Approximation Properties Of Several Positive Operators

This dissertation is concerned with the following problems in the field of operator approximation theory:the pointwise simultaneous approximation of the linear combinations of Baskakov operators;the characterization of the saturation approximation order and the strong Steckin type inequality for the linear combinations of Bernstein operators on the space C[0,1] and Bernstein-Kantorovich operators on the space L_p[0,1](1≤p≤∞);the convergence and approximation orders of the localized Bernstein,Szasz-Mirakjan and Baskakov operators. Moreover,the Jackson type inequality for multiplier operators on Herz-type Hardy spaces is also investigated here.The thesis consists of eight chapters.ChapterⅠgives a brief introduction to the contents concerned.In ChapterⅡ,we study the relation between the approximation order for the derivatives of the linear combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated.By adding auxiliary operators and canceling the moments of lower order,we also establish the positive theorem and the approximation equivalence theorem. Meanwhile,when r≥2 and 0＜λ＜1-1/r,the relation between the approximation order of the linear combinations of Baskakov operators and the smoothness of the functions approximated is studied,and the positive theorem and the approximation equivalence theorem are also established.Thus the characterization problem of the non-saturation approximation order of the linear combinations of Baskakov operators has been thoroughly solved.ChapterⅢdeals with a new type of K-functional and the precise expression of the moments of Bernstein type operators.In addition,the relation between the saturation approximation order for the linear combinations of the Bernstein type operators and the smoothness for the functions approximated is studied under the uniform as well as L_p approximation. By the optimal approximation method of algebraic polynomials,we also establish the equivalence theorems of the saturation approximation of the linear combinations of Bernstein type operators.Thus the characterization problem of their saturation approximation orders is solved.Meanwhile,with the aid of the characters of optimal approximation polynomials,the equivalence of this new type of K-functional and the well-known Ditzian-Totik modulus of smoothness has been proposed for the spaces of C[0,1],L_p[0,1](1≤p＜∞) and L_∞[0,1].Based on ChapterⅢ,in ChapterⅣ,the inverse problem of the saturation approximation for the linear combinations of the Bernstein type operators is further investigated.By the use of the equivalence of this new type of K-functional and the Ditzian-Totik modulus of smoothness,the strong type of Steckin inequality and its lower estimate are established. Thus the characterization problems of the uniform and L_p approximation orders of the linear combinations of the Bernstein type operators are solved.To reduce the amount of computation and to avoid unnecessary data collection,in ChapterⅤ,we construct some localized deformation Bernstein operators and improve the previous Berry-Esseen theorem in probability theory given by V.V.Petrov.By this modified Berry-Esseen theorem,we study the convergence of these new localized Bernstein operators and give their approximation orders.We also give the sufficient and necessary conditions of their convergence to the approximated function itself.In ChapterⅥ,the central limit theorem in probability theory is improved and the uniform pointwise estimate is obtained.By using a new analysis method,the convergence and the approximation orders of the localized Szasz-Mirakjan are established and similar problems to the localized Baskakov operators are also solved.In ChapterⅦ,to further reduce the amount of computation and to avoid unnecessary data collection in application,other new types of localized Szasz-Mirakjan and Baskakov operators are constructed respectively.With a mathematical analysis method,the pointwise approximation theorems of these new Szasz-Mirakjan operators under different situations are established respectively.Meanwhile,another form of uniform estimate of the kernel of Baskakov operators is also given following a probabilistic method,which leads to the pointwise approximation theorem of these new localized Baskakov operators.In the last Chapter,we firstly study some new approximation problems for Herz-type Hardy spaces.The Jackson type inequality for multiplier operators and its application are investigated.The aforesaid inequality is able to be applied to some important operators in Fourier analysis,for instance,Bochner-Riesz operator of order greater than critical value, generalized Bochner-Riesz average operator,and generalized Abel-Poisson operator.