The Approximation Properties Of Beta Type Mixed Summation-Integral Operators

The approximation of operators is an important branchs of approximation theory of functions,which have closely related with the improvement of modern computational mathematics. The study of linear positive operators is one of central issues of the approximation of operators. Mathematicians have studied the approximation properties of many famous operators by the untiring efforts and achieved a series of beautiful and classical results, which settled the stablebase for the studies on their transformation operators and other type operators. The mixed summation-integral type operators include two well-known operators which have important theoretical and practical values in the modern approximation theory of functions. Nevertheless, they are not the simple combination by the two operators. The research of such operators are more difficult and complicated. On the one hand, the construction of the mixed summationintegraltype operators are complex. On the other hand, the seldom results about this type of operators increase the computation works and difficulty.In this paper we investigate and study the two mixed summation-integral type operators, that is Szasz-Beta operators S_n(f, x) and Baskakov-Beta operators B_n(f, x), which have Beta basis functions. They are defined aswhere W_n(x, t) = sum from v=1 to∞s_n,v(x)b_n,v(t) + s_n,0(x)δ(t),δ(t) is Dirac-Delta function,whereThe two operators are linear positive operators. For the operators S_n(f, x), there were some directresults, an error estimate in simultaneous approximation, asymptotic formula of Voronovskaja type and global direct approximation theorems using the moduli of smoothness of second order. And only asymptotic formula and estimation of error in simultaneous approximation were studiedfor the operators B_n(f,x). This paper continue to study the approximation properties on the basis of the former research. The paper has two sections, in the first section we extend the former study and obtain the pointwise approximation theorems, using the equivalence of unifiedmoduli of sommthness and K-functional, and get the strong converse inequality of type B for Szasz-Beta operators by the new K-functional. In the second section we obtain the approximationproperties in the L_p(1≤p≤∞) spaces for the Baskakov-Beta operators, using the equivalence of moduli of smoothness and K-functional, and get the direct, converse theroems of pointwise approximation. Our main results can be stated as follows:Theorem 1 If f∈C_B[0,∞), 0<α<2, 0≤λ≤1,δ_n²(x) = (?)²(x) +1/n, (?)(x) =(?) n≥4, thenTheorem 2 Suppose 0≤λ≤1,0<α<2, f∈C_λ,α⁰, n≥6, there exists a constant K > 1, for l≥Kn, we haveTheorem 3 For f∈L_p[0,∞) (1≤p≤∞), 0 <α< 2, (?)(x) = (?), n≥5,thenTheorem 4 For f∈C[0,∞), 0<α<2, 0≤λ≤1,δ_n²(x) = (?)²(x) + 1/n,(?)(x) =(?), n≥3, then...