缺项及随机级数的边界性质

缺项及随机级数的边界性质 THE PROPERTIES OF THE BOUNDARY OF LACUNARY AND RANDOM SERIES 孙道椿 Sun Daochun 1 first-author 武汉大学数学系 武汉大学数学系 Department of mathematies,Wuhan University Department of mathematies,Wuhan University 本文首先证明了一类缺项幂级数在其收敛圆周上的任意一段弧的像都是不可求长曲线。顺便可知,它的收敛圆周必定是自然边界,即无法解析开拓到收敛圆外。然后证明了一类随机幂级数几乎必然具有上述性质。 Choose arbitrarily an interval I=[a,b][O,2π] .Let l（g（x）; I）be the image of a continuous complex-valued function g（x）（x∈I）.It has beem pro ved that the continuous curve l（f（e i x）;I）is non rectifiable for some lacun- ary power series f（z）and the continuous curves l（f（e b x,w）; I）are almost surely non rectifiable for some random power series f（z,w）. 缺项级数 随机级数 不可求长曲线 自然边界 零一律 lacunary series random series non-rectifiable natural boundary zero one law 国家自然科学基金(61672102) 1991-01-01 2021-04-01 1