| The approximation theory of functions is one of the important branches of modern mathematics. It began in the 19th century when two famous theorems were established. One says that the continuous functions can be approximated by certain polynominals which was established by Weierstrass in 1885, and the other that establishes characterization for approximation by polynomials was obtained by Chebyshev in 1859. It grew prosperous in the 20th century and became an independent subject. People have generated a series of theories and methods in order to use the simple and computable functions to approximate general functions. For example, the best approximation, Fourier approximation, approximatio by trigonometric polynomials, approximation by algebra polynominals, linear operators approximation, interpolation approximation, rational approximation, approximation by reciprocals of polynomials. Miintz approximation and so on.Approximation by linear operators got the full development in the last century, and a set of mature theories were established. For applicable purpose, many mathematicians brought forward a lot of useful operators, particularly some typical positive linear operators.The thesis will do research on Bernstein-Durrmeyer operators, and their application on neural network.The object is as follows:(1) We do research on weighted Bernstein-Durrmeyer operators, and get the correspongding results in weighted spaces.(2) We generalize the suitable space of Bernstein-Durrmeyer operators. That is, generalize from Lp to weighted space Lp, so that extend the application range, and use these operators to deal with certain problems.(3) We give applications of the operators to neural network. The thesis will be divided into five sections.1 IntroductionIn this section the basic conceptions of the Bernstein operators, Bernstein-Durrmeyer operators, the weighted Bernstein-Durrmeyer operators, the weighted Bernstein-Durrmeyer operators on simplex, and the two important tools, modulli ofsmoothness and K-function will be introduced.2 An integral inequality on basic Bernstein polynomialsLet f(x) C[0,1], Bn(f,x) is the Bernstein polynominal of degree n which is defined as follows:where Pn,k{x) = (kn)xk(1- x)n-h, k = 0,1,2, ... , n, is called as the basic Bernstein polynominals.The Basic Bernstein polynominals not only have much applications in analysis, but also have ultimate relationship with probablity theory. For example, we can use basic Bernstein polynominal to prove Weierstrass approximation theorem. There are many inequalities based on the basic Bernstein polynominals and playing an important role. We will establish an integral-type inequality.3 Approximation by Bernstein-Durrmeyer operators with Jacobi weightDurrmeyer defined the Bernstein-Durrmeyer operators Mn(f.x) as follows:There were many significant results on these operators. Berens and Xu introduced the Bernstein-Durnneyer operators with Jacobi weight:where Cn,k = < a,b < 1. It is observed that one can construct translation network operators having approximation property in Lp space established more useful than Mn(f,x). People established the direct and converse theorems of Mn(f,x) in Lp space, and we will give the corresponding results for Dn(f,x) in Lp space in this section.4 The translation network based on Bernstein-Durrmeyer operatorsApproximation problem of translation network includes wavelet analysis and neural network approximation, and is one of the problems be settled by nonlinear approximation. The key point here is to construct limited linear combinations by a translation of the given functions.For the condition of period functions, people have studied the corresponding approximation problem. It is well knowm that the basic Bernstein polynomials can be served as approximation tool. Therefore it is possible to construct translation network operators and approximate some useful functions.The results of this section will show that Bn*(f, x) can be used not only to approximate functions in Lp but also to construct translati... |
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