论文标题
傅立叶系列在本地田地的环形环上的曲线融合，并应用于Gabor Systems
Pointwise Convergence of Fourier Series on the Ring of Integers of Local Fields with an Application to Gabor Systems
论文作者
论文摘要
我们在$ p $ adiC字段$ \ q_p $的整数上构建了一个可集成功能的简单示例。另一方面，我们证明了$ l^p（\ d，w）$，$ 1 <p <\ infty $中的傅立叶系列功能的重点融合，其中$ \ d $是本地字段$ k $ and $ w $的整数圈，$ w $是muckenhoupt $ a_p $ class的重量。此结果包括特殊情况，当$ \ d $是$ \ q_p $的整数或字段$ \ mathbb {f} _q（（（x））$的字段$ \ mathbb {f} _q（（（x））$，在有限的字段$ \ mathbb {f} _q $上，尤其是当$ \ d $ \ d $ \ d $ whalsh-paley或wallad $ whalsh-paley groups时，为了实现这一目标，我们为与$ l^p（\ d，w）$中功能的傅立叶部分总和运算符相对应的最大运算符的加权估计。作为一个应用程序，我们以$ \ d \ d \ times \ d $上的$ a_2 $权重和生成Gabor System的$ \ d \ times \ d $的$ \ d \ times \ d $上的$ a_2 $ the $ \ d \ d \ times \ d $上的$ a_2 $重量来表征Gabor Systems的Schauder基础属性。给出了一些示例来说明此结果。特别是，我们构建了一个完整且最小的Gabor系统的示例，但并未成为$ l^2（k）$的chauder基础。
We construct a simple example of an integrable function on the ring of integers of the $p$-adic field $\Q_p$ having an almost everywhere divergent Fourier series. On the other hand, we prove the pointwise convergence of the Fourier series of functions in $L^p(\D,w)$, $1<p<\infty$, where $\D$ is the ring of integers of a local field $K$ and $w$ is a weight in the Muckenhoupt $A_p$ class. This result includes, as special cases, when $\D$ is the ring of integers of $\Q_p$ or the field $\mathbb{F}_q((X))$ of formal Laurent series over a finite field $\mathbb{F}_q$, and in particular, when $\D$ is the Walsh-Paley or dyadic group $2^ω$. To achieve this, we establish a weighted estimate for the maximal operator corresponding to the Fourier partial sum operators for functions in $L^p(\D,w)$. As an application, we characterize the Schauder basis property of the Gabor systems in a local field $K$ of positive characteristic in terms of the $A_2$ weights on $\D\times\D$ and the Zak transform $Zg$ of the window function $g$ that generates the Gabor system. Some examples are given to illustrate this result. In particular, we construct an example of a Gabor system which is complete and minimal, but fails to be a Schauder basis for $L^2(K)$.
