Characterizations of the stability and orthonormality of a multivariate matrix refinable function D with arbitrary matrix dilation M are provided in terms of the eigenvalue and 1-eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of D is equivalent to the order of the vanishing moment conditions of the matrix refinement mask {Pa}. The restricted transition operator associated with the matrix refinement mask {Pa } is represented by a finite matrix (Ami-j)i,j, with A -Idet(M)I-1 E, P-j Q3 P,E and P,-j Q3 P, being the Kronecker product of matrices P,-j and P,. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function 4 is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.

/pdf/multivariate-matrix-refinable-functions-with-arbitrary-1l9wmtt0s9.pdf

Multivariate matrix refinable functions with arbitrary matrix dilation

This paper investigates spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions.
Let $\Phi=(\phi_1,\ldots,\phi_r)^T$ be an $r \times 1$ vector of compactly supported functions in $L_2(\Rs)$ satisfying $\,\Phi = \sum_{\ga\in\Zs} a(\ga) \Phi({M\kern .1em \cdot}-\ga)$, where M is an expansive integer matrix. The smoothness of $\Phi$ is measured by the Sobolev critical exponent $\gl (\Phi) := \sup \bigl\{ \gl: \int_{\Rs} |\hat \phi_j(\xi)|^2(1+|\xi|^\gl)^2\,d\xi<\infty, \, 1 \le j\le r \bigr\}$. Suppose %the mask $a$ is finitely supported, i.e., %the set $M$ is similar to %a diagonal matrix $\hbox{diag}(\gs_1,\ldots,\gs_s)$ with $|\gs_1|=\cdots=|\gs_s|$ and $\supp a := \{\ga\in\Zs: a(\ga) \ne 0\}$ is finite. For $\mu=(\mu_1,\ldots,\mu_s)\in\NN_0^s$, define $\gs^{-\mu} := \gs_1^{-\mu_1} \cdots \gs_s^{-\mu_s}$. Let $A := \sum_{\ga\in\Zs} a(\ga)/|\det M|$ and $b(\ga) := \sum_{\gb\in\Zs} \overline{a(\gb)} \otimes a(\ga+\gb)/|\det M|$, $\ga\in\Zs$, where $\,\otimes\,$ denotes the (right) Kronecker product. Suppose that the highest total degree of polynomials reproduced by $\Phi$ is $k-1$ and $\spec(A)$ (the spectrum of $A$) is $\{\eta_1,\eta_2, \ldots,\eta_r\}$ with $\eta_1=1$ and $\eta_j \ne 1, 2\le j\le r$. Set $$ E_k := \{ \eta_j \overline{\gs^{-\mu}}, \overline{\eta_j} \gs^{-\mu}: |\mu|

Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets

https://sites.ualberta.ca/~rjia/Paper01-05/CJR.pdf

Convergence of Vector Subdivision Schemes in Sobolev Spaces

In this paper we survey recent results on approximation power of refinable vectors of functions Let Φ = (φ1,    , φr) be an r × 1 vector of compactly supported functions in Lp(IR) (1 ≤ p ≤ ∞) The first part of this paper is devoted to an investigation of approximation power of S(Φ), the shift-invariant space generated from Φ We review results on characterizations of the approximation order of S(Φ) and describe approximation schemes that achieve the optimal approximation order We also give a self-contained treatment of various equivalent forms of the Strang-Fix conditions We say that Φ is refinable if Φ = ∑ α∈Z s a(α)Φ(M · − α), where M is an expansive s× s integer matrix, and the refinement mask a is finitely supported The second part of this paper is dedicated to a study of accuracy of Φ We review results on characterizations of the accuracy of Φ in terms of the mask in both time and frequency domains We also discuss the relationship between the accuracy of Φ and the sum rules associated with the mask Examples are provided to illustrate the general theory † Supported in part by NSERC Canada under Grant OGP 121336 Approximation Power of Refinable Vectors of Functions §

Approximation Power of Refinable Vectors of Functions

This paper gives an overview of recent achievements of the multiwavelet theory. The construction of multiwavelets is based on a multiresolution analysis with higher multiplicity generated by a scaling vector. The basic properties of scaling vectors such as L-stability, approximation order and regularity are studied. Most of the proofs are sketched.

From wavelets to multiwavelets

A second look at the authors' [BDR1], [BDR2] characterization of the approximation order of a Finitely generated Shift-Invariant (FSI) subspace $S(\Phi)$ of L
 2(R
 d
 ) results in a more explicit formulation entirely in terms of the (Fourier transform of the) generators $\varphi\in\Phi$ of the subspace. Further, when the generators satisfy a certain technical condition, then, under the mild assumption that the set of 1-periodizations of the generators is linearly independent, such a space is shown to provide approximation order k if and only if $\mathop{\rm span}\nolimits\{\varphi(\cdot-j): |j| < k, \varphi\in\Phi\}$ contains a $\psi$ (necessarily unique) satisfying $D^j\widehat{\psi}(\alpha)=\delta_j\delta_\alpha$ for |j|<k , $\alpha\in 2\pi{\Bbb Z}^d$ . The technical condition is satisfied, e.g., when the generators are $O(|\cdot|^{-\rho})$ at infinity for some $\rho$ >k+d. In the case of compactly supported generators, this recovers an earlier result of Jia [J1], [J2].

/pdf/approximation-orders-of-fsi-spaces-in-l2-rd-wohm2zidfo.pdf

Approximation Orders of FSI Spaces in L2(Rd)

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Approximation Orders of FSI Spaces in L2(Rd)