On Properties Of Pointwise Approximation Of Some Operators Of Probabilistic Type

This paper focuses on a study about properties of pointwise approximation to certain discontinuous functions by operators of probabilistic type. It comprises two parts. Part one, composed of Chapter 2-4, is devoted to an asymptotic estimate of pointwise approximation to locally bounded function with certain growth conditions by univariate operators of probabilistic type at its discontinuity of the first kind. Part two (Chapter 5-7) explores the theorem of pointwise approximation to some bivariate function by bivariate averaging operators of probabilistic type at its discontinuity of the first kind.Chapter 1 reviews the literature of history and present advances in the researches of properties of pointwise approximation to discontinuous functions by linear operators,followed by a summary of the main contents of this paper.By a counter example, Chapter 2 shows that Theorem 4 in [29] given by Gupta and Kumar does not hold, and is on the asymptotic estimation of pointwise approximation to functions of locally bounded variation on [0,+∞) by the Modified Baskakov operators.Then, by means of the Bojanic-Cheng's method and some probabilistic tools, the author obtains an asymptotic estimation on this type of approximation, which corrects the mistaken estimate in [29] given by Gupta and Kumar.In Chapter 3, a study is made of properties of pointwise approximation of Modified Gamma operators for a general class of functions B_loc((0,+∞),e^β/t,t^P)( re.Definition 1.2) at their discontinuity of the first kind. Using the improved Bojanic-Cheng's method and some probabilistic tools combined with the new metricΩ_X(f,λ),the author makes an asymptotic estimation of pointwise approximation to f∈B_loc((0,+σ),e^β/t,t^P by Modified Gamma operators. The result contains the function of locally bounded variation as a special case. Furthermore, discussions are given about properties of pointwise approximation of Modified Gamma operators for functions which have a locally bounded derivative on (0,+∞), and an asymptotic esti- mation is obtained of the rate of convergence which is best possible in the asymptotical sense.Chapter 4 makes use of classical analytic methods to establish an inequality on the integral of the density function of normal distribution. In addition, the Bojanic-Cheng's method is adopted to reach an asymptotic estimation of pointwise approximation of Gauss-Weierstrass operators W_n(f,x) for functions f∈B_loc((-∞,+∞),e^-βt,e^βt) at their discontinuity of the first kind. The finding includes the function of locally bounded variation as a special case, and is better than that obtained by S.Guo and M. K. Khan in [12]. Chapter 4 ends with a study on properties of pointwise approximation of operators W_n(f,x) for functions which have a locally bounded derivative on R,along with an asymptotic estimation of the rate of convergence as a research finding.In Chapter 5, the concept of a discontinuity of the first kind of bivariate function is introduced before a class of bivariate functions on R² is given and denoted by IB(R²)(re.Definition 5.3).Then, an attempt is made to explore properties of pointwise approximation of bivariate Gauss-Weierstrass tensor product operator W_n[f(u,v);x,y] for the function f∈IB(R²),and establish its approximation theorem. By this approximation theorem combined with the classical central limit theorem of probability theory, it is easy to obtain the approximation theorem of general bivariate tensor product operator T_n[f(u,v);x,y] for the function f∈IB(R²),which reaches Corollary 1.1 given by M. K. Khan in [34].Based on the study of properties of pointwise approximation of bivariate Gauss-Weierstrass tensor product operator W_n[f(u, v);x,y] for the function f∈IB(R²),Chapter 6 studies properties of pointwise approximation of bivariate Gauss-Weierstrass averaging operator (?)_n[f(u,v);x,y] for the function f∈IB(R²).It is certainly more difficult to study properties of pointwise approximation of bivariate averaging operator (?)_n[f(u,v);x,y] than those of bivariate tensor product operator W_n[f(u,v);x,y] since it involves the problem of direction. Supplemented by affine transformation, the similar methods used in Chapter 5 are applied to establish the approximation theorem of bivariate averaging operator (?)_n[f(u,v);x,y] for the function f∈IB(R²).In Chapter 7, the probability theory, especially the central limit theorem for in- dependent identically distributed 2-dimensional random vector, is applied to arrive at the limit theorem of general bivariate averaging operator M_n[f(u,v);x,y] for bivariate Gauss-Weierstrass averaging operator (?)_n[f(u,v);x,y] , and obtain the theorem of pointwise approximation of general bivariate averaging operator M_n[f(u,v);x,y] for function f∈IB(R²),of which Theorem 1.4 established by M. K. Khan in [34] is its corollary.