关于随机padé逼近的存在性和收敛性

On the Existence and Convergence of Random Pads Approximants

首次对随机Pade逼近进行了研究。考虑了随机形式幂级数 f(z,ω)=a_0ξ_0(ω)+a_iξ_i(ω)z+…(1)的Pade逼近。其中a_i(i=0,1,…)是全不为零的实数序列,ξ_i(ω)是独立的连续随机变量。 首先证明了(1)的任意Pade逼近的a.s.存在性。其次,考虑了一类形如 (2) 的随机准解析函数Pade逼近的a.s.依勒贝格测度收敛性。

The problem of random Pade approximants is a new one which nobody has investigated before. In this paper a formal power series(1)is considered where ai (i= 0,1,…) are arbitrary nonzero real numbers,ξ:(ω) are continuous random variables. At first, it is proved that there exists a.s. arbitrary Pade approximants of (1). Secondly, the author investigates some random quasianalytic functions in the following form(2)where ξn(ω) are complex random variable sequences, and it is proved that the [(N+J)/N] Pade approximants to (2) converges a.s. in measure within any bounded region of the complex plane as N approaches infinity.