Rndin─Shapiro函数的Bouligand维数

THE BOULIGAND DIMENSION OF THE RUDIN-SHAPIRO FUNCTION

设Ｐｎ，Ｑｎ为Ｒｕｄｉｎ－Ｓｈａｐｉｒｏ多项式，ψ为相应的Ｒｕｄｉｎ－Ｓｈａｐｉｒｏ函数，本文引入ψ的伴随函数△与，讨论ψ的分析性质与算术性质。为讨论ψ的分形性质，我们确定了△及的Ｈｏｌｄｅｒ指数，利用该结果及插值技巧确定了ψ的函数图象的Ｂｏｕｌｉｇａｎｄ维数与ｐａｃｋｉｎｇ维数均为３／２．从而，该函数可作为一维Ｂｒｏｗｎ运动的模拟。

Let Pn and Qn be the Rudin-Shapiro polynomials and let ψ be the associated Rudin-Shapiro function. We study first the arithmetic and analytic properties of ψ by studing two auxiliary functions △ and . To discuss the fractal structure of ψ, we determine the Hlder exponent of △ and. Then by using interpolating technique, we prove that the Bouligand dimension and the Packing dimension of the graph of ψ are 3/2. The result shows that we can simulate one dimensional Brown Movement by the Rudin-Shapiro function.