Approximation By Modified Summation Integral Mixed Type Operators In The Lp Spaces

Beta operators are important and applied widely in probability theory and approximation theory. The generalized summation integral mixed type operators with Beta basis functions are widely studied. At present, the investigations for the properties of these operators are only limited to the functions of bounded variation. Some authors studied the rate of point-wise rate of convergence, asymptotic formula of Voronovskaja type, and some direct results about these type of operators.In this paper, we will study the direct, inverse and equivalent theorems of modified summation integral mixed type operators in the L_p spaces, and obtain the following results.Theorem 1(Direct results) For f∈L_p[0,∞) (1≤p≤∞), (?)~2(x) = x(1+x), we havewhereTheorem 2(The Bernstein-Type inequality) For f∈L_p[0,∞) (1≤p≤∞), one hasTheorem 3(Inverse results) For f∈L_P[0,∞) (1≤p≤∞), 0≤α≤2, we have Theorem 4(Equivalence results) For f∈L_p[0,∞)(1≤p≤∞),0≤α<2, the following two statements axe equivalent:(1)(2)...