Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pointwise Convergence of Fourier Series on the Ring of Integers of Local
  Fields with an Application to Gabor Systems   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 $2^\omega$. 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 $L^2(K)$.

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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex