Abstract
We derive the complete asymptotic expansion for the quasiinterpolants of Gauss Weierstrass operators \(W_{n}\) and their left quasi interpolants \(W_{n}^{[r]}\) with explicit representation of the coefficients. The results apply to all locally integrable real functions \(f\) on \(\mathbb{R}\) satisfying the growth condition \(f(t)=O\left(e^{ct^{2}}\right)\ \)as \(\ |t|\rightarrow+\infty\), for some \(c>0\). All expansions are shown to be valid also for simultaneous approximation.
Authors
U. Abel
Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
R. Păltănea
Keywords
Approximation by integral operators, rate of convergence, degree of approximation, asymptotic expansions.
Paper coordinates
U. Abel, O. Agratini, R. Păltănea, A complete asymptotic expansion for the quasi-interpolants of Gauss–Weierstrass operators, Mediterranean Journal of Mathematics 15 (2018), pp. 154-156, https://doi.org/10.1007/s00009-018-1195-8
requires subscription: https://doi.org/10.1007/s00009-018-1195-8
About this paper
Journal
Mediteranean Journal of Mathematics
Publisher Name
Springer
Print ISSN
Online ISSN
1660-5454
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[1] Abel, U.: Asymptotic expansions for Favard operators and their left quasi interpolants. Stud. Univ. Babes-Bolyai Math. 56, 199–206 (2011)
[2] Abel, U., Ivan, M.: Simultaneous approximation by Altomare operators. Proceedings of the 6th international conference on functional analysis and approximation theory, Acquafredda di Maratea (Potenza), Italy, 2009, Palermo: Circolo Matematico di Palermo, Rend. Circ. Mat. Palermo, Serie II, Suppl. 82, 177–193 (2010)
[3] Abel, U., Butzer, P.L.: Complete asymptotic expansion for generalized Favard operators. Constr. Approx. 35, 73–88 (2012). https://doi.org/10.1007/s00365-011-9134-y
[4] Altomare, F., Milella, S.: Integral-type operators on continuous function spaces on the real line. J. Approx. Theory 152, 107–124 (2008). https://doi.org/10.1016/j.jat.2007.11.002
[5] Altomare, F., Campiti, M.: Korovkin-type approximation theory and its applications. de Gruyter Studies in Mathematics 17, W. de Gruyter, Berlin, (1994)
[6] Henry, W.: Gould, combinatorial identities. Morgantown Print & Bind, Morgantown (1972)
[7] Sablonniere, P.: Weierstrass quasi-interpolants. J. Approx. Theory 180, 32–48 (2014). https://doi.org/10.1016/j.jat.2013.12.003
[8] Sikkema, P.C.: On some linear positive operators. Indag. Math. 32, 327–337 (1970)