# On the rate of convergence of a positive approximation process

## Abstract

In this paper we are dealing with a class of summation integral operators on unbounded interval generated by a sequence $$(L_{n})_{n\geq1}$$ of linear and positive operators. We study the degree of approximation in terms of the moduli of smoothness of first and second order. Also we present the relationship between the local smoothness of functions and the local approximation. By using probabilistic methods, new features of $$L_{n}f$$ are pointed out such as the approximation property at discontinuity points and the monotonicity property under some additional assumptions of the function $$f$$ . Also the rate of convergence of these operators for functions of bounded variation is given.

## Authors

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

?

## Paper coordinates

O. Agratini, On the rate of convergence of a positive approximation process, Nihonkai Math. J., 11 (2000) no. 1, pp. 47-56.

## About this paper

##### Journal

Nihonkai Mathematical Journal

##### Publisher Name

Niigata University, Department of Mathematics

1341-9951

##### Online ISSN

google scholar link

[1] O. Agratini, On a sequence of linear and positive operators, Facta Universitatis (Nis), Series: Mathematics and Informatics, 13(1998), in print.
[2] W. Feller, An Introduction to Probability Theory and Its Applications, II, Wiley, 1966.
[3] S.S. Guo and M.K. Khan, On the rate of convergence of some operators on functions of bounded variation, Journal of Approximation Theory, 58(1989), 90-101.
[4] R.A. Khan, Some probabilistic methods in the theory of approximation operators, Acta Mathematica Academiae Scientiarum Hungaricae, 35(1980), 1-2, 193-203.
[5] B. Levikson, On the behaviour of a certain class of approximation operators for discontinuous functions, Acta Mathematica Academiae Scientiarum Hungaricae, 33(1979), 3-4, 299-306.
[6] G.G. Lorentz, Bernstein Polynomials, Mathematical Expositions, No.8, University of Toronto Press, Toronto, 1953.
[7] A. Lupas , The approximation by some positive linear operators, In: Proceedings of the International Dortmund Meeting on Approximation Theory (IDoMAT 95)-edited by M.W. M\”uller, M. Felten, D.H. Mache, Mathematical Research, Vol.86, pp.201-229, Akademie Verlag, Berlin, 1995.
[8] D.D. Stancu, Use of probabilistic methods in the theory of uniform approximation of continuous functions, Rev. Roum. Math. Pures et Appl., 14(1969), 673-691.