Abstract
A Gupta-type variant of Shepard operators is introduced and convergence results and pointwise and uniform direct and converse approximation results are given. An application to image compression improving a previous algorithm is also discussed.
Highlights
In the last decades Shepard operators have been object of several papers, thanks to their properties interesting in classical approximation theory and in scattered data interpolation problems
Pointwise and uniform approximation error estimates, converse results, bridge theorems, saturation statements, simultaneous approximation results can be found for example in [1,2,3,4,5,6,7]
The aim of the present paper is to give a positive answer to the above question, introducing a generalization of Gupta-type of Shepard operator depending on a real positive parameter
Summary
In the last decades Shepard operators have been object of several papers, thanks to their properties interesting in classical approximation theory and in scattered data interpolation problems. Pointwise and uniform approximation error estimates, converse results, bridge theorems, saturation statements, simultaneous approximation results can be found for example in [1,2,3,4,5,6,7]. Applications of Shepard operators to scattered data interpolation problems, image compression and CAGD can be found for example in [8,9,10,11,12,13,14,15,16,17]. Convergence results and uniform and pointwise approximation error estimates for such operator are given in Theorems 2.1–2.2 in Sect.
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