By using the central limit theorem in probability theory,the problem of simultaneous approximation is investigated by Beta operators and its derivatives for functions whose derivatives have only discontinuity points of the first kind,the pointwise result of the error is obtained.

Use the new K-funtional which is defined in the paper and prove:for Beta operators there have constants R>1,when l≥Rn,ω~2_(φ)f,1n≤C‖β_nf-f‖+‖β_lf-f‖+4n‖f‖,φ(x)=x,where C is a constant which is independent on n.

Investigate the degree of approximation by Beta operators for functions which have only discontinuity points of the first kind on (0,∞),use the pointwise modulus of continuity ω x(f,t) and normal operator technique,obtain the approximation result on functions which have only discontinuity points of the first kind by Beta operators,extend the result on functions of bounded variation.

Lower estimates of convergence rate for Beta operators;

The author uses Laplace and Bojanic method to study the rate of convergence of functions and their derivatives by Beta operator and obtain degree of approximation.

Using the relation between the weighted modulus of smoothness and the weighted main-part modulus of smoothness,we discuss the pointwise direct and equivalent approximation theorem with Jacobi weight for Beta operator.

The Voronovskaja type expansion fomula of the modified Baskakov-Beta operators;

By using the probability theory,the pointwise approximations for bounded variation functions of the modified Baskakov-Beta operators are studied.

In this paper,we introduce the modified Beta Operators which use integer on finite intervals and discuss their necessary and sufficient conditions of convergence.