A class of nonharmonic Fourier series
A class of nonharmonic Fourier series
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DOI：
10.1090/s0002-9947-1952-0047179-6
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发表时间：
1952-02
影响因子：
1.3
通讯作者：
R. Duffin;A. Schaeffer
中科院分区：
文献类型：
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作者：
R. Duffin;A. Schaeffer
1.导论.序列{Xn}，n= 0，？一，二，我们说真实的或复数具有均匀密度1，如果存在常数L和a，|Xn-nl？L和对于n% m，lXn-Xml> 5> 0。这是一个比密度更具限制性的概念，因为，只考虑那些n> 0的Xn，很明显，一个均匀密度为1的序列具有由波利亚定义的密度等于1，但匡威则不成立。对于任何d> 0，均匀密度d的序列将在本文的后面部分定义。如果f（z）是指数型-y的整函数，0< ry< ar，即f（z）= O（e&zI）在所有方向上一致为zl-oo，则f（z）完全由其在任何一致密度为1的序列上的值确定。指数型整函数的某些性质以一种自然的方式从均匀密度序列扩展到真实的轴或平行于真实的轴的带的所有点。例如，作者证明了[6]，如果一个指数型-y的整函数，o< ry<？r在一个密度为1的序列上一致有界，则它在整个真实的轴上一致有界.它在平行于真实的轴的每一条中也有一个边界。这一结果被应用于幂级数系数的问题。本文证明了均匀密度序列的一个进一步性质。它表明，如果f（z）是一个整函数的指数型，y，0？ty<i/r，属于真实的轴上的L2（-s，Io），并且{X}是均匀密度为1的序列，则比率{En}如果（Xn）|2}/如果% Y f（x）I 2dx具有与函数无关的正上界和下界。一个本质上等价的陈述是，如果g（t）CL 2（-y，Iy），其中O<^ y<-7r，并且{AXn}是均匀密度为1的序列，则存在与函数g（t）无关的正常数A和B，使得
1. Introduction. A sequence {Xn}, n= 0,? 1, 2,..., of real or complex numbers we shall say has uniform density 1 if there are constants L and a such that| Xn-nl? L and lXn-Xml> 5> 0 for n% m. This is a more restrictive notion than density, for, considering only those Xn for which n> 0, it is clear that a sequence of uniform density 1 has a density as defined by Polya equal to 1, but the converse is not true. Sequences of uniform density d are defined in a later part of the present paper for any d> 0. If f (z) is an entire function of exponential type-y, 0< ry< ar, that is, f (z)= O (e&zI) uniformly in all directions as zl-oo, then f (z) is completely determined by its values at any sequence of uniform density 1. Some properties of entire functions of exponential type extend in a natural way from a sequence of uniform density to all points of the real axis or of a strip parallel to the real axis. For example, the authors have shown [6] that if an entire function of exponential type-y, o< ry<? r, is uniformly bounded at a sequence of uniform density 1, then it is uniformly bounded on the entire real axis. It also has a bound in every strip parallel to the real axis. This result was applied to ques-tions concerning the coefficients of power series. In the present paper a further property of sequences of uniform density is proved. It is shown that if f (z) is an entire function of exponential type, y, 0? ty< i7r, belonging to L2 (-s, Io) on the real axis and {X} is a sequence of uniform density 1, then the ratio {En If (Xn)| 2}/If% Y f (x) I 2dx has positive upper and lower bounds independent of the function. An essentially equivalent statement is that if g (t) CL2 (-y, Iy) where O<^ y<-7r, and {AXn} is a sequence of uniform density 1, then there are positive constants A and B independent of the function g (t) such that