Lebesgue functions of rational interpolations of non-band-limited functions
Margit Pap, Ákos Pilgermajer · 2017
This paper concentrates on the Lebesgue functions of rational interpolation of non-band-limited continuous time signals. Approximation based on sampling and interpolation are cornerstones of applied mathematics. In the last years rational interpolations has been in the focus of the investigations, because they have better approximation properties then the polynomial interpolations. The Whittaker-Kotelnikov-Shannon sampling theorem is for band-limited signals and requires the a priori knowledge of the band-width. In [1], [2] new rational interpolation operators were developed for the transfer function of non-band-limited signals, which can be used also in cases when the band-width is not known a priori. The construction of these operators is based on the discrete orthogonality of the Malmquist-Takenaka systems. Combining these interpolations one can give exact interpolation on the real line for a large class of rational functions among them for the Runge test function. Our aim is to study the properties of the Lebesgue function of these rational interpolation operators.