阿贝尔泛级数
Abel universal series
摘要：
给定一个趋于1的序列=（rn）$\in$ [0，1），我们考虑由开单位圆D中的全纯函数f组成的Abel泛级数的集合UA（D，），使得对于包含在单位圆T中的任何不同于T的紧集K，集合{z $\rightarrow$ f（rn $\bullet$）|K：n $\in$ N}在K上的连续函数空间C（K）中稠密.已知集合U_A（D_i）是H（D）中的残差。我们证明了它不与任何其它经典的泛全纯函数集重合。特别地，它甚至不能与全纯函数的集合相比较，对于任何不同于T的紧集K $\subset$ T，其泰勒多项式在0处在C（K）中稠密。证明了Abel泛级数类在微分算子作用下不是不变的。最后，Abel泛级数可以被视为作用于H（D）的扩张算子序列T n的泛向量：f $\rightarrow$ f（r n $\bullet$）。从而研究了（Tn）n的多重普适性和（公共）频繁普适性等动力学性质。所有的证据都是建设性的。
英文摘要：
Given a sequence = (r n) n $\in$ [0, 1) tending to 1, we consider the set U A (D,) of Abel universal series consisting of holomorphic functions f in the open unit disc D such that for any compact set K included in the unit circle T, different from T, the set {z $\rightarrow$ f (r n $\bullet$)| K : n $\in$ N} is dense in the space C(K) of continuous functions on K. It is known that the set U A (D,) is residual in H(D). We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at 0 are dense in C(K) for any compact set K $\subset$ T different from T. Moreover we prove that the class of Abel universal series is not invariant under the action of the differentiation operator. Finally an Abel universal series can be viewed as a universal vector of the sequence of dilation operators T n : f $\rightarrow$ f (r n $\bullet$) acting on H(D). Thus we study the dynamical properties of (T n) n such as the multi-universality and the (common) frequent universality. All the proofs are constructive.
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