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Limiting properties of Inverse Beta and generalized Bleimann–Butzer–Hahn operators

Published online by Cambridge University Press:  24 October 2008

José A. Adell  and
Jesús De La Cal
José A. Adell
Affiliation:
Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Jesús De La Cal
Affiliation:
Departamento de Matemática Aplicada y Estadística e Investigación Operativa, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
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Abstract

In this paper, we consider limiting properties concerning linear operators of probabilistic type. Specifically, we show that gamma operators are limits of inverse Beta operators and that Bleimann–Butzer–Hahn, Szász and Baskakov operators are limits of generalized Bleimann–Butzer–Hahn operators. By duality, these results are closely related to the convergence in the total variation distance of the probability measures involved. In each case, rates of convergence are given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 3 , November 1993 , pp. 489 - 498
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Adell, J. A., Badia, F. G. and De La Cal, J.. Beta-type operators preserve shape properties, to appear in Stochastic Process. Appl.Google Scholar
[2]Adell, J. A., De La Cal, J. and Miguel, M. San. Inverse Beta and generalized Bleimann–Butzer–Hahn operators, to appear in J. Approx. Theory.Google Scholar
[3]Barbour, A. D. and Hall, P.. On the rate of Poisson convergence. Math. Proc. Cambridge Philos. Soc. 95 (1984), 473–480.CrossRefGoogle Scholar
[4]Bleimann, G., Butzer, P. L. and Hahn, L.. A Bernstein-type operator approximating continuous functions on the semi-axis. Indag. Math. 42 (1980), 255–262.CrossRefGoogle Scholar
[5]Chow, Y. S. and Teicher, H.. Probability Theory: Independence, Interchangeability, Martingales (Springer-Verlag, 1978).CrossRefGoogle Scholar
[6]De La Cal, J.. On Stancu–Mühlbach operators and some connected problems concerning probability distributions. J. Approx. Theory 74 (1993), 59–68.CrossRefGoogle Scholar
[7]De La Cal, J. and Luquin, F.. A note on limiting properties of some Bernstein-type operators. J. Approx. Theory 68 (1992), 322–329.CrossRefGoogle Scholar
[8]De La Cal, J. and Luquin, F.. Approximating Szász and Gamma operators by Baskakov operators, to appear in J. Math. Anal. Appl.Google Scholar
[9]Feller, W.. An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, 1966).Google Scholar
[10]Hall, P.. On measures of the distance of a mixture from its parent distribution. Stochastic Process. Appl. 8 (1979), 357–365.CrossRefGoogle Scholar
[11]Johnson, N. L. and Kotz, S.. Discrete Distributions (Houghton-Mifflin, 1969).Google Scholar
[12]Johnson, N. L. and Kotz, S.. Continuous Univariate Distributions I (Wiley, 1970).Google Scholar
[13]Johnson, N. L. and Kotz, S.. Continuous Univariate Distributions II (Wiley, 1970).Google Scholar
[14]Keilson, J. and Steutel, F. W.. Mixtures of distributions, moment inequalities and measures of exponentiality and normality. Ann. Probab. 2 (1974), 112–130.CrossRefGoogle Scholar
[15]Khan, M. K. and Peters, M. A.. Lipschitz constants for some approximation operators of a Lipschitz continuous function. J. Approx. Theory 59 (1989), 307–315.CrossRefGoogle Scholar
[16]Khan, R. A.. Some probabilistic methods in the theory of approximation operators. Acta Math. Acad. Sci. Hungar. 39 (1980), 193–203.CrossRefGoogle Scholar
[17]Khan, R. A.. A note on a Bernstein-type operator of Bleimann, Butzer and Hahn. J. Approx. Theory 53 (1988), 295–303.CrossRefGoogle Scholar
[18]Khan, R. A.. Some properties of a Bernstein-type operator of Bleimann, Butzer and Hahn. In Progress in Approximation Theory, Nevai, P. and Pinkus, A., eds. (Academic Press, 1991). pp. 497–504.Google Scholar
[19]Lupas, A. and Müller, M.. Approximationseigenschaften der Gammaoperatoren.Math. Z. 98 (1967), 208–226.CrossRefGoogle Scholar
[20]Müller, M.. Die Folge der Gammaoperatoren. Dissertation, Stuttgart, 1967.Google Scholar
[21]Stancu, D. D.. Use of probabilistic methods in the theory of uniform approximation of continuous functions. Rev. Roumaine Math. Pures Appl. 14 (1969), 673–691.Google Scholar