We present a concrete method to build discrete biorthogonal systems such that the wavelet filters have any number of vanishing moments. Several algorithms are proposed to construct multivariate biorthogonal wavelets with any general dilation matrix and arbitrary order of vanishing moments. Examples are provided to illustrate the general theory and the advantages of the algorithms. §

We introduce an equivalence relation on the set of single wavelets of L (R) associated with an arbitrary dilation matrix. The corresponding equivalence classes are characterized in terms of the support of the Fourier transform of wavelets and it is shown that each of these classes is non-empty.

The regularity of a univariate compactly supported reenable function is known to be related to the spectral properties of an associated transfer operator. In the case of multivariate reenable functions with a general dilation matrix A, although factorization techniques, which are typically used in the univariate setting, are no longer applicable, we derive similar results that also depend on th...

The regularity of a univariate compactly supported reenable function is known to be related to the spectral properties of an associated transfer operator. In the case of multivariate reenable functions with a general dilation matrix A, although factorization techniques, which are typically used in the univariate setting, are no longer applicable, we derive similar results that also depend on th...

We generalize the Second Oversampling Theorem for wavelet frames and dual wavelet frames from the setting of integer dilations to real dilations. We also study the relationship between dilation matrix oversampling of semi-orthogonal Parseval wavelet frames and the additional shift invariance gain of the core subspace.

Abstract. In this note, we discuss dilation-theoretic matrix parametrizations of contractions and positive matrices. These parametrizations are then applied to some problems in quantum information theory. First we establish some properties of positive maps, or entanglement witnesses. Two further applications, concerning concrete dilations of completely positive maps, in particular quantum opera...

Based on the method for constructing tight wavelet frames of [RS2], we show that one can construct, for any dilation matrix, and in any spatial dimension, tight wavelet frames generated by compactly supported functions with arbitrarily high smoothness. AMS (MOS) Subject Classifications: Primary 42C15, Secondary 42C30

We study in this paper nonlinear subdivision schemes in a multivariate setting allowing arbitrary dilation matrix. We investigate the convergence of such iterative process to some limit function. Our analysis is based on some conditions on the contractivity of the associated scheme for the differences. In particular, we show the regularity of the limit function, in L and Sobolev spaces.