Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Closure of dilates of shift-invariant subspaces   Let $V$ be any shift-invariant subspace of square summable functions. We
prove that if for some $A$ expansive dilation $V$ is $A$-refinable, then the
completeness property is equivalent to several conditions on the local
behaviour at the origin of the spectral function of $V$, among them the origin
is a point of $A^*$-approximate continuity of the spectral function if we
assume this value to be one. We present our results also in the more general
setting of $A$-reducing spaces. We also prove that the origin is a point of
$A^*$-approximate continuity of the Fourier transform of any semiorthogonal
tight frame wavelet if we assume this value to be zero.

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The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
5
The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
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The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph