Generalized sampling operators with derivative samples

Abstract

The generalized sampling operator is able to approximate bounded continuous functions. It is modeled on the sampling expansion for band-limited functions given by the Whittaker-Kotel'nikov-Shannon theorem. During the decades, some variations of this classical theorem have been proposed. One of them (dating back to Jagerman and Fogel and, in a more general form, to Linden and Abramson) takes into consideration also the derivative samples for the reconstruction of bandlimited functions, with a consequent benefit of a larger sampling rate compared to the Whittaker-Kotel'nikov-Shannon theorem. Motivated by this new reconstruction, we modify the generalized sampling operator including the samplings of derivatives up to a generic order to approximate non necessarily band-limited functions. One of the main features of this new operator (which we call an Hermite-type sampling operator) is the faster order of approximation. Besides the convergence and its rate, we study well-posedness, regularity, simultaneous approximation and Voronovskayatype formula. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data