| This paper study the approximation characteristic of the various operators of probabilistic type for continuous functions, differentiable functions, functions of bounded variation and absolutely continuous functions; study the properties of the presevation of monotonicity, the presevation of Lipschitz class for operator family of probabilistic type; study approximation operators of probabilistic type by other sequence (family) of operators of probabilistic type; and study the first boundary value problem for a class of quasilinear degenerate elliptic equation.The paper contains four chapters. Chapter 1, in ?. 2, we determine the exact upper bounds of a series of important basis functions of approximation operators, these basis functions include the Bernstein, Szasz-Mirakjan, Baskakov, the Meyer-Konig and Zeller basis functions, the basis functions of tensor product operators formed by the above-mentioned basis functions, and the multivariate Bernstein, Meyer-Konig and Zeller basis functions over a simplex. The results in this section are better than the results of Quo [30, 31], Wang [32], Gupta [33], Love [36]. In ?. 3, ?. 4, By means of probabilistic methods and Bojanic-Cheng's method combining with analysis technique, we study the rates of convergence of some Bezier type operators for functions of bounded variation, and obtain the optimal asymptotic estimates, these operators manily include the Bernstein-Bezier operators Br^a, Kantorovich-Bezier operators and Durrmeyer-Bezier operators. We obtain the following result:Theorem I. 5 Let f 6 BV[Q, 1]). // a > 1, then for every x ?(0, 1) andIVn > 1. we haveAa is a positive constantdepending only on a, \f(gx) is total variation of gx on [a, b],Moreover, we prove that the estimations (A. 1) and (A. 2) are asymptotically optimal. The result in (A. 1) by taking a = 1 is better than the Cheng's result [10] and the Guo and Khan's result [15]. In ?. 5, by using new probabilistic techniques we study pointwise approximation for bouned functions by a class of operators of Bernstein probabilistic type, and get the exact estimates. Our investigations subsume the case of functions of bounded variation as special case. In ?. 6, we study rates of convergence of sequence of Szasz operators and Beta operators for a class absolute continuous functions, and obtain the optimal asymptotic estimates. Our results improve Bojanic and Khan's result [26] by replacing their approximation order 0(n~l) with the optimal approximation order 0(n~3/2); improve Davis's result [28] by replacing his approximation order 0(n~2/3) with the optimal approximation order O(n~3/2).In Chapter 2, we introduce a class of generalized Feller-Trotter type operators Ln and study its approximation behaviors. In ?. 2 we get the rates of convergence of operators Ln for continuous functions and differentiable functions. In ?. 3, using the idea of control functions on interval, we study the approximation behavior of operators Ln for unbounded functions class C(I1, ga, gb), and obtain the following Theorem :Theorem 2. 6 Vf C C(I1,ga,9b), x € I\, Ln(f, x) converge (converge uniformly) to f(x) if and only if Ln(gg(t) + tg's(x), x) converge (converge uniformly) to gs(x) + xg's(x) (s = a, 6), where ga, gi, are control functions.In ?. 4, we obtain the locally saturation theorem of Feller operators, and apply it to set up the locally satturation theorem for probability express of semigroups operators of class (Co). This result subsume the Fang's result [49] as special case. In ?. 5, we modify the method of parabolas of Bajsanski-Bojanic, by that and Pfeifer's Lemma [63], we set up a global saturation theorem for a class of generalized Feller-Trotter type operators.In Chapter 3, we study approximation operators of probabilistic type by other operator sequences (family) and study preservation Lipschitz class for operators of probabilistic tpye. In ?. 2, by S ?A distribution, we introduce Bernstein-Trotter operators of probabilistic tpye, study its limiting op...
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