For the finite weighted Hilbert transform we consider two different product integration rules, the VP rule and the L-rule, based on the same nodes and obtained by approximating the density function with filtered de la Vallée Poussin and classical Lagrange interpolation polynomials, respectively. The L-rule is well known and widely studied.

The aim of this talk is to show how classical approximation tools such as Lagrange interpolation and more generally de la Vallée Poussin type interpolation, both of them based on Chebyshev zeros of first kind, can be fruitfully applied for resizing an arbitrary digital image. By means of such operators, we get image scaling methods running for any scale factor or desired size, in both downscaling and upscaling. The performance of such interpolation methods is discussed by several numerical experiments and some theoretical estimates of the mean squared error.

According to the European Charter for Researchers «all researchers should ensure [...] that the results of their research are disseminated and exploited, e.g. communicated, transferred into other research settings or, if appropriate, commercialised ...». Therefore, it's part of the researchers' mission to raise the general public awareness with respect to science. This need is further emphasized by a survey of Eurobarometer 2010: society is strongly interested in science but, at the same time, is often scared by the risks connected with new technologies.