On sequences of polynomials and the distribution of their zeros
On sequences of polynomials and the distribution of their zeros
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关于多项式序列及其零点分布
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发表时间：
1943
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通讯作者：
O. Szâsz
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作者：
O. Szâsz
定理2。如果数列(1)在圆\z\ <R内一致收敛，并且根znv在半平面%z^0内对于每一个n，则数列(1)在每一个有限域内一致收敛于一个函数F(z)它的最大属为2，并且F(z)的根zv满足I>,|-<^。而在定理1中，一致收敛的假设可以用有限极限点的无穷多点收敛性和|cwi |，•••，|cw*-的有界性来代替。i |, w = l, 2，•••，定理2的演绎需要在\z\ <R内一致收敛。本文给出了定理2的一个新的证明，用一个较弱的假设代替了定理2的一致收敛，假设了序列| cni |， | cn21在有限域上的无穷多点收敛和有界性。我们进一步推广了关于零点位置的假设（遵循Weisner[5]的类似评论），仅假设Pn(z)的零点位于包含原点在其边界上的半平面上，但其他部分随n变化。最后我们将结果推广到整个函数的某些序列。
THEOREM 2. If the sequence (1) converges uniformly in a circle \z\ <R, and if the roots znv lie in the half-plane %z^0 for each n, then the sequence (1) converges uniformly in every finite domain to an entire function F(z) which is at most of genus 2, and the roots zv of F(z) satisfy I>,|-<^. While in Theorem 1 the assumption of uniform convergence could be replaced by convergence at infinitely many points with a finite limit point and by boundedness of the sequences: |cwi | , • • • , |cw*-.i | , w = l, 2, • • • , the deduction of Theorem 2 required uniform convergence in \z\ <R. We give here a new proof for Theorem 2 with a weaker hypothesis assuming instead of uniform convergence only convergence at infinitely many points in some finite domain and boundedness of the sequences | cni | , | cn21. We further generalize the assumption on the location of the zeros (following a similar remark of Weisner [5]), assuming only that the zeros of Pn(z) lie in a halfplane containing the origin on its boundary, but otherwise varying with n. Finally we extend the results to certain sequences of entire functions.