伯氏多項式在統計上之應用

本篇論文主要探討提供函數圖形的模型，用伯氏多項式來描述，並從一維度推廣到n維度。本篇論文架構如下，第一節是介紹伯氏多項式相關的背景。第二節說明伯氏多項式圖形與係數的關係，其中命題1提供伯氏多項式幾何性質的充分條件，命題2以伯氏多項式的觀點去描述連續函數的幾何性質。第3節是用機率的方法去說明高維度伯氏多項式的逼近定理，並且從一維度推廣到n維度。第4節用分析的方法來敘述函數均勻收斂理論。第5節高維度伯氏多項式圖形與係數的關係，以二變數為例去討論係數在某些條件限制下函數圖形的變化情形，可用來描述曲面上的幾何形狀，應用在曲面迴歸分析估計。第6節是討論，在此篇論文之後對於連續函數圖形如何用伯氏多項式來描述，大致上有一個有系統的方法，對曲面圖形也有相當好的描述方法，對曲面圖形的估計有莫大的幫助。

關鍵字

伯氏多項式

並列摘要

This paper mainly searches that provides the model of the function graph, describes with Bernstein polynomial, and promotes from a dimension to the n dimension. The structure of this paper as follow. The first section introduces Bernstein polynomial correlation background. The second section explains the shape of function graph and Bernstein polynomial coefficient relations. Proposition 1 provides sufficiency of Bernstein polynomial geometry character. Proposition 2 describes the continuous function geometry character by Bernstein polynomial viewpoint. The third section uses the probability method showed the high dimension Bernstein polynomial approximation theo- rem. The fourth section explains the function uniform convergence theory with the analysis method. The fifth section explains the shape of function graph and the high dimension Bernstein polynomial coefficient relations. Discusses the coefficient take two variables as the example in certain condition limit minor function graphs change situations, may use to describe in the curved surface geometry shape, applies in curved surface regression analysisestimate. The sixth section is a discussion. After this paper regarding how to describe the continuous function graph uses Bernstein polynomial, roughly has the system method, also has the quite good description method to the curved surface graph, has the greatest help to the curved surface graph estimate.