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Search: MSC category 42C40 ( Wavelets and other special systems )

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1. CMB 2013 (vol 57 pp. 254)

Christensen, Ole; Kim, Hong Oh; Kim, Rae Young
On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle
The unitary extension principle (UEP) by Ron and Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators, and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

Keywords:Bessel sequences, frames, extension of wavelet Bessel system to tight frame, wavelet systems, unitary extension principle
Categories:42C15, 42C40

2. CMB 2013 (vol 56 pp. 729)

Currey, B.; Mayeli, A.
The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames
In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is a unitary representation of a wavelet group and $\Gamma$ is the abstract pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the form $\tau(\Gamma)\Psi$ where $\tau$ is a super-representation of $\pi$. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.

Keywords:frame, dilation, wavelet, Baumslag-Solitar group, shearlet
Categories:43A65, 42C40, 42C15

3. CMB 2013 (vol 56 pp. 745)

Fu, Xiaoye; Gabardo, Jean-Pierre
Dimension Functions of Self-Affine Scaling Sets
In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$.

Keywords:scaling set, self-affine tile, orthonormal multiwavelet, dimension function
Category:42C40

4. CMB 2011 (vol 55 pp. 424)

Yang, Jianbin; Li, Song
Convergence Rates of Cascade Algorithms with Infinitely Supported Masks
We investigate the solutions of refinement equations of the form $$ \phi(x)=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\:\phi(Mx-\alpha), $$ where the function $\phi$ is in $L_p(\mathbb R^s)$$(1\le p\le\infty)$, $a$ is an infinitely supported sequence on $\mathbb Z^s$ called a refinement mask, and $M$ is an $s\times s$ integer matrix such that $\lim_{n\to\infty}M^{-n}=0$. Associated with the mask $a$ and $M$ is a linear operator $Q_{a,M}$ defined on $L_p(\mathbb R^s)$ by $Q_{a,M} \phi_0:=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\phi_0(M\cdot-\alpha)$. Main results of this paper are related to the convergence rates of $(Q_{a,M}^n \phi_0)_{n=1,2,\dots}$ in $L_p(\mathbb R^s)$ with mask $a$ being infinitely supported. It is proved that under some appropriate conditions on the initial function $\phi_0$, $Q_{a,M}^n \phi_0$ converges in $L_p(\mathbb R^s)$ with an exponential rate.

Keywords:refinement equations, infinitely supported mask, cascade algorithms, rates of convergence
Categories:39B12, 41A25, 42C40

5. CMB 2009 (vol 53 pp. 133)

Moritoh, Shinya; Tomoeda, Kyoko
A Further Decay Estimate for the Dziubański-Hernández Wavelets
We give a further decay estimate for the Dziubański-Hernández wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley-Wiener theorem for ultradifferentiable functions.

Keywords:wavelets, ultradifferentiable functions
Categories:42C40, 46E10

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