扩张周期函数族的多项基准则
A multi-term basis criterion for families of dilated periodic functions
摘要：
在本文中，我们制定了一个具体的方法来确定是否一个系统的扩张周期函数形成一个Riesz基在$L^2(0,1)$。这种方法依赖于Hedenmalm，Lindqvist和Seip大约20年前开发的一个通用框架，该框架将基础问题转化为一个关于相应解析乘数的零点和极点的本地化问题。我们的结果改进了以前制定的各种标准，它给出了充分条件可逆的乘子的傅里叶系数的尖锐估计。我们的重点是通过分析或精确的数值逼近的假设的具体验证。然后，我们研究的基础问题的配置文件在附近的一个非基础的家庭所产生的周期性跳函数。对于这些配置文件之一，$p$ -正弦函数，我们确定了一个阈值的基础上的问题，提高了最近发现的积极的答案。
英文摘要：
In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in $L^2(0,1)$. This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients. Our focus is on the concrete verification of the hypotheses by means of analytical or accurate numerical approximations. We then examine the basis question for profiles in a neighbourhood of a non-basis family generated by periodic jump functions. For one of these profiles, the $p$-sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.
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