Abstract
In this paper we introduce and study a sequence of positive linear operators acting on suitable spaces of measurable functions on [0,+∞[, including L p([0,+∞[) spaces, 1 ≤ p < +∞, as well as continuous function spaces with polynomial weights. These operators generalize the Szász–Mirakjan–Kantorovich operators and they allow to approximate (or to reconstruct) suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,+∞[ that do not constitute a subdivision of it. We also give some estimates of the rates of convergence by means of suitable moduli of smoothness.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Altomare F.: Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 5, 92–164 (2010) free available online at http://www.math.techmion.ac.il/sat/papers/13/
Altomare F., Campiti M.: Korovkin-type approximation theory and its applications, de Gruyter Studies in Mathematics 17. Walter de Gruyter & Co., Berlin (1994)
Altomare F., Cappelletti Montano M., Leonessa V.: On a generalization of Kantorovich operators on simplices and hypercubes. Adv. Pure Appl. Math. 1(3), 359–385 (2010)
Altomare F., Leonessa V.: On a sequence of positive linear operators associated with a continuous selection of Borel measures. Mediterr. J. Math. 3, 363–382 (2006)
Bauer H.: Probability theory, de Gruyter Studies in Mathematics 23. Walter de Gruyter & Co., Berlin (1996)
Becker M.: Global approximation theorems for Szász–Mirakjan and Baskakov operators in polynomial weight spaces. Indiana Univ. Math. J. 27(1), 127–142 (1978)
Bustamante J., Morales de la Cruz L.: Korovkin type theorems for weighted approximation. Int. J. Math. Anal. 26(1), 1273–1283 (2007)
Butzer P.L.: On the extensions of Bernstein polynomials to the infinite interval. Proc. Amer. Math. Soc. 5, 547–553 (1954)
Cheney E.W., Sharma A.: Bernstein power series. Canadian J. of Math. 16(2), 241–264 (1964)
DeVore R.A., Lorentz G.G.: Constructive Approximation, Grundlehren der mathematischen Wissenschaften 303. Springer, Berlin (1993)
Ditzian Z., Totik V.: Moduli of smoothness, Springer Series in Computational Mathematics 9. Springer, New-York (1987)
Duman O., Della Vecchia M.A., Della Vecchia B.: Modified Szász–Mirakjan–Kantorovich operators preserving linear functions. Turk. J. Math. 33, 151–158 (2009)
Favard J.: Sur les multiplicateurs d’interpolations. J. Math. Pures Appl. 23(9), 219–247 (1944)
Gonska H.: Positive operators and approximation of functions: selected topics. Conf. Semin. Mat. Univ. Bari 288, 28 (2002)
Holhoş A.: The rate of approximation of functions in an infinite interval by positive linear operators. Stud. Univ. Babeş–Bolyai Math. 55(2), 133–142 (2010)
Mirakjan G.M.: Approximation of continuous functions with the aid of polynomials. (Russian), Dokl. Akad. Nauk SSSR 31, 201–205 (1941)
Păltănea R.: Approximation theory using positive linear operators. Birkhäuser, Boston (2004)
Swetits J.J., Wood B.: Quantitative estimates fo L p approximation with positive linear operators. J. Approx. Theory 38, 81–89 (1983)
Szász O.: Generalization of Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Stds. 45, 239–245 (1950)
Totik V.: Approximation by Szász–Mirakjan type operators. Acta Math. Hungarica 41, 291–307 (1983)
Totik V.: Approximation by Szász–Mirakjan–Kantorovich operators in L p(p > 1). Analysis Math. 9, 147–167 (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Altomare, F., Cappelletti Montano, M. & Leonessa, V. On a Generalization of Szász–Mirakjan–Kantorovich Operators. Results. Math. 63, 837–863 (2013). https://doi.org/10.1007/s00025-012-0236-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-012-0236-z