New Results on Rational Approximation

Abstract

First asymptotic relations of Voronovskaya-type for rational\ud operators of Shepard-type are shown. A positive answer in some senses\ud to a problem on the pointwise approximation power of linear operators\ud on equidistant nodes posed by Gavrea, Gonska and Kacso is given. Direct and converse results, computational aspects and Gruss-type inequalities\ud are also proved. Finally an application to images compression is discussed, showing the outperformance of such operators in some senses

On Shepard–Gupta-type operators

“…From Theorem 2.2 , for , we obtain This is the first pointwise estimate for Shepard operator on an equispaced mesh and it reflects the interpolatory character of at the knots , and the constants preservation property. A similar estimate was obtained for a generalization of Shepard operator in [ 9 ]. The result in ( 7 ) is interesting; indeed the Shepard operator is strongly influenced by the mesh distribution and pointwise error estimates, for Shepard operators on nonuniformly spaced meshes present a function depending on the mesh thickness at the r.h.s.…”

“…In Sect. 2.2 an application to image compression is examined improving an analogous algorithm in [ 9 ] and numerical experiments confirming the outperformance of such technique compared with other algorithms are also shown.…”

Rational Operators Based on q-Integers

“…. , n. Shepard operators are widely studied in classical approximation theory and in scattered data interpolation problems (see, e.g., [1,2,7,8,10,22,23]). If the nodes mesh in (4) is equispaced, then direct and converse results are well-known for S n (see, e.g.…”

“…Here extending the idea of Lupaş, we construct simple rational operators based on q-integers and prove they are a good tool to approximate functions from C([0, 1]), achieving pointwise approximation error estimates improving (1)- (2). In Section 2 the q−analogue of Shepard operators are considered and uniform convergence results and pointwise approximation error estimates are given in Theorems 1-10.…”

Weighting Shepard-type operators

“…Following Amato and Della Vecchia (2015a) if λ = 0, we can also get pointwise approximation error estimates, we omit details.…”