74,949 research outputs found
Uncertainty constants and quasispline wavelets
In 1996 Chui and Wang proved that the uncertainty constants of scaling and
wavelet functions tend to infinity as smoothness of the wavelets grows for a
broad class of wavelets such as Daubechies wavelets and spline wavelets. We
construct a class of new families of wavelets (quasispline wavelets) whose
uncertainty constants tend to those of the Meyer wavelet function used in
construction.Comment: 27 page
On The Continuous Steering of the Scale of Tight Wavelet Frames
In analogy with steerable wavelets, we present a general construction of
adaptable tight wavelet frames, with an emphasis on scaling operations. In
particular, the derived wavelets can be "dilated" by a procedure comparable to
the operation of steering steerable wavelets. The fundamental aspects of the
construction are the same: an admissible collection of Fourier multipliers is
used to extend a tight wavelet frame, and the "scale" of the wavelets is
adapted by scaling the multipliers. As an application, the proposed wavelets
can be used to improve the frequency localization. Importantly, the localized
frequency bands specified by this construction can be scaled efficiently using
matrix multiplication
Wavelets and graph -algebras
Here we give an overview on the connection between wavelet theory and
representation theory for graph -algebras, including the higher-rank
graph -algebras of A. Kumjian and D. Pask. Many authors have studied
different aspects of this connection over the last 20 years, and we begin this
paper with a survey of the known results. We then discuss several new ways to
generalize these results and obtain wavelets associated to representations of
higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets"
associated to a higher-rank graph. Here, we generalize this construction to
build wavelets of arbitrary shapes. We also present a different but related
construction of wavelets associated to a higher-rank graph, which we anticipate
will have applications to traffic analysis on networks. Finally, we generalize
the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a
third family of wavelets associated to higher-rank graphs
Generalized Morse Wavelets as a Superfamily of Analytic Wavelets
The generalized Morse wavelets are shown to constitute a superfamily that
essentially encompasses all other commonly used analytic wavelets, subsuming
eight apparently distinct types of analysis filters into a single common form.
This superfamily of analytic wavelets provides a framework for systematically
investigating wavelet suitability for various applications. In addition to a
parameter controlling the time-domain duration or Fourier-domain bandwidth, the
wavelet {\em shape} with fixed bandwidth may be modified by varying a second
parameter, called . For integer values of , the most symmetric,
most nearly Gaussian, and generally most time-frequency concentrated member of
the superfamily is found to occur for . These wavelets, known as
"Airy wavelets," capture the essential idea of popular Morlet wavelet, while
avoiding its deficiencies. They may be recommended as an ideal starting point
for general purpose use
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