Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 755-767

A CHARACTERIZATION OF THE CONVERGENCE IN VARIATION FOR THE GENERALIZED SAMPLING SERIES

Laura Angeloni, Danilo Costarelli and Gianluca Vinti

University of Perugia, Department of Mathematics and Computer Science
Via Vanvitelli 1, 06123, Perugia, Italy; laura.angeloni 'at' unipg.it

University of Perugia, Department of Mathematics and Computer Science
Via Vanvitelli 1, 06123, Perugia, Italy; danilo.costarelli 'at' unipg.it

University of Perugia, Department of Mathematics and Computer Science
Via Vanvitelli 1, 06123, Perugia, Italy; gianluca.vinti 'at' unipg.it

Abstract. In this paper, we study the convergence in variation for the generalized sampling operators based upon averaged-type kernels and we obtain a characterization of absolutely continuous functions. This result is proved exploiting a relation between the first derivative of the above operator acting on f and the sampling Kantorovich series of f'. By such approach, also a variation detracting-type property is established. Finally, examples of averaged kernels are provided, such as the central B-splines of order n (duration limited functions) or other families of kernels generated by the Fejér and the Bochner–Riesz kernels (bandlimited functions).

2010 Mathematics Subject Classification: Primary 41A30, 41A05, 47A58, 26A46.

Key words: Convergence in variation, generalized sampling series, sampling-Kantorovich series, averaged kernel, variation detracting-type property, absolutely continuous functions.

Reference to this article: L. Angeloni, D. Costarelli and G. Vinti: Ann. Acad. Sci. Fenn. Math. 43 (2018), 755-767.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4343