指数函数项级数与Hadamard乘积

The series of exponential terms and Hadamard product

在本文中,证明了下述定理,设sum sum k=1 to ∞ c_k,sum from j=1 to ∞ d_j均为绝对收敛级数且其各项均不为零.设(λ_k),(μ_j)为两个有界复数序列,每一序列的任惠两项均不相等,并设μ=|μ_1|>|μ_2|≥…。于是,F(z)=e·f(z)·g(z)均为正指型函数,这里f(z),g(z)分别表示f(z)g(z)的相伴函数·表示Hadamard乘积算子.设f(z)的指示图为Ⅰ:令μ_j=ρ_je_j^(iφ)(ρ_j>0,0≤φ_j<2π)(j=1,2,…)。把Ⅰ旋转角—φ_j,并进行比ρj的相似变换得到凸集Ⅰ_j。如果二重序列(λ_kμ_j)(k=1,2,…,j=1,2,…)的任何两项均不相等,则F(z)的指标图Ⅰ 为点集Ⅰ_1,Ⅰ_2,…的凸壳 从此定理出发,在适当条件下,证明F(z)与f(z)有公共Julia线以及二者同为多零点整函数。

In this paper we prove the following.Theorem.Suppose that ∑(c_k)k from 1 to ∞ and ∑(d_j) j from 1 to ∞ are both absolutely convergent with non-zero terms Suppose that (λ_k),(μ_j) are both bounded sequecnes and thatany two terms in each sequence are not equal, ane that μ=|μ_1|>|μ_2|≥|μ_3|≥…。Then f(z) =∑(c_ke~λk^z) k from 1 to ∞ g(z) = ∑(d_je~μj^k) j from 1μ to ∞, F(z) = e~?·f(z)·e(z) all are functions ofpositive exponential type, ehere f(z), g(z) are functions associated with f(z)g(z) respectively and '·' is a Hadamard product operator .Let the indicatordiagram of f(z) be I and μ_j= ρ_je^(iφj)(ρ_j>0,0≤φ_j<2π)(j=1, 2, …), we denote the set obtained from I by rotation an angle-φ_j and under a similar transformation of ratio ρ_j by Ij if any two terms of the double sequence (λ_kμ_j)(k=1,2,…;j=1, 2,…)are not eqnal, then the iudicator diagram I, of F(z) is theconvex hull of the set I_1,I_2,…。 From this theorem we prove that unber suitable couditious, F(z) and f(z)have common lincs of Julia aud they are both integral functions with manyzeros.