Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonhomogeneous Wavelet Systems in High Dimensions   It is of interest to study a wavelet system with a minimum number of
generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in
[11] that for any $d\times d$ real-valued expansive matrix M, a homogeneous
orthonormal M-wavelet basis can be generated by a single wavelet function. On
the other hand, it has been demonstrated in [21] that nonhomogeneous wavelet
systems, though much less studied in the literature, play a fundamental role in
wavelet analysis and naturally link many aspects of wavelet analysis together.
In this paper, we are interested in nonhomogeneous wavelet systems in high
dimensions with a minimum number of generators. As we shall see in this paper,
a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system
with almost all properties preserved. We also show that a nonredundant
nonhomogeneous wavelet system is naturally connected to refinable structures
and has a fixed number of wavelet generators. Consequently, it is often
impossible for a nonhomogeneous orthonormal wavelet basis to have a single
wavelet generator. However, for redundant nonhomogeneous wavelet systems, we
show that for any $d\times d$ real-valued expansive matrix M, we can always
construct a nonhomogeneous smooth tight M-wavelet frame in $L_2(R^d)$ with a
single wavelet generator whose Fourier transform is a compactly supported
$C^\infty$ function. Moreover, such nonhomogeneous tight wavelet frames are
associated with filter banks and can be modified to achieve directionality in
high dimensions. Our analysis of nonhomogeneous wavelet systems employs a
notion of frequency-based nonhomogeneous wavelet systems in the distribution
space. Such a notion allows us to separate the perfect reconstruction property
of a wavelet system from its stability in function spaces.

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The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph