The Pointwise Approximation Theorems For The Meyer-K(?)nig-Zeller Type Operators

Function approximation theory is an age-old subject, which has substantial content and strong sense of practicality. It is one of the most flourishing in mathematics. It not only studies the simple function (polynomial function, linear operator, etc.) optimal approximation question, but also studies other functions (irrational function, exponential function, piecewise polynomial, etc.) optimal approximation question. At the same time, it is not only closely relatedwith various research directions such as algebra, functional analysis, harmonic analysis, wavelet analysis and so on, but also is the basic foundation and the method basis for the computationalmathematics, the applied mathematics and optimization the theory of the scientific project computer. In the 1950s, along with increasing influence of the functional analysis on the approximation theory research and application, the operator approximation becomes an importantresearch direction in the approximation theory. The operator approximation mainly studies the convergence property of linear operator series, its convergence rate and related question. The study of approximation direct and inverse theorem, equivalence theorem and strong converseinequaliy on some famous linear operator (Bernstein operators, Baskakov operators, their Durrmeyer transformation and Kantorovich transformation) is an important subject in operator approximation theory, and is very meaningful in the theory and the application domain.Of the positive operators that are used in the literature one of the most challenging are the Meyer-Konig and Zeller operators M_n(f, x), which are given byIn recent years there are many results about their approximation properties and their transformations.Ditzian introduced the unified moduli of smoothnessω_?λ²(f,t)(0≤λ≤1) and gave an interesting direct result for the Bernstein operators which united the results withω²2(f, t) andω_?λ²(f,t). Since it is difficult to handle the estimates of the moments for the Meyer-Konig and Zeller type operators, the direct and inverse results for Durrmeyer-type modifications of the Meyer-Kdnig and Zeller operators are not perfect, and we have not seen the result of strong converse inequality for the operators.In this paper, we will consider a new modification of Meyer-Kdnig-Zeller-Durrmeyer type operators M_n(f,x):whereFirstly, we obtain the estimates of the second and fourth-order moments for these operators.Moreover, using the equivalence between the unified moduli of smoothnessω_?λ²(f,t)and the Peetre K-functional K_?λ²(f,t²) (0≤λ≤1), we give the direct, inverse and equivalence theorems for the operators. Now we state the equivalence theorem:Theorem A For f∈C[0,1), 0 <α< 2,0≤λ≤1, n≥2, one hasSecondly, we give the strong converse inequality of B-type in terms of new K-functional K_?λ²(f,t²)(0≤λ≤1,0 <α< 2) for the Meyer-Konig-Zeller-Durrmeyer type operators.Theorem B Suppose 0≤λ≤1, 0<α<2, f∈C_λ,α⁰, there exists a constant K > 1, for l≥K_n, we haveFinally, there were only L_p-approximation (1≤p≤∞) theorems for Meyer-Konig-Zeller-Kantorovich-type modification M_n^*(f, x):The aim of the present paper is to study the classical positive estimate in terms of the (?)-modulus of smoothness, as well as a corresponding converse. As a result, one can get the classical pointwise equivalence estimate. The equivalence theorem is as follows:Theorem C For f∈C[0, 1), 0 <α< 2, n≥2, there are...