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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1161–1175 | Cite as

Extension of Saturation Theorems for the Sampling Kantorovich Operators

  • Benedetta Bartoccini
  • Danilo CostarelliEmail author
  • Gianluca Vinti
Article

Abstract

In this paper, we extend the saturation results for the sampling Kantorovich operators proved in a previous paper, to more general settings. In particular, exploiting certain Voronovskaja-formulas for the well-known generalized sampling series, we are able to extend a previous result from the space of \(C^2\)-functions to the space of \(C^1\)-functions. Further, requiring an additional assumption, we are able to reach a saturation result even in the space of the uniformly continuous and bounded functions. In both the above cases, the assumptions required on the kernels, which define the sampling Kantorovich operators, have been weakened with respect to those assumed previously. On this respect, some examples have been discussed at the end of the paper.

Keywords

Inverse results Sampling Kantorovich series Order of approximation Generalized sampling operators Saturation order 

Mathematics Subject Classification

41A25 41A05 41A30 47A58 

Notes

Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors are partially supported by the “Department of Mathematics and Computer Science” of the University of Perugia (Italy). Moreover, the second author of the paper has been partially supported within the 2017 GNAMPA-INdAM Project “Approssimazione con operatori discreti e problemi di minimo per funzionali del calcolo delle variazioni con applicazioni all’imaging”.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of PerugiaPerugiaItaly

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