连续函数的多项式逼近与随机复合
Polynomial approximations to continuous functions and stochastic compositions
摘要：
本文用随机方法研究了Bernstein算子$B_n$的迭代行为，当迭代次数$k$趋于无穷大而$n$保持不变或当$n$也趋于无穷大时，连续函数$f \in C[0,1]$化为一次$n$多项式.在第一种情况下，潜在的随机过程是所谓的赖特-费舍尔模型，而在第二种情况下，潜在的随机过程是赖特-费舍尔扩散。这两个过程可能是数学遗传学中最基本的过程。通过使用马尔可夫链理论和随机组合，我们解释了一个定理，由于Kelisky和Rivlin的概率，并通过使用随机微积分，我们计算出一个公式的应用程序$B_n$多次$k=k(n)$的多项式$f$时，$k(n)/n$趋于一个常数。
英文摘要：
This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator $B_n$ taking a continuous function $f \in C[0,1]$ to a degree-$n$ polynomial when the number of iterations $k$ tends to infinity and $n$ is kept fixed or when $n$ tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright-Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright-Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of $B_n$ a number of times $k=k(n)$ to a polynomial $f$ when $k(n)/n$ tends to a constant.
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