shepard operators \[ S_n(f,x):= {\sum^n_{k=-n}f(x_k)(x- x_k)^{-2}\over \sum^n_{k=-n}(x- x_k)^{-2}} \] on \(\mathbb{R}\) with respect to a weight \(w\), where for \(k=0,\pm1,\dots,\pm n\) the \(x_k:=\lambda_nk/n\) are equidistributed nodes with \(\lambda_n>0\). As a corollary of their main theorem they find that for a wide class of weights \(w\) and for every \(f\) which is continuous on \(\mathbb{R}\) with \(\lim_{|x|\to\infty} w(x)f(x)= 0\) there exists a sequence \((\lambda_n)\) such that \[ \sup_{x\in\mathbb{R}} |w(x)(f(x)- S_n(f,x))|\to 0\quad (n\to\infty). \] In particular, the result implies that for this class of weights the rational functions are dense in the space of continuous functions satisfying the above growth condition, in contrast to the polynomial case in which the Akhiezer-Babenko condition is necessary for such density.