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Bounded semigroups of matrices
In this note are proved two conjectures of Daubechies and Lagarias. The first asserts that if Z is a bounded set of matrices such that all left infinite products converge, then 8 generates a boundedExpand
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The finiteness conjecture for the generalized spectral radius of a set of matrices
The generalized spectral radius\g9(∑) of a set ∑ of n × n matrices is \g9(∑) = lim supk→∞\g9k(∑)1k, where \g9k(∑) = sup{ϱ(A1A2…Ak): each Ai ∈ ∑}. The joint spectral radius\g9(∑) is \g9(∑) = limExpand
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Self-affine tiles in ℝn
Abstract A self-affine tile in R n is a set T of positive measure with A ( T )=∪ d ∈ D ( T + d ), where A is an expanding n × n real matrix with |det( A )|= m an integer, and D ={ d ,  d 2 , ...,  dExpand
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Arbitrarily smooth orthogonal nonseparable wavelets in R 2
For each $r\in\N$, we construct a family of bivariate orthogonal wavelets with compact support that are nonseparable and have vanishing moments of order r or less. The starting point of theExpand
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Spectral Sets and Factorizations of Finite Abelian Groups
Aspectral setis a subsetΩofRnwith Lebesgue measure 0<μ(Ω)<∞ such that there exists a setΛof exponential functions which form an orthogonal basis ofL2(Ω). The spectral set conjecture of B. FugledeExpand
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Haar Type Orthonormal Wavelet Bases in R2
AbstractK.-H. Grochenig and A. Haas asked whether for every expanding integer matrix A ∈ Mn(ℤ) there is a Haar type orthonormal wavelet basis having dilation factor A and translation lattice ℤn. TheyExpand
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Let A be an expanding n n integer matrix with j det A j m A standard digit set D for A is any complete set of coset representatives for Z A Z Associated to a given D is a set T A D which is theExpand
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Orthonormal bases of exponentials for the n-cube
Any set that gives such an orthogonal basis is called a spectrum for . Only very special sets in R are spectral sets. However, when a spectrum exists, it can be viewed as a generalization of FourierExpand
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Lattice tiling and the Weyl—Heisenberg frames
Abstract. Let {\cal L} and {\cal K} be two full rank lattices in $ {\Bbb R}^d $. We prove that if $ {\rm v}({\cal L} ) = {\rm v}({\cal K}) $, i.e. they have the same volume, then there exists aExpand
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Integral self-affine tiles in ℝn I. Standard and nonstandard digit sets
We investigate the measure and tiling properties of integral self-affine tiles, which are sets of positive Lebesgue measure of the form T(A,@) = { £ * x A~'d^: all d}€@}, where AeMn(Z) is anExpand
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