The Study On Problems About Random Polynomials

Deterministic approximation theory has developed into a rather mature period.In practical application,the points are not exact,because of environment and conditions.Therefore,the problem of random interpolation and random fitting is very important and interesting.The random interpolation problem means that for given random points(x_i,y_i(ξ)),i=0,1,…n,there exist only one random polynomial of degree n p(x,ξ)=a₀(ξ)+a₁(ξ)x+…+a_n(ξ)xⁿ.such that p(x,ξ) = yi(ξ),i = 0,1,…n.Bloch and polya has introduced the concept of random polynomial F(z,ω)=(?)All the coefficients are random variables[1].1. Random interpolation polynomialWe introduce concept of random Lagrange interpolation polynomial and random Newton interpolation polynomial,and study characteristicsof expection and variance.We analyse difference and similarities of random Lagrange polynomial and random Newton polynomial,from comparing ways of formation of these polynomials.At last,we study random Hermite interpolation polynomial.We discuss numerical characteristics of this kind of random polynomial,after introducing concept of it.Some examples are given to show expectation can approximate sample functions,and variance can measure effectiveness of expectation.2 Random Bezier curveWe introduce concept of random Bezier curve,and study expectation and variance of random Bezier curve,and some examples are cited to explain random fitting.