Regularity of anisotropic refinable functions

TL;DR

This paper develops a method to precisely compute the H"older regularity of anisotropic refinable functions, extending previous univariate and isotropic results to the more complex anisotropic multivariate setting.

Contribution

It introduces an exact formula for H"older regularity in anisotropic cases and provides an efficient algorithm for its computation, advancing understanding of multivariate wavelet regularity.

Findings

Exact H"older exponent formula for anisotropic refinable functions

Efficient algorithm for spectral analysis of transition matrices

Analysis of higher and local regularity, and convergence rates

Abstract

This paper presents a detailed regularity analysis of anisotropic wavelet frames and subdivision. In the univariate setting, the smoothness of wavelet frames and subdivision is well understood by means of the matrix approach. In the multivariate setting, this approach has been extended only to the special case of isotropic refinement with the dilation matrix all of whose eigenvalues are equal in the absolute value. The general anisotropic case has resisted to be fully understood: the matrix approach can determine whether a refinable function belongs to $C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p < \infty$, but its H\"older regularity remained mysteriously unattainable. It this paper we show how to compute the H\"older regularity in $C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p < \infty$. In the anisotropic case, our expression for the exact H\"older exponent of a refinable function reflects the impact of the variable moduli of the eigenvalues of the corresponding dilation matrix. In the isotropic case, our results reduce to the well-known facts from the literature. We provide an efficient algorithm for determining the finite set of the restricted transition matrices whose spectral properties characterize the H\"older exponent of the corresponding refinable function. We also analyze the higher regularity, the local regularity, the corresponding moduli of continuity, and the rate of convergence of the corresponding subdivision schemes. We illustrate our results with several examples.

Equations

$$\phi (x)=\sum _{k\in {\mathbb{Z}}^s}{c}_k\phi (Mx-k),x\in {\mathbb{R}}^s,$$

$${\mathbb{R}}^s=\underset{i=1}{\overset{q(M)}{⨁}}{J}_i$$

$$B^{-1}MB\quad=\quad\left(\begin{array}[]{cccc}M|_{J_{1}}&0&\cdots&0\\ 0&M|_{J_{2}}&&\vdots\\ \vdots&&\ddots&0\\ 0&\cdots&0&M|_{J_{q}}\end{array}\right),$$

$$G=\left\{\sum _{i=1}^{\infty }{M}^{-i}{d}_i:d_i\in D(M)\right\}.$$

$$\chi (x)=\sum _{d\in D(M)}\chi (Mx-d)x\in {\mathbb{R}}^s;$$

$$G_{d_1\dots d_n}=M_{d_1}^{-1}\cdots M_{d_n}^{-1}G,d_1,\dots ,d_n\in D(M).$$

$$\sum _{k\in {\mathbb{Z}}^s}{c}_{Mk-d}=1,d\in D(M).$$

$$\mathbf{T}f(x)=\sum _{k\in {\mathbb{Z}}^s}{c}_{k}f(Mx-k),x\in {\mathbb{R}}^s.$$

$$K\quad=\quad\Bigl{\{}\ x\in{\mathbb{R}}^{s}\quad:\quad x\ =\ \sum_{j=1}^{\infty}\ M^{-j}\gamma_{j},\ \ \gamma_{j}\in\hbox{supp}(c),\ \ c=\{c_{k}\}_{k\in{\mathbb{Z}}^{s}}\Bigl{\}}$$

$$K\subseteq \Omega +G=\bigcup _{k\in \Omega }(k+G).$$

$$v:G\rightarrow{\mathbb{R}}^{N},\quad v(x)\ =\ v_{f}(x)\ =\ \Bigl{(}\,f(x+k)\,\Bigr{)}_{k\in\Omega}\,,\quad x\in G\,.$$

$$\Bigl{\{}\,f\in L_{1}({\mathbb{R}}^{s})\quad:\quad{\rm supp}\,f\,\subset\,\Omega+G\,,\Bigr{\}}$$

$$(\mathbf{A}v)(x)=T_{d}v(Mx-d),x\in M^{-1}(d+G),d\in D(M),$$

$$(T_d{)}_{ab}=c_{Ma-b+d},a,b\in \Omega ,d\in D(M).$$

$$\mathcal{T}=\{T_d:d\in D(M)\}.$$

$$v(x)=T_{d}v(Mx-d),x\in M^{-1}(d+G),d\in D(M).$$

$$V\ =\ \Bigl{\{}\,w\,=\,(w_{1},\ldots,w_{N})^{T}\in{\mathbb{R}}^{N}\quad:\quad\sum_{j=1}^{N}w_{j}\,=\,1\,\Bigr{\}}.$$

$$\sum _{k\in {\mathbb{Z}}^s}\phi (x+k)\equiv 1x\in {\mathbb{R}}^s.$$

$$W\quad=\quad\Bigl{\{}\,w\ =\ (w_{1},\ldots,w_{N})^{T}\in{\mathbb{R}}^{N}\quad:\quad\sum_{j=1}^{N}w_{j}=0\,\bigr{\}}\,.$$

$$U\ =\ {\rm span}\,\Bigl{\{}\,v(y)\,-\,v(x)\quad:\quad x,y\,\in G\,\Bigr{\}},\quad n={\rm dim}\,U.$$

$$\mathcal{A}=\mathcal{T}{|}_U=\{A_d:d\in D(M)\}$$

$$U_{i}\ =\ {\rm span}\,\Bigl{\{}v(y)\,-\,v(x)\quad:\quad x,y\in G,\ y-x\in J_{i}\Bigr{\}}\,,\quad i=1,\ldots,q(M).$$

$$v(y^{\prime})-v(x^{\prime})\ =\ A_{d}\,\bigl{(}v(My^{\prime}-d)\,-\,v(Mx^{\prime}-d)\bigr{)}\ =\ A_{d}\,\bigl{(}v(y)\,-\,v(x)\bigr{)}\,.$$

$$\rho (\mathcal{A})=\underset{k\to \infty }{\lim }\underset{A_{d_i}\in \mathcal{A},i=1,\dots ,k}{\max }\Vert A_{d_1}\cdots A_{d_k}{\Vert }^{1/k}.$$

$$\rho({\mathcal{A}})\quad=\quad\inf\,\Bigl{\{}\,\beta\geq 0\ :\ \exists\ \|\cdot\|\ \hbox{in}\ {\mathbb{R}}^{n}\ \hbox{such that}\ \|A\|<\beta,\ A\in{\mathcal{A}}\,\Bigr{\}}.$$

$${\rho }_i=\rho (\mathcal{A}{|}_{U_i}),i=1,\dots ,q(M).$$

$$\alpha_{\varphi}\ =\ \sup\,\bigl{\{}\alpha\geq 0\ :\ \|\varphi(\cdot+h)-\varphi\|_{C({\mathbb{R}}^{s})}\leq C\|h\|^{\alpha},\ \ h\in{\mathbb{R}}^{s}\bigr{\}}$$

$${\alpha }_{\phi }=\underset{i=1,\dots ,q(M)}{\min }{\log }_{1/r_i}{\rho }_i.$$

$$\alpha_{\varphi,J}\quad=\quad\sup\,\bigl{\{}\alpha\geq 0\ :\ \|\varphi(y)-\varphi(x)\|\leq C\|y-x\|^{\,\alpha}\,,\ y-x\in J\,\bigr{\}}\,.$$

$${\alpha }_{\phi ,J_i}={\log }_{1/r_i}{\rho }_i,i=1,\dots ,q(M).$$

Full text

We show how to compute the Hölder regularity in $C({\mathbb{R}}^{s})$ or $L_{p}({\mathbb{R}}^{s})$, $1\leq p<\infty$. In the anisotropic case, our expression for the exact Hölder exponent of a refinable function reflects the impact of the variable moduli of the eigenvalues of the corresponding dilation matrix. In the isotropic case, our results reduce to the well-known facts from the literature. We also analyze the higher regularity of anistropic refinable functions. We illustrate our results with several examples.

Keywords: multivariate subdivision, wavelets and frames, refinable functions, Hölder regularity, anisotropic dilation matrix, transition matrix, joint spectral radius, invariant polytope algorithm, discrete shearlet transform

**Classification (MSCS): 65D17, 15A60, 39A99 **

1 Introduction

We study the multivariate refinement equation

[TABLE]

with a compactly supported sequence $c=\{c_{k}\ :\ k\in{\mathbb{Z}}\}$ of coefficients $c_{k}\in{\mathbb{R}}$ and with a general integer dilation matrix $M\in{\mathbb{Z}}^{s\times s}$ all of whose eigenvalues are larger than one in the absolute value. We do not make any assumptions on the stability of the integer shifts of $\varphi$.

In this paper, we characterize continuous and $L_{p}$ solutions of (1). Our main contribution is the exact expression for the Hölder exponent of $\varphi$ in $C({\mathbb{R}}^{s})$ and in $L_{p}({\mathbb{R}}^{s})$, $1\leq p<\infty$, see Theorems 1 and 5. In the anisotropic case, the Hölder exponent of $\varphi$ reflects the influence of the invariant subspaces of $M$ corresponding to its different by modulus eigenvalues. In the isotropic case, when all the eigenvalues of $M$ are equal in the absolute value, our results reduce to the well-known ones from the literature. We also analyze the higher regularity of continuous $\varphi$.

It is well known that compactly supported solutions (refinable functions) of (1) can generate systems of multivariate wavelets or frames, see e.g [11, 12, 19, 44, 48]. Refinable functions are building blocks for the limits of subdivision algorithms widely used in approximation and for generating curves and surfaces, see e.g. [5, 13, 25, 49, 65]. Refinable functions naturally appear in recent applications in combinatorics, multigrid, number theory, and probability  [7, 8, 21, 27, 43, 52, 57].

In the univariate case, $M\geq 2$ is an integer, there are several efficient methods for determining the regularity of refinable functions. In [14, 22, 26] the authors compute precisely the Sobolev exponent of $\varphi\in L_{2}({\mathbb{R}})$. The so-called matrix approach yields the Hölder exponent of $\varphi\in C({\mathbb{R}})$ and, in addition, provides a detailed analysis of its moduli of continuity and of its local regularity [17, 20, 53, 58]. An obstacle to the practical use of the matrix approach is the NP-hardness of the joint spectral radius computation. This problem, however, was successfully resolved for a large class of problems by recent results in [30, 31, 47] where the authors presented fast and efficient methods of the joint spectral radius computation. Indeed, the invariant polytope algorithm [30] estimates the joint spectral radius for the corresponding transition matrices of size up to $20$ and, in most cases, even determines its precise value.

The generalization of the matrix approach to the multivariate case turned out to be a difficult task in the case of general dilation matrices. The special case of isotropic dilation is currently fully understood, see [5, 6, 10, 14, 22, 25, 26, 32, 33, 34, 37, 38, 40, 42, 59]. Several partial results in the anisotropic case are also available: for characterizations of continuity and $L_{p}$, $1\leq p<\infty$, regularity of $\varphi$ see e.g. [4, 35]; for estimates for the Hölder exponent of $\varphi$ see e.g. [4, 35].

The reason for the difficulty of the anisotropic case is natural and hardly avoidable. In the univariate case, say $M=2$, the distance between two points $x,y\in{\mathbb{R}}$ can be expressed in terms of their binary expansions. The distance between the values $\varphi(x)$ and $\varphi(y)$ depends on the behavior of the products of certain square matrices derived from $c_{k}$, $k\in{\mathbb{Z}}$. These two observations establish a correlation between $|x-y|$ and $|\varphi(x)-\varphi(y)|$, which leads to the formula for the Hölder exponent of $\varphi$. This summarizes the essence of the matrix approach. In the multivariate case, one can similarly estimate the distance between $\varphi(x)$ and $\varphi(y)$ by suitable matrix products. The problem occurs at an unexpected point: the expression for the distance between $x,y\in{\mathbb{R}}^{s}$. One can try to use the corresponding $M$-adic expansions with a certain set of digits from ${\mathbb{Z}}^{s}$, but such expansions do not provide a clear estimate for the distance between $x$ and $y$. Indeed, unless the matrix $M$ is isotropic, multiplication by a high power of $M$ can enlarge distances differently in different directions. Hence, the points $M^{\ell}x$ and $M^{\ell}y$, $\ell\in{\mathbb{N}}$, whose $M$-adic expansions are essentially the same, may have different asymptotic behavior as $\ell\to\infty$. Remarkably simple examples show that a direct analogue of the isotropic formula for the Hölder exponent does not hold in the anisotropic case. Moreover, unless $M$ is isotropic, this formula never holds for Lipschitz refinable functions, see section 3.1.

Nevertheless, there are ways of treating the anisotropic case. In [3, 15, 16, 62], the authors consider special anisotropic Hölder and Sobolev spaces. We put emphasize on incorporating the spectral properties of the dilation matrix $M$ into the expression for the Hölder exponent of $\varphi$. Furthermore, we get rid of the $M$-adic expansions and base our analysis on geometric properties of tilings generated by $M$.

Our paper is organized as follows. In section 3, we characterize the continuity and determine the Hölder regularity of multivariate refinable functions, see Theorems 1 and 2. In subsection 3.2, we provide an algorithm for construction of continuous solutions of (1). We consider several examples and list several important special cases of Theorems 1 in subsection 3.1. The crucial steps of the proofs and actual proofs of Theorems 1 and 2 are given in subsections 3.3 and 3.5. We illustrate our results on two numerical examples in subsection 3.6. Both examples deal with continuity and Hölder regularity of refinable functions, the first one being of shearlet type [18]. In section 4, we show how to factorize smooth refinable functions and compute the Hölder exponents of their directional derivatives. In section 5, we analyze the existence of $L_{p}$-solutions of (1). We show that a direct analogue of the formula for the Hölder exponent (i.e. replacing the joint spectral radius by the $p$-radius) does not hold in $L_{p}$, $1\leq p<\infty$. To characterize the $L_{p}$ Hölder exponent of $\varphi$, we consider extended transition matrices, see subsections 5.3 and 5.4.

2 Background and notation

We use the standard notation for the function spaces $C,C^{k},L_{p}$, $1\leq p<\infty$. The space of vector-valued functions $f:X\to{\mathbb{R}}^{n}$ with components in $L_{p}$ is denoted by $L_{p}(X,{\mathbb{R}}^{n})$. We write $L_{p}(X)$, if the range space is fixed. The Schwartz space of smooth rapidly decreasing functions over ${\mathbb{R}}^{s}$ is denoted by ${\mathcal{S}}$, and ${\mathcal{S}}^{\prime}$ is the space of tempered distributions (distributions over ${\mathcal{S}}$ or distributions of slower growth). By $\mu(X)$ we denote the Lebesgue measure of a set $X\subset{\mathbb{R}}^{n}$ and by by $|\cdot|$ either a modulus of a complex number or the cardinality of a finite set. The norm $\|\cdot\|$ in finite dimensional spaces is always а Euclidean, unless stated otherwise.

2.1 Spectral properties of the dilation matrix

We assume that the integer dilation matrix $M\in{\mathbb{Z}}^{s\times s}$ is expansive, i.e., all its eigenvalues are larger than $1$ in the absolute value. Hence, $m=|{\rm det}\,M|\geq 2$. Exactly $n_{i}$ among the eigenvalues $|\lambda_{1}|\leq\cdots\leq|\lambda_{s}|$ of $M$ are in the absolute value equal to $r_{i}$, $i=1,\ldots,q(M)\leq s$. If $M$ is isotropic, then $q(M)=1$. For $i=1,\ldots,q(M)$, let $J_{i}\subseteq{\mathbb{R}}^{s}$, $\hbox{dim}(J_{i})=n_{i}$, be the eigenspaces of $M$ corresponding to the eigenvalues of modulus $r_{i}$. The space ${\mathbb{R}}^{s}$ is a direct sum

[TABLE]

of the subspaces $J_{1},\ldots,J_{q(M)}$. There exists an invertible transformation $B:{\mathbb{R}}^{s}\rightarrow{\mathbb{R}}^{s}$ such that $M$ has the following block diagonal structure

[TABLE]

where the operator $M|_{J_{i}}$ has all its eigenvalues equal to $r_{i}$ in the absolute value.

2.2 Dilation matrix and tiles

The matrix $M$ splits the integer lattice ${\mathbb{Z}}^{s}$ into $m$ equivalence (quotient) classes defined by the relation $x\sim y\,\Leftrightarrow\,y-x\in M\,{\mathbb{Z}}^{s}$. Choosing one representative $d_{i}\in{\mathbb{Z}}^{s}$ from each equivalence class, we obtain a set of digits $D(M)=\{d_{i}\ :i=0,\ldots,m-1\}$. We always assume that $0\in D(M)$. The standard choice is to take $D(M)={\mathbb{Z}}^{s}\cap M[0,1)^{s}$.

For every integer point $d\in{\mathbb{Z}}^{s}$, we denote by $M_{d}$, the affine operator $M_{d}\,x\,=\,Mx-d,\,x\in{\mathbb{R}}^{s}$. We use the notation $0.d_{1}d_{2}\ldots=\displaystyle\sum_{i=1}^{\infty}M^{-i}d_{i}$, $d_{i}\in D(M)$. Consider the following set

[TABLE]

By [28, 29], for every expansive integer matrix $M$ and for an arbitrary set of digits $D(M)$, the set $G$ is a compact set with a nonempty interior and possesses the properties:

a) the Lebesgue measure $\mu(G)\in{\mathbb{N}}$;

b) $\ G\,=\,\bigcup\limits_{d\in D(M)}M_{d}^{\,-1}\,G$, the sets $M_{d}^{\,-1}\,G$ have intersections of zero measure;

c) the indicator function $\chi=\chi_{G}(x)$ of $G$ satisfies the refinement equation

[TABLE]

d) $\displaystyle\sum_{k\in{\mathbb{Z}}^{s}}\chi(x+k)\,\equiv\,\mu(G)$, i.e., integer shifts of $\chi$ cover ${\mathbb{R}}^{s}$ with $\mu(G)$ layers;

e) $\mu(G)=1$ if and only if the function system $\{\chi(\cdot+k)\}_{k\in{\mathbb{Z}}^{s}}$ is orthonormal.

If $\mu(G)=1$, then $G$ is called a tile. The integer shifts of a tile define a tiling.

Definition 1**.**

A tiling generated by an integer expansive matrix $M$ and by a set of digits $D(M)$ is a collection of sets ${\mathcal{G}}=\{k+G\}_{k\in{\mathbb{Z}}^{s}}$ such that

a) the union of the sets in ${\mathcal{G}}$ covers ${\mathbb{R}}^{s}$ and $\mu\left((\ell+G)\cap(k+G)\right)=0$, $\ell\not=k$;

b) $\displaystyle\ G\,=\,\cup_{d\in D(M)}M_{d}^{\,-1}\,G$.

Not every $M$ possesses a digit set $D(M)$ such that $G$ is a tile. Those situations, however, are rare. For instance, a digit set generating a tile always exists in cases $s=2,3$ and also for arbitrary $s$ with an extra assumption $\,|{\rm det}\,M|>s$, which is quite general for integer expansive matrices [45]. See [4, 46] for more details. Thus, in this paper, we assume that $G$ is a tile.

We denote

[TABLE]

Then ${\mathcal{G}}^{n}\,=\,M^{-n}{\mathcal{G}}=\{M^{-n}(k+G)\}_{k\in{\mathbb{Z}}^{s}}=\,\{G_{d_{1}\ldots d_{n}}\ :\ d_{1},\ldots,d_{n}\in D(M)\}$, $n\in{\mathbb{N}}$.

2.3 Refinable functions and the transition operator

A compactly supported distribution $\varphi\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$ satisfying equation (1) is called a refinable function. It is well known that the solution of (1) such that $\int_{{\mathbb{R}}^{s}}\varphi(x)\,dx\neq 0$ exists if and only if $\displaystyle\sum_{k\in{\mathbb{Z}}^{s}}c_{k}=m$. We assume further that the coefficients of (1) satisfy sum rules of order one

[TABLE]

These conditions are necessary for existence of stable refinable functions [5]. Consider the *transition operator * ${\boldsymbol{T}}:\,{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})\,\to\,{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$ defined by

[TABLE]

For every compactly supported function $f\in{\mathcal{S}}^{\prime}$ such that $\int_{{\mathbb{R}}^{s}}f(x)\,dx=1$, the sequence $\{{\boldsymbol{T}}^{j}f\}_{j\in{\mathbb{N}}}$ converges to $\varphi$ in the space ${\mathcal{S}}^{\prime}$ [4]. The space of distributions supported on the set

[TABLE]

is invariant under ${\boldsymbol{T}}$. Hence, for $f\in{\mathcal{S}}^{\prime}(K)$, we have ${\boldsymbol{T}}^{j}f\in{\mathcal{S}}^{\prime}(K)$ for all $j\in{\mathbb{N}}$. Therefore, the limit $\varphi\in{\mathcal{S}}^{\prime}(K)$. Thus, ${\rm supp}\,\varphi\,\subseteq\,K$, see [4, Proposition 2.2].

Definition 2**.**

A finite set $\Omega\subset{\mathbb{Z}}^{s}$ is a minimal subset of ${\mathbb{Z}}^{s}$ with the property

[TABLE]

We denote $\,N\,=\,|\Omega|$.

It is shown easily that $M_{d}^{-1}(\Omega+G)\,\subseteq\,\Omega+G$ for every $d\in D(M)$.

The main idea of the matrix approach is to pass from a function $f:{\mathbb{R}}^{s}\to{\mathbb{R}}$ supported on $K$ to the vector-valued function

[TABLE]

Then the transition operator (5) restricted to the space

[TABLE]

becomes the self-similarity operator ${\boldsymbol{A}}:L_{1}(G,{\mathbb{R}}^{N})\rightarrow L_{1}(G,{\mathbb{R}}^{N})$ defined by

[TABLE]

where $T_{d}$ are the $N\times N$ transition matrices defined by

[TABLE]

The rows and columns of the matrices $T_{d}$ are enumerated by elements from the set $\Omega$. We denote

[TABLE]

The refinement equation becomes the self-similarity equation ${\boldsymbol{A}}v=v$ for the vector-valued function $v(x)=v_{\varphi}(x)$ defined by (7) with $f=\varphi$, i.e.

[TABLE]

2.4 Important subspaces of ${\mathbb{R}}^{N}$

We consider the following affine subspace of the space ${\mathbb{R}}^{N}$

[TABLE]

It is well known [4, 5] that every compactly supported refinable function $\varphi\in{\mathcal{S}}^{\prime}$ such that $\int_{{\mathbb{R}}^{s}}\varphi(x)\,ds=1$ possesses the partition of unity property:

[TABLE]

Hence, if $\varphi$ is continuous, then $v(x)\in V$ for all $x\in G$. In particular $v(0)\in V$. For summable refinable function, $v(x)\in V$ for almost all $x\in G$. We denote the linear part of $V$ by

[TABLE]

Finally, every continuous refinable function defines the difference space

[TABLE]

Since $v(x)\in V$ for all $x\in G$, we have $U\subseteq W$, and, therefore, $n\leq N-1$. The sum rules (4) imply that the column sums of each matrix $T_{d}$ are equal to one. Therefore, $T_{d}V\subseteq V$ and $T_{d}W\subseteq W$. Thus, $V$ is a common affine invariant subspace of the family ${\mathcal{T}}$ and $W$ is its common linear invariant subspace.

Since $U$ is invariant under all $T_{d},\,d\in D(M)$, the restrictions $A_{d}=T_{d}|_{U}$ of the operators $T_{d}$, $d\in D(M)$, to the subspace $U$ are well defined. For a fixed basis of $U$, we denote by

[TABLE]

the set of the associated $n\times n$ matrices. If the family ${\mathcal{T}}$ is irreducible on $W$, i.e., the operators in ${\mathcal{T}}$ do not share a common nontrivial invariant subspace of W, then ${\mathcal{A}}={\mathcal{T}}|_{W}$. We also consider the following subspaces of the space $U$

[TABLE]

Note that $U_{i}$ are nonempty, due to the interior $\hbox{int}(G)$ of $G$ being nonempty. It is seen easily that the spaces $\{U_{i}\}_{i=1}^{q(M)}$ span the whole space $U$, but their sum may not be direct. Indeed, the subspaces $\{U_{i}\}_{i=1}^{q(M)}$, unlike the subspaces $\{J_{i}\}_{i=1}^{q(M)}$, may have nontrivial intersections. For example, they can all coincide with $U$. Lemma 1 shows that all $U_{i}$ are invariant under ${\mathcal{A}}$.

Lemma 1**.**

If $J$ is an invariant subspace of $M$, then $L\,=\,{\rm span}\,\{v(y)-v(x)\ :\ y-x\in J\}$ is a common invariant subspace for ${\mathcal{A}}$.

Proof.

If $u\in L$, then $u$ is a linear combination of several vectors of the form $v(y)-v(x)$ with $y-x\in J$. For $x^{\prime}=M^{-1}(x+d)$, $y^{\prime}=M^{-1}(y+d)$, $d\in D(M)$, we have $y^{\prime}-x^{\prime}\in J$ and

[TABLE]

Hence, A_{d}\bigl{(}v(y)-v(x)\bigr{)}\in L for each pair $(x,y)$, and, therefore, $A_{d}u\in L$ for all $u\in L$. ∎

2.5 Joint spectral radius

Definition 3**.**

The joint spectral radius of a finite family ${\mathcal{A}}$ of linear operators $A_{d}$ is

[TABLE]

This limit always exists and does not depend on the operator norm [60]. The joint spectral radius measures the simultaneous contractibility of the operators from ${\mathcal{A}}$. Indeed, $\rho({\mathcal{A}})<1$ if and only if there exists a norm in ${\mathbb{R}}^{n}$ in which all $A\in{\mathcal{A}}$ are contractions. In general,

[TABLE]

We denote

[TABLE]

3 Continuous solutions and Hölder regularity

In this section, in Theorem 1, we characterize the continuity of a solution $\varphi$ of the refinement equation (1) in terms of the spectral properties of ${\mathcal{A}}$ and determine the exact Hölder exponent

[TABLE]

of $\varphi$. Although the definition of $U$ in (12) depends on $\varphi$, Propositions 1 and 2 remove this dependency. Moreover, the space $U$ can be determined explicitly using Algorithm 1 in subsection 3.2 without the knowledge of $\varphi$. If this algorithm fails, then there exists no continuous solution of the corresponding refinement equation. The special cases of Theorem 1 are considered in subsection 3.1, for its summary see Remark 2. The crucial result for the proof of Theorem 1 is Theorem 2. The main steps of the proof of Theorem 2 are summarized in subsection 3.3 and the proofs of Theorems 1 and 2 are given in subsection 3.5. We illustrate our results with examples in subsection 3.6. For the readers convenience, we first list some of the above mentioned crucial results.

Proposition 1**.**

Let $v_{0}\in V$ be an eigenvector of $T_{0}$ associated to the eigenvalue $1$. If $\varphi\in C({\mathbb{R}}^{s})$, then $U$ is the smallest by inclusion common invariant subspace of the matrices $T_{d},\,d\in D(M)$, that contains the $m$ vectors $\,T_{d}v_{0}-v_{0},\,d\in D(M)$.

For the proof of Proposition 1 see the arXive version of this paper [9]. Our proof is the multivariate analogue of [17, Proposition 3].

Remark 1**.**

Recall that $0\in D(M)$, which justifies the notation $T_{0}$. The existence of the eigenvector $v_{0}\in V$ of $T_{0}$ associated to the eigenvalue $1$ follows by by (11), i.e. $T_{0}v(0)\,=\,v(0)$ and by the continuity $\varphi$ implying that $v(0)\in V$.**

Similarly to [17], Proposition 1 yields a useful characterization of $U$.

Proposition 2**.**

Let $v_{0}\in V$ be an eigenvector of $T_{0}$ associated to the eigenvalue $1$. The space $U$ is the minimal common invariant subspace of the $m$ matrices $T_{d},\,d\in D(M)$, and $U$ contains the $m$ vectors $T_{d}v_{0}-v_{0},\,d\in D(M)$.

If the eigenvalue $1$ of $T_{0}$ is not simple, Proposition 3 in subsection 3.2 guarantees that there exists at most one eigenvector $v_{0}\in V$ associated with one of these eigenvalues $1$ such that $\rho({\mathcal{A}})<1$. Thus, the subspace $U$ can be computed, unless the refinement equation does not possess a continuous solution. For simplicity, we make the following assumption.

Assumption 1**.**

The matrix $T_{0}$ has a simple eigenvalue $1$.

Recall that $\rho_{i}\ =\ \rho({\mathcal{A}}|_{U_{i}})$, where the subspaces $U_{1},\ldots,U_{q(M)}$ are defined in (14).

Theorem 1**.**

A refinable function $\varphi$ belongs to $C({\mathbb{R}}^{s})$ if and only if $\rho({\mathcal{A}})<1$. In this case,

[TABLE]

The proof of (15) is based on Theorem 2. To state it, we define the Hölder exponent of $\varphi$ along a linear subspace $J\subset{\mathbb{R}}^{s}$ by

[TABLE]

Theorem 2**.**

If $\varphi\in C({\mathbb{R}}^{s})$, then

[TABLE]

Remark 2**.**

The identity (15) emphasizes the influence of the spectral structure of the dilation matrix $M$ on the regularity of the solution $\varphi$. Recall that, in the univariate case, the Hölder exponent is given by $\alpha_{\varphi}\,=\,\log_{1/r}\,\rho({\mathcal{A}})$, where $M=r\geq 2$ is the corresponding dilation factor. In the multivariate case, the Hölder exponent is equal to the minimum of several such values taken over different dilation coefficients $r_{i}$ on the corresponding subspaces $J_{i}$ of $M$. In special, favorable multivariate cases, the expression in (15) becomes $\alpha_{\varphi}=\log_{1/\rho(M)}\,\rho({\mathcal{A}})$ and, thus, resembles the univariate case. This happens, for instance, when the matrix $M$ is isotropic, i.e. $|\lambda_{1}|=\ldots=|\lambda_{s}|=\rho(M)$, in particular, when $M=r\,I$, $r\geq 2$. Another favorable situation is when the matrices in ${\mathcal{A}}$ do not possess any common invariant subspace. However, the need for the minimum in (15) is not exceptional. It is of crucial importance e.g. for anisotropic refinable Lipschitz continuous functions $\varphi$, see Corollary 3 in subsection 3.1.**

3.1 Special cases of Theorem 1 and examples

To compare the result of Theorem 1 with the known results from the literature, we need to define the stability of $\varphi$.

Definition 4**.**

A compactly supported $f\in L_{\infty}({\mathbb{R}}^{s})$ is stable, if there exists $0<C_{1}\leq C_{2}<\infty$ such that for all ${\boldsymbol{c}}\in\ell_{\infty}({\mathbb{Z}}^{s})$,

[TABLE]

The univariate case ($\mathbf{s=1}$). In this case, the dilation factor is $M\geq 2$ and $M=m=r$. Theorem 1 becomes a well-known statement that $\alpha_{\varphi}=\log_{1/r}\rho({\mathcal{A}})$. If $\varphi$ is stable, then we have $\rho({\mathcal{A}})=\rho({\mathcal{T}}|_{U})=\rho({\mathcal{T}}|_{W})$ even if $U\neq W$ (see [5]). The space $U$ was completely characterized in [54] and it was shown that every refinement equation can be factorized to the case $U=W$. In the multivariate case, however, there is no general factorization procedure and, even in the stable case, we cannot expect $U=W$, see Example 1 below.

The case $\mathbf{s\geq 2}$ with isotropic dilation matrix. Since $q(M)=1$, it follows that $U_{1}=U$. Theorem 1 then implies the following well-known fact.

Corollary 1**.**

If $M$ is isotropic, then $\alpha_{\varphi}\,=\,\log_{\,1/\rho(M)}\rho({\mathcal{A}})$.

The irreducible case with $\mathbf{s\geq 2}$. The dilation matrix $M$ can be anisotropic, i.e. the number of different in modulus eigenvalues of $M$ is $q(M)>1$. We say that the set of matrices ${\mathcal{A}}={\mathcal{T}}|_{\,U}$ is irreducible, if they do not possess any common invariant subspace. Another corollary of Theorem 1 states the following.

Corollary 2**.**

If the family ${\mathcal{A}}$ is irreducible, then $\alpha_{\varphi}\,=\,\log_{\,1/\rho(M)}\rho({\mathcal{A}})$.

The irreducibility assumption fails however in many important cases. For instance, if $\varphi$ is a tensor product of two refinable functions, then ${\mathcal{A}}$ is always reducible.

Example 1**.**

Let $\varphi_{1},\varphi_{2}\in C^{1}({\mathbb{R}})$ be two univariate refinable function with dilations $M_{1}=2$ and $M_{2}=3$ and refinement coefficients ${\boldsymbol{c}}_{1}\in\ell_{0}({\mathbb{Z}})$ and ${\boldsymbol{c}}_{2}\in\ell_{0}({\mathbb{Z}})$, respectively. Then the function $\varphi=\varphi_{1}\otimes\varphi_{2}$ satisfies the refinement equation with $M={\rm diag}\,(2\,,\,3)$ and  ${\boldsymbol{c}}={\boldsymbol{c}}_{1}\otimes{\boldsymbol{c}}_{2}$. Due to $\varphi_{1},\varphi_{2}\in C^{1}({\mathbb{R}})$, we have $\rho_{1}=\rho({\mathcal{A}}|_{U_{1}})=\frac{1}{2},\ \rho_{2}=\rho({\mathcal{A}}|_{U_{2}})=\frac{1}{3}$. By Theorem 1, $\alpha_{\varphi}\,=\,\min\,\{\log_{1/2}\rho_{1}\,,\,\log_{1/3}\rho_{2}\}=1$, which is natural, because $\varphi\in C^{1}({\mathbb{R}}^{2})$. On the other hand, \rho({\mathcal{A}})=\max\,\bigl{\{}\rho_{1}\,,\,\rho_{2}\bigr{\}}=\frac{1}{2}. Hence, $\log_{\,1/\rho(M)}\rho({\mathcal{A}})=\log_{1/3}\frac{1}{2}\,=\,0.630092\ldots$. Thus, in this case, $\alpha_{\varphi}\,>\,\log_{\,1/\rho(M)}\rho({\mathcal{A}})$. Note that, if $\varphi_{1}$ and $\varphi_{2}$ are both stable, then so is $\varphi$. Nevertheless, unlike in the univariate case, the Hölder exponent of $\varphi$ is not determined by the value $\log_{\,1/\rho(M)}\rho({\mathcal{A}})$. **

After Example 1 one may hope that the case of reducible family ${\mathcal{A}}$ is exceptional, and the equality $\alpha_{\varphi}=\log_{\,1/\rho(M)}\rho({\mathcal{A}})$ actually holds for most refinable functions. On the contrary, the result of Corollary 3 shows that the the situation when the isotropic formula fails is rather generic.

Corollary 3**.**

If the matrix $M$ is anisotropic and the refinable function $\varphi\not=0$ is Lipschitz continuous, then $1=\alpha_{\varphi}>\log_{1/\rho(M)}\rho({\mathcal{A}})$ and the family ${\mathcal{A}}$ is reducible.

Proof.

Assume that $1\,\leq\,\log_{\,1/\rho(M)}\rho({\mathcal{A}})$, or, equivalently, $\rho({\mathcal{A}})\leq 1/\rho(M)$. Since $M$ is anisotropic, factorization (2) contains $q(M)\geq 2$ blocks, and, hence, $r_{i}<\rho(M)$ for some $i\in\{1,\ldots,q(M)\}$. By Theorem 2, we have $\alpha_{\varphi,J_{i}}\,=\,\log_{1/r_{i}}\rho({\mathcal{A}})\,>\,\log_{1/\rho(M)}\rho({\mathcal{A}})\,\geq\,1$. Therefore, $\varphi$ is constant on every affine subspace $u+U_{i}$, $u\in{\mathbb{R}}^{n}$. Hence, $\varphi\equiv 0$, because it is compactly supported. The reducibility of ${\mathcal{A}}$ follows by Corollary 2. ∎

Thus, we see that at least for all anisotropic smooth refinable functions, the simple formula for the Hölder exponent fails and the minimum in (15) is significant.

The case of a dominant invariant subspace. In practice, this case is much more generic than the irreducible case.

Definition 5**.**

A subspace $U^{\prime}\subset U$ is called dominant for a family of operators ${\mathcal{A}}$ if

$(i)$

$U^{\prime}$* is a common invariant subspace of ${\mathcal{A}}$,*

$(ii)$

$U^{\prime}$* is contained in all common invariant nontrivial subspaces of ${\mathcal{A}}$ and*

$(iii)$

$\rho({\mathcal{A}}|_{U^{\prime}})\,=\,\rho({\mathcal{A}})$.

Take a basis of a dominant subspace $U^{\prime}$ and complement it to a basis of $U$. Let $B$ be the $n\times n$ matrix containing these basis elements of $U$. Then every matrix $A_{d}\in{\mathcal{A}}$ in this basis has a block lower triangular form

[TABLE]

By Definition 5,

[TABLE]

Furthermore, since any common invariant subspace of ${\mathcal{A}}$ contains $U^{\prime}$, it follows that the joint spectral radius of ${\mathcal{A}}$ restricted to any common invariant subspace is equal to $\rho({\mathcal{A}})$. Therefore, we have proved the following result.

Corollary 4**.**

If the family ${\mathcal{A}}$ possesses a dominant subspace, then $\,\alpha_{\varphi}\,=\,\log_{\,1/\rho(M)}\rho({\mathcal{A}})$.

3.2 Construction of $U$ and of the continuous refinable function $\varphi$.

In this section, we answer two crucial questions: how to determine the space $U$ and how to construct the corresponding continuous refinable function $\varphi$. In the univariate case, the algorithm for determining the space $U$ was elaborated in [17]. In this section, we present its multivariate analogue.

Algorithm for construction of the space $U$.

Algorithm 1: For a given set ${\mathcal{T}}=\{T_{d}\in{\mathbb{R}}^{N\times N}\ :\ d\in D(M)\}$ of transition matrices

$1$.Step: Compute $v_{0}$ in $T_{0}\,v_{0}=v_{0}$ and normalize $({\bf 1},v_{0})\,=\,1$, where ${\bf 1}=(1,\ldots,1)^{T}\in{\mathbb{R}}^{N}$.

$2$.Step: Define U^{(1)}={\rm span}\,\bigl{\{}T_{d}v_{0}\,-\,v_{0}\ :\ d\in D(M)\setminus\{0\}\bigr{\}}.

$3$.Step: Repeat

[TABLE]

while ${\rm dim}(U^{(k)})\ <\ {\rm dim}(U^{(k+1)})$.

[TABLE]

If $1$.Step is impossible, i.e., the eigenvector $v_{0}$ does not exist, then, by Remark 1, the solution of the refinement equation (1) is not continuous. Note that $1\leq k\leq N-1$ is dictated by ${\rm dim}\,U\,\leq\,N-1$ and that, by construction, at least one extra element is added to $U^{(k+1)}$ before the algorithm terminates. Proposition 1, stated at the beginning of section 3, implies that the space $U$ in Algorithm 1 coincides with the space $U$ in Definition 12.

Algorithm for construction of a continuous $\varphi$.

Due to the fact that the rational $M$-adic points are dense in $G$, the slight modification of Algorithm 1 yields a method for the step-by-step construction of the vector-valued function $v=v_{\varphi}$ defined on $G$ or, equivalently, of the function $\varphi$.

Algorithm 2: For a given set ${\mathcal{T}}=\{T_{d}\in{\mathbb{R}}^{N\times N}\ :\ d\in D(M)\}$ of transition matrices

$1$.Step: Compute $(0,v_{0})$ such that $T_{0}\,v_{0}=v_{0}$ and normalize $({\bf 1},v_{0})\,=\,1$.

$2$.Step: Define $V^{(0)}\,=\,\{(0,v_{0})\}$.

$3$.Step: For $k=1,2,3,\ldots$

[TABLE]

end

If the function $v$ is continuous, Algorithm 2 determines $V^{(k)}$ in a unique way.

The next result ensures that $U$ is well defined even if the eigenvalue $1$ of the matrix $T_{0}$ is not simple.

Proposition 3**.**

The matrix $T_{0}$ has at most one eigenvector $v_{0}\in V$ associated with the eigenvalue $1$ such that $\rho({\mathcal{A}})<1$. If such $v_{0}\in V$ exists, then $\varphi$ is continuous and $v_{0}=v_{\varphi}(0)$.

Proof.

By Algorithm 2, there exists a refinable function $\varphi$ with $v_{0}=v_{\varphi}(0)$. If there is another eigenvector $\tilde{v}_{0}\in V$ with this property, then, by Algorithm 2, it generates another refinable function $\tilde{\varphi}$ for which $\tilde{v}_{0}=v_{\tilde{\varphi}}(0)$. By the uniqueness of the solution of the refinement equation, these two solutions may only differ by a constant, hence, the vectors $v_{0}$ and $\tilde{v}_{0}$ are collinear. The continuity of $\varphi$ follows from Theorem 1. ∎

3.3 Road map of our main results

We would like to emphasize that, to tackle the anisotropic case, we use geometric properties of tilings rather than the $M$-adic expansions of points in ${\mathbb{R}}^{s}$ (the latter being a successful strategy in the isotropic case). Our key contribution is Theorem 2 that finally reveals the delicate dependency of the Hölder exponent of a refinable function on its Hölder exponents along the subspaces $J_{i}$, $i=1,\ldots,q(M)$. Due to the importance of Theorem 2, we would like to give here a preview of its proof.

Step 1. Extend the vector-valued function $v$ in (7) defined on the tile $G$ to the whole ${\mathbb{R}}^{s}$, see (19). Lemma 4 yields, for $x,y\in{\mathcal{G}}$ (i.e. $x,y\in G-j$ for some $j\in{\mathbb{N}}$), the estimate $\|\tilde{v}(y)-\tilde{v}(x)\|\,\leq\,\|v(y+j)-v(x+j)\|$. The extension of $v$ is motivated by the fact that parts of the line segment $[x,y]$ can lie outside of $G$, due to its possible fractal structure.

Step 2. Lemma 3 shows that, for a tiling ${\mathcal{G}}$ of ${\mathbb{R}}^{s}$, the total number of the subsets of the tiling intersected by a line segment is proportional to the length of that segment.

Step 3. Due to Step 2, Lemma 2 and Proposition 4 imply that, for $k\in{\mathbb{N}}$, any line segment $[x,y]$ in ${\mathbb{R}}^{s}$, $y-x\,\in\,J_{i}$, consists of several line segments such that 1) the endpoints of each of those line segments belong to one subset of the tiling ${\mathcal{G}}^{k}$; 2) the total number of those line segments is bounded by $C\,\|M^{k}(x-y)\|\,\asymp\,r_{i}^{k}\|x-y\|$.

Step 4. The difference between the values of the function $v$ at the endpoints of each of those subsegments of $[x,y]$ is bounded from above by $C_{1}\,(\rho_{i}+\varepsilon)^{k}$ for some $\varepsilon>0$. Hence, by Step 1, the same is true for $\tilde{v}$. Therefore, by the triangle inequality, $\|\tilde{v}(y)-\tilde{v}(x)\|\,\leq\,C_{2}(\rho_{i}+\varepsilon)^{k}$. For $k$ such that $r_{i}^{k}\,\asymp\,1/\|x-y\|$, we obtain $\|\tilde{v}(y)-\tilde{v}(x)\|\,\leq\,C_{3}\|x-y\|^{\,\alpha(\varepsilon)}$, where $\alpha(\varepsilon)$ approaches $\log_{1/r_{i}}\rho_{i}$ as $\varepsilon$ goes to [math].

3.4 Auxiliary results for Theorems 1 and 2

The proofs of our main results, Theorems 1 and 2, are based on an important observation formulated in Proposition 4. We also make use of the following basic properties of the joint spectral radius and two auxiliary lemmas.

Theorem A1 [60]. For a family of operators ${\mathcal{A}}$ acting in ${\mathbb{R}}^{n}$ and for any $\varepsilon>0$, there exists a norm $\|\cdot\|_{\varepsilon}$ in ${\mathbb{R}}^{n}$ such that $\|A\|_{\varepsilon}\,<\,\rho({\mathcal{A}})+\varepsilon$ for all $A\in{\mathcal{A}}$.

Theorem A2 [2]. *For a family of operators ${\mathcal{A}}$ acting in ${\mathbb{R}}^{n}$ there exists $u\in{\mathbb{R}}^{n}$ and a constant $C(u)>0$ such that $\max_{A_{d_{i}}\in{\mathcal{A}}}\|A_{d_{1}}\cdots A_{d_{k}}u\|\geq C(u)\,\rho({\mathcal{A}})^{\,k},\ k\in{\mathbb{N}}$. Moreover, if ${\mathcal{A}}$ is irreducible, then $\max_{A_{d_{i}}\in{\mathcal{A}}}\|A_{d_{1}}\cdots A_{d_{k}}\|\leq C\,\rho({\mathcal{A}})^{\,k},\ k\in{\mathbb{N}}$, for some constant $C>0$. *

Lemma 2**.**

Assume that the segment $[0,1]$ is covered with $\ell$ distinct closed sets. Then there exist $\ell+1$ points $0=a_{0}\leq\ldots\leq a_{\ell}=1,$ such that for each $i=0,\ldots,\ell-1$, the points $a_{i},a_{i+1}$ belong to one of these sets,.

Proof.

Let the first set contain the point $a_{0}=0$. Choose $a_{1}$ to be the maximal (in the natural ordering of the real line) point of the first set. If $a_{1}\neq 1$, then $a_{1}$ must belong to another set of the tiling. Choose $a_{2}$ to be the maximal point of this set. Repeat until $a_{\ell_{0}}=1$ for some $\ell_{0}\in{\mathbb{N}}$. We have $\ell_{0}\leq\ell$, since the sets are distinct. If $\ell_{0}<\ell$, we extend the sequence $a_{0}\leq\ldots\leq a_{\ell_{0}}$ by the points $a_{\ell_{0}+1},\ldots,a_{\ell}=1$. ∎

Next we show that a segment of a given length intersects finitely many sets of the tiling ${\mathcal{G}}$.

Lemma 3**.**

For a tiling ${\mathcal{G}}$, there exists a constant $C>0$ such that every line segment $[x,y]\in{\mathbb{R}}^{s}$ intersects at most $\,C\max\{1\,,\,\|y-x\|\,\}$ sets of ${\mathcal{G}}$.

Proof.

It suffices to prove that the number of sets $G+k\subset{\mathcal{G}}$ intersected by a segment of length $1$ is bounded above by $C>0$. It will imply that the number of sets $G+k\subset{\mathcal{G}}$ intersected by a segment of length $\|y-x\|>1$, is bounded by $C\,\|y-x\|$, and the claim follows. Thus, let a segment $[x,y]$ be of length $1$. If $(G+k)\cap[x,y]\not=\emptyset$ for some $k\in{\mathbb{N}}$, then the set $G+k$ is contained in $[x,y]+B_{r}(0)$, where $B_{r}(0)$ is the Euclidean ball of radius $r={\rm diam}\,(G)$. Denote by $V$ the volume of $[x,y]+B_{r}(0)$, then the total number of sets $G+k\subset{\mathcal{G}}$ intersecting $[x,y]$ is bounded by $C=\frac{V}{\mu(G)}=V<\infty$, due to $\mu(G)=1$. ∎

To deal with line segments $[x,y]$, $x,y\in G$, that do not completely belong to $G$, we extend the continuous vector-valued function $v=v_{\varphi}$ in (7) which is defined on $G$ to the whole ${\mathbb{R}}^{s}$. Define

[TABLE]

In Lemma 4 and in Proposition 4, we compare the properties of $v$ and $\tilde{v}$.

Lemma 4**.**

Let $x,y\in G-j$, $j\in{\mathbb{Z}}^{s}$. Then $\|\tilde{v}(y)\,-\,\tilde{v}(x)\|\,\leq\,\|v(y+j)\,-\,v(x+j)\|$.

Proof.

Let $j\in{\mathbb{Z}}^{s}$. By (7) and due to the compact support of $\varphi$, the $k$-th component of $\tilde{v}(y)\,-\,\tilde{v}(x)$ is given by

[TABLE]

Hence, we have $\|\tilde{v}(y)-\tilde{v}(x)\|\leq\|v(y+j)-v(x+j)\|$. ∎

Proposition 4**.**

Let $\varphi\in C({\mathbb{R}}^{s})$ be refinable, $x,y\in{\mathbb{R}}^{s}$ and $k\in{\mathbb{N}}$. There exist

[TABLE]

(with $C>0$ from Lemma 3), integers $\{d_{1}^{(i)},\ldots,d_{k}^{(i)}\}_{i=0}^{\ell-1}$ from $D(M)$, positive numbers $\{\alpha_{i}\}_{i=0}^{\ell-1}$ whose sum is equal to one, and sets of points $\{x_{i}\}_{i=0}^{\ell-1}$, $\{y_{i}\}_{i=0}^{\ell-1}$ from $G$ such that $y_{i}-x_{i}\,=\,\alpha_{i}\,M^{k}(y-x)$ for all $i=0,\ldots,\ell-1$, and

[TABLE]

Proof.

For $[x,y]\subset{\mathcal{G}}^{k}$, by Lemma 2, there exist $\ell+1$ points $\{x=a_{0}\leq a_{1}\leq\ldots\leq a_{\ell}=y\}\subset[x,y]$ such that each pair of successive points $a_{i},a_{i+1}$ belongs to only one set $G_{d_{1}^{(i)}\ldots d_{k}^{(i)}}-j^{(i)}$, $j^{(i)}\in{\mathbb{N}}$, of the tiling ${\mathcal{G}}^{k}$. First we give an estimate for $\ell$. Since $\ell$ elements of the tiling ${\mathcal{G}}^{k}\,=\,M^{-k}{\mathcal{G}}$ cover a segment of length $\|y-x\|$, the same number of elements of the tiling ${\mathcal{G}}$ cover a segment of length $\|M^{k}(y-x)\|$. Therefore, Lemma 2 yields $\,\ell\,\leq\,C\,\max\{1\,,\,\|M^{k}(y-x)\|\,\}$. Furthermore, the set $G_{d_{1}^{(i)}\ldots d_{k}^{(i)}}-j^{(i)}\subset G-j^{(i)}$, $i=0,\ldots,\ell-1$. By Lemma 4, we obtain

[TABLE]

Due to $a_{i}+j^{i}\,,a_{i+1}+j^{(i)}\in G_{d_{1}^{(i)}\ldots d_{k}^{(i)}}$, $i=0,\ldots,\ell-1$, the points

[TABLE]

belong to $G$. Thus, by (11), we obtain

[TABLE]

For each $i=0,\ldots,\ell-1$, we define the number $\alpha_{i}$ from the equality $\|a_{i+1}-a_{i}\|\,=\,\alpha_{i}\,\|y-x\|$. It follows that $\,\displaystyle\sum_{i=1}^{\ell-1}\,\alpha_{i}\,=\,1$ and that $y_{i}-x_{i}\,=\,M^{k}(a_{i+1}-a_{i})\,=\,\alpha_{i}M^{k}(y-x)$. ∎

3.5 Proofs of Theorems 1 and 2

In this subsection we prove Theorems 1 and 2. We start with Theorem 2 as its proof is a crucial part of the proof of Theorem 1. Note that for both Theorems 1 and 2 the assumption that $\varphi\in C({\mathbb{R}}^{s})$ implies, e.g. by [4], that $\rho({\mathcal{A}})<1$. We will not reprove this result here.

Proof of Theorem 2.

Let $\varepsilon\in(0,1-\rho_{i})$ and $i\in\{1,\ldots,q(M)\}$. We first show that $\alpha_{\varphi,J_{i}}\geq\log_{1/r_{i}}\,\rho_{i}$. For arbitrary points $x,y\in G$ such that $y-x\in J_{i}$ and $\|y-x\|<1$, define $k$ to be the smallest integer such that $\,\|M^{k}(y-x)\|\,\geq\,1$. Since $y-x\in J_{i}$, it follows that

[TABLE]

where the constant $C>0$ depends only on $M$. By (19), Theorem A1 and by Proposition 4, for these $x,y$ and $k$, there exist a constant $C_{1}>0$ depending on $G$ and the integer

[TABLE]

such that (note that $y-x\in J_{i}$ implies, by Proposition 4, that $x_{j},y_{j}$ in (21) satisfy $y_{j}-x_{j}\in J_{i}$, $j=0,\ldots,\ell-1$)

[TABLE]

with the constant $C_{3}$ independent of $k$. By the choice of $k$, we have $\|M^{k-1}(y-x)\|<1$ and, hence, $\|M^{k}(y-x)\|\,\leq\,\|M\|\,\|M^{k-1}(y-x)\|\,<\,\|M\|$. Thus,

[TABLE]

Combining the above estimate with (22) (i.e. $k\geq-\frac{\log\|y-x\|}{\log(r_{i}+\varepsilon)}+C_{4}$), we get, due to $\rho_{i}+\varepsilon<1$,

[TABLE]

with $\,\alpha(\varepsilon)\,=\,\log_{1/(r_{i}+\varepsilon)}(\rho_{i}+\varepsilon)$ and with $C(\varepsilon)>0$. Letting $\varepsilon\to 0$, we obtain the claim.

Next we establish the reverse inequality $\alpha_{\varphi,J_{i}}\leq\log_{1/r_{i}}\,\rho_{i}$. Let $\varepsilon\in(0,r_{i})$ and $d_{1},\ldots,d_{k}\in D(M)$, $k\in{\mathbb{N}}$. By Theorem A2, there exist $u\in U_{i}$ and a constant $C(u)>0$ such that $\|A_{d_{1}}\cdots A_{d_{k}}\,u\|\,\geq\,C(u)\rho_{i}^{\,k}$. Since the subspace $U_{i}$ is spanned by the differences $v(y)-v(x),\ y-x\in J_{i}$, $x,y\in G$, there exist $x_{j},y_{j}\in G$, $y_{j}-x_{j}\in J_{i}$, $j=1,\ldots,n_{i}$ ($n_{i}$ dimension of $U_{i}$), such that \displaystyle u\,=\,\sum_{j=1}^{n_{i}}\gamma_{j}\bigl{(}v(y_{j})-v(x_{j})\bigr{)}, $\gamma_{j}\in{\mathbb{R}}$. Denote $x_{j}^{(k)}=M_{d_{1}}^{-1}\cdots M_{d_{k}}^{-1}x_{j}\,$ and $\,y_{j}^{(k)}=M_{d_{1}}^{-1}\cdots M_{d_{k}}^{-1}y_{j}$. Thus, $\,x_{j}^{(k)},y_{j}^{(k)}\in G_{d_{1}\ldots d_{k}}$ and there exists $C(\varepsilon)>0$ such that

[TABLE]

Moreover, we have

[TABLE]

Consequently, at least one of the $n_{i}$ numbers $\|v(y_{j}^{(k)})-v(x_{j}^{(k)})\|,\ j=1,\ldots,n_{i}$, is larger than or equal to $\frac{C(u)}{\sum_{j}|\gamma_{j}|}\,\rho_{i}^{\,k}$. Combining this estimate with (24) (i.e. $k\leq-\frac{\log\|y_{j}^{(k)}-x_{j}^{(k)}\|}{\log(r_{i}-\varepsilon)}+C_{1}$ with the constant $C_{1}$ independent of $k$), we obtain

[TABLE]

where $\alpha(\varepsilon)=\log_{1/(r_{i}-\varepsilon)}\,\rho_{i}$ and the constant $C>0$ does not depend on $k$. Since $\|y_{j}^{(k)}-x_{j}^{(k)}\|\to 0$ as $k\to\infty$, there exist arbitrary small segments $[x_{j}^{(k)}\,,\,y_{j}^{(k)}]$ on which the variation of the function $v$ is at least a constant times the length of that segment to the power of $\log_{1/(r_{i}-\varepsilon)}\,\rho_{i}$. Therefore, $\alpha_{\varphi,J_{i}}\leq\log_{1/(r_{i}-\varepsilon)}\,\rho_{i}$. Since $\varepsilon$ is arbitrary, the claim follows. ∎

Proof of Theorem 1.

We only show that the condition $\rho({\mathcal{A}})<1$ is sufficient for continuity of $\varphi$. Let $\varepsilon\in(0,1-\rho({\mathcal{A}}))$. Note that the set of rational $M$-adic points

[TABLE]

is dense in $G$. To determine the values of $v=v_{\varphi}$ in (7) on $Q$ use the algorithm from subsection 3.2. We first show that $v$ is uniformly bounded on $Q$. Denote $C_{0}=\max\{\|v(x)-v(y)\|\ :\ x,y\in Q_{1}\}$. Then, for every $x=0.d_{1}\ldots d_{k}\,\in Q_{k}\subset Q$, we have

[TABLE]

Note that, by construction, $v(0.d_{j+1})-v(0)\,=\,T_{d_{j+1}}v_{0}-v_{0}\,\in\,U$ for $j=1,\ldots,k-1$. Therefore, by Theorem A1, \bigl{\|}T_{d_{1}}\cdots T_{d_{j}}\bigl{(}v(0.d_{j+1})-v(0)\bigr{)}\bigr{\|}\,\leq\,C_{1}(\rho+\varepsilon)^{j}, $j=1,\ldots,k-1$. Thus, we obtain

[TABLE]

where the constant $C_{1}>0$ is independent of $k$. Hence, $\|v(x)\|\,\leq\,\|v(0)\|\,+\,\frac{C_{1}}{1-\rho({\mathcal{A}})-\varepsilon}$ which proves the uniform boundedness of $v$ on $G$.

The values of $v$ on $Q$ define the function $\varphi$ on $\tilde{Q}$, where $\tilde{Q}=\cup_{k\in{\mathbb{Z}}^{s}}(k+Q)$ is the set of all rational $M$-adic points of ${\mathbb{R}}^{s}$. The so constructed $\varphi:\tilde{Q}\rightarrow{\mathbb{R}}$ is supported on $K\cap\tilde{Q}$. Using $\varphi$, define the extension $\tilde{v}:\tilde{Q}\to{\mathbb{R}}$ of $v$ in (7). We show next that $\tilde{v}$ is uniformly continuous on $\tilde{Q}$, which implies that its extension to ${\mathbb{R}}^{s}$ is continuous. Take arbitrary points $x,y\in\tilde{Q}$. By Proposition 4 and the same argument as in the first part of the proof of Theorem 2, we obtain

[TABLE]

where $k$ is the smallest number such that $\|M^{k}(y-x)\|\geq 1$. Note that the value $\|v\|_{C(Q)}$ is finite because $v$ is bounded $Q$. Since $\|M\|^{k}\geq 1/\|y-x\|$, i.e. $k$ goes to $\infty$ as $\|y-x\|$ goes to zero, $\tilde{v}$ is uniformly continuous on $\tilde{Q}$, which completes the proof of continuity. Thus, if $\rho({\mathcal{A}})<1$, then $\varphi\in C({\mathbb{R}}^{s})$.

By Theorem 2, the Holder exponent $\alpha_{\varphi}$ of $\varphi$ on shifts along the subspace $J_{i}$ is equal to $\alpha_{i}=\log_{1/r_{i}}\,\rho_{i}$. We pass to a basis in the space ${\mathbb{R}}^{s}$, in which all the subspaces $J_{i}$ are orthogonal to each other. Using a natural expansion $h=h_{1}+\ldots+h_{q(M)},\ h_{i}\in J_{i}$ we obtain for arbitrary $\varepsilon>0$

[TABLE]

where $\alpha=\min\limits_{i=1,\ldots,q(M)}\alpha_{i}$. Consequently, $\alpha_{\varphi}\,=\,\min\limits_{i=1,\ldots,q(M)}\alpha_{i}$. ∎

3.6 Examples

There are several types of anisotropic dilation matrices, e.g. hyperbolic dilations or products of parabolic scaling and shear matrices, used in hyperbolic wavelet transform or discrete shearlet transform, see [1, 18, 61].

Example 2**.**

In [18], the authors present a general method for constructing interpolatory subdivision schemes with dilations which are products of parabolic scaling and shear matrices. We consider one of these examples with the anisotropic dilation matrix M=\left(\begin{array}[]{cc}2&1\\ 0&3\end{array}\right) and the refinement coefficients

[TABLE]

Note that the corresponding sequence of coefficients is supported on the set $\{(0,j)\ :\ j=0,\ldots,10\}$. By [28], for the digit set $D(M)=\{(0,0),(1,0),(1,1),(1,2),(2,1),(2,2)\}$, the set $G$ in (3) is a tile. Using the result of [4], we determine the set $\Omega$ of size $19$. For the corresponding $19\times 19$ transition matrices $T_{d}$, $d\in D(M)$, using Algorithm 1 from subsection 3.2 we construct the set $U$ starting with

[TABLE]

We obtain

[TABLE]

Due to $\hbox{dim}(W)=18$ and $\hbox{dim}(U)=16$, $U$ is a proper subspace of $W$. Denote ${\mathcal{A}}=\{A_{d}=T_{d}|U\ :\ d\in D(M)\}$. We computed the joint spectral radius of the set ${\mathcal{A}}$ using the invariant polytope algorithm from [30] and obtained that the joint spectral radius is attained at the finite product of length $1$

[TABLE]

Since $M$ is anisotropic, there exist two non-zero subspaces $U_{1},U_{2}\subseteq U$. We construct these subspaces (each of dimension $15$) using Step $3$ of Algorithm 1 from subsection 3.2 with

[TABLE]

respectively. Note that, due to the continuity of $\varphi$, we have $v(0,0)=v_{0}$. Furthermore,

[TABLE]

and

[TABLE]

For the restrictions ${\mathcal{A}}|_{U_{1}}$ and ${\mathcal{A}}|_{U_{2}}$, we obtain $\rho_{1}=\rho_{2}=\rho({\mathcal{A}})$. Therefore, by Theorem 1,

[TABLE]

The next example shows that in some cases $W=U=U_{1}=U_{2}$.

Example 3**.**

We consider the dilation matrix M=\left(\begin{array}[]{rr}2&1\\ 1&-1\end{array}\right) and the refinement coefficients

[TABLE]

The dilation matrix has eigenvalues $\lambda_{1}=\frac{1-\sqrt{13}}{2}$ and $\lambda_{2}=\frac{1+\sqrt{13}}{2}$. By [28], for the digit set $D(M)=\{(0,0),(1,0),(2,0)\}$, the set $G$ in (3) is a tile. Using the result of [4], we determine

[TABLE]

The corresponding transition matrices $T_{(0,0)}$, $T_{(1,0)}$ and $T_{(2,0)}$ are given by

[TABLE]

and

[TABLE]

respectively. The matrix $T_{(0,0)}$ has one eigenvalue $1$ with the corresponding eigenvector

[TABLE]

Using Algorithm 1 from subsection 3.2, we obtain $U$ with $\hbox{dim}(U)=12$. Thus, $W=U$ and, therefore, ${\mathcal{A}}=\{A_{d}=T_{d}|_{W}\ :\ d\in D(M)\}$. We computed the joint spectral radius of the set ${\mathcal{A}}$ using the invariant polytope algorithm from [30] and obtained that the joint spectral radius is attained at the finite product $(A_{(0,0)}A_{(1,0)})^{2}A_{(0,0)}^{2}A_{(2,0)}$ of length $7$, i.e.

[TABLE]

We verified that the matrix family ${\mathcal{A}}$ is irreducible in this case. Hence, the only non-zero common invariant subspace of the matrices in ${\mathcal{A}}$ is $U$. Thus, $U_{1}=U_{2}=U=W$. Therefore, $\rho_{1}=\rho_{2}=\rho({\mathcal{A}})$ and Theorem 1 implies that

[TABLE]

4 Higher order regularity

In this section, we show that the derivatives of the multivariate refinable function $\varphi\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$ satisfy a system of nonhomogeneous refinement equations. The differentiability of $\varphi\in C({\mathbb{R}}^{s})$ is then equivalent to continuity of the solutions of all these equations, see Theorem 3. The main idea is that the directional derivatives of $\varphi$ along the eigenvectors of the dilation matrix $M$ satisfy certain refinement equations and the directional derivatives along the generalized eigenvectors (of the Jordan basis) of $M$ satisfy nonhomogeneous refinement equations, see Proposition 5.

Definition 6**.**

A multivariate nonhomogeneous refinement equation is a functional equation of the form

[TABLE]

where ${\boldsymbol{T}}$ is the transition operator in (5) and $g$ is a compactly supported function or distribution.

For more details on nonhomogeneous refinement equations see e.g. [23, 39, 41, 63] and references therein.

Let $E=\{e_{1},\ldots,e_{s}\}$ be the Jordan basis of the matrix $M$ in ${\mathbb{R}}^{s}$. The Jordan basis consists of the eigenvectors of $M$, which satisfy $Me_{i}=\lambda e_{i}$, and of the generalized eigenvectors, which satisfy $Me_{i}=\lambda e_{i}+e_{i-1}$. Consider an $\ell\times\ell$ Jordan block of $M$ corresponding to an eigenvalue $\lambda$. With a slight abuse of notation, denote by

[TABLE]

the Jordan basis corresponding to this Jordan block. In the following we study the properties of the directional derivatives of the refinable function $\varphi\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$, which belong to the following subspaces of ${\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$.

Definition 7**.**

For a vector $a\in{\mathbb{R}}^{s}$, we denote by

[TABLE]

the space of compactly supported distributions, whose mean along every straight line \bigl{\{}x=at+b\ :\ t\in{\mathbb{R}}\bigr{\}}, $\ b\in{\mathbb{R}}^{s}$, parallel to $a$, is equal to zero.

By \nabla\varphi\,=\,\bigl{(}\frac{\partial\varphi}{\partial x_{1}},\ldots,\frac{\partial\varphi}{\partial x_{s}}\bigr{)} we denote the total derivative (gradient) of $\varphi$ and by \frac{\partial\,\varphi}{\partial\,a}\,=\,\bigl{(}a\,,\,\nabla\varphi\bigr{)}, its directional derivative along a nonzero vector $a\in{\mathbb{R}}^{s}$. Due to the compact support of $\varphi\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$, its directional derivative $\frac{\partial\,\varphi}{\partial\,a}$ belongs to ${\mathcal{S}}^{\prime}_{a}({\mathbb{R}}^{s})$.

The next result shows that a directional derivative of a refinable function $\varphi$ along an eigenvector of the dilation matrix $M$ is also a refinable function and satisfies the refinement equation (25). A directional derivative of $\varphi$ along a generalized eigenvector of $M$ satisfies the nonhomogeneous refinement equation (26).

Proposition 5**.**

Let $\varphi\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$ and $\lambda$ be an eigenvalue of $M$. If $\varphi={\boldsymbol{T}}\varphi$, then \varphi_{i}=\bigl{(}e_{i},\nabla\varphi\bigr{)}\in{\mathcal{S}}^{\prime}_{i}:={\mathcal{S}}_{e_{i}}^{\prime}({\mathbb{R}}^{s}), $e_{i}\in E_{\lambda}$, satisfy the refinement equation

[TABLE]

and the nonhomogeneous refinement equations

[TABLE]

Conversely, the system of equations (25)-(26) possesses a unique up to a normalization solution $(\varphi_{1},\ldots,\varphi_{\ell})\in{\mathcal{S}}^{\prime}_{1}\times\ldots\times{\mathcal{S}}^{\prime}_{\ell}$. Moreover, if $\varphi\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$ satisfies \bigl{(}e_{i},\nabla\varphi\bigr{)}=\varphi_{i}, $e_{i}\in E_{\lambda}$, then $\varphi={\boldsymbol{T}}\varphi$ along the lines parallel to $e_{i}\in E_{\lambda}$.

Proof.

By induction on $\ell$, we show that, if $\varphi\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$ satisfies the refinement equation $\varphi={\boldsymbol{T}}\varphi$, then \varphi_{i}=\bigl{(}e_{i},\nabla\varphi\bigr{)}, $i=1,\ldots,\ell$, satisfy (25)-(26). For $\ell=1$, due to $Me_{1}=\lambda e_{1}$, we have

[TABLE]

For $i\in\{2,\ldots,\ell\}$, due to $Me_{i}=\lambda e_{i}+e_{i-1}$ and (26), we obtain

[TABLE]

Conversely, by [39], the system (25)-(26) possess a unique up to normalization solution. We show that the compactly supported primitive $\varphi$ of $\varphi_{i}$, $i=1,\ldots,\ell$, along $e_{i}\in E_{\lambda}$ satisfies $\varphi={\boldsymbol{T}}\varphi$. Indeed, since $Me_{1}=\lambda e_{1}$ and by (25), we obtain

[TABLE]

which implies that the gradient of the function ${\boldsymbol{T}}\varphi-\varphi$ is orthogonal to $e_{1}$. Hence, the function ${\boldsymbol{T}}\varphi-\varphi$ is constant along all lines parallel to $e_{1}\in E_{\lambda}$. On the other hand, $\varphi$ is compactly supported, consequently ${\boldsymbol{T}}\varphi-\varphi=0$ along all lines parallel to $e_{1}\in E_{\lambda}$. For $i\in\{2,\ldots,\ell\}$, due to $Me_{i}=\lambda e_{i}+e_{i-1}$ and $\varphi_{i}=\lambda{\boldsymbol{T}}\varphi_{i}+{\boldsymbol{T}}\varphi_{i-1}$, we obtain

[TABLE]

Analogous considerations about ${\boldsymbol{T}}\varphi-\varphi$ along $e_{i}\in E_{\lambda}$ imply the claim. ∎

Remark 3**.**

If, for the eigenvalue $\lambda$ of the dilation matrix $M$, the set $E_{\lambda}$ does not contain any generalized eigenvectors, then the system (25)-(26) reduces to homogeneous refinement equations $\varphi_{i}\ =\ \lambda\,{\boldsymbol{T}}\varphi_{i}$, $e_{i}\in E_{\lambda}$. **

The main result of this section, Theorem 3, states that $\varphi\in C^{1}({\mathbb{R}}^{s})$ if and only if the (nonhomogeneous) refinement equations in Proposition 5 corresponding to the Jordan basis $E\subset{\mathbb{R}}^{s}$ of the dilation matrix $M$ have continuous solutions $\varphi_{i}\in{\mathcal{S}}^{\prime}_{e_{i}}({\mathbb{R}}^{s})$, $e_{i}\in E$. The directional derivatives $\varphi_{i}$, $i=1,\ldots,s$, determine the total derivative $\nabla\varphi$ of $\varphi$. Moreover, $\varphi_{i}$, $i=1,\ldots,s$, can be constructed and their Hölder exponents can be computed as described in section 3 (see Remark 4). Thus, the higher regularity of any refinable function $\varphi$ can be analyzed by this recursive reduction to a set of continuous refinable functions.

Theorem 3**.**

Let $\varphi\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{s})$. There exist continuous solutions $\varphi_{i}\in{\mathcal{S}}^{\prime}_{e_{i}}({\mathbb{R}}^{s}),\,e_{i}\in E_{\lambda}$, of (25) – (26) for each eigenvalue $\lambda$ of the dilation matrix $M$ if and only if $\varphi\in C^{1}({\mathbb{R}}^{s})$ satisfies $\varphi={\boldsymbol{T}}\varphi$ and $\frac{\partial\,\varphi}{\partial\,e_{i}}\,=\,\varphi_{i}$, $e_{i}\in E$.

Proof.

If $\varphi\in C^{1}({\mathbb{R}}^{s})$ is a compactly supported solution of the refinement equation $\varphi={\boldsymbol{T}}\varphi$, then, by Proposition 5, its directional derivatives $\varphi_{i}:=\frac{\partial\,\varphi}{\partial\,e_{i}}\in C({\mathbb{R}}^{s})$ along $e_{i}\in E$ satisfy equations (25)-(26). Conversely, if the equations (25)-(26) possess continuous solutions, then, by Proposition 5, $\varphi$ is in $C^{1}({\mathbb{R}}^{s})$, $\frac{\partial\,\varphi}{\partial\,e_{i}}=\varphi_{i}$, $e_{i}\in E$, and satisfies $\varphi={\boldsymbol{T}}\varphi$. ∎

Corollary 5**.**

Suppose that $E$ does not contain any generalized eigenvectors, i.e., the matrix $M$ has a basis of eigenvectors; then

$(i)$

If $\varphi\in C^{1}({\mathbb{R}}^{s})$ is refinable, then $\varphi_{i}\,=\,\frac{\partial\,\varphi}{\partial\,e_{i}}\in C({\mathbb{R}}^{s})$ satisfy $\varphi_{i}\ =\ \lambda_{i}\,{\boldsymbol{T}}\varphi_{i}$, $i=1,\ldots,s$.

$(ii)$

Conversely, if the solutions $\varphi_{i}\in{\mathcal{S}}_{e_{i}}^{\prime}$ of $\varphi_{i}\ =\ \lambda_{i}\,{\boldsymbol{T}}\varphi_{i}$, $i=1,\ldots,s$, are continuous, then the solution of $\varphi={\boldsymbol{T}}\varphi$ belongs to $C^{1}({\mathbb{R}}^{s})$. Moreover, $\frac{\partial\varphi}{\partial e_{i}}=\varphi_{i}$, $i=1,\ldots,s$.

Remark 4**.**

The system of refinement equations (25)-(26) is solved and analysed in the same way described in subsection 3.2. First we solve the equation $\varphi_{1}=\lambda{\boldsymbol{T}}\varphi_{1}$. We find $v_{\varphi_{1}}(0)$ as an eigenvector of the matrix $T_{0}$ with the eigenvalue $1/\lambda$. If $T_{0}$ does not have this eigenvalue, then equation $\varphi_{1}=\lambda{\boldsymbol{T}}\varphi_{1}$ does not have a solution, and hence $\varphi\notin C^{1}({\mathbb{R}}^{s})$. Then we define the space $U_{\lambda,1}$ as the minimal common invariant subspace of ${\mathcal{T}}$ containing the vectors $\lambda T_{d}v_{\varphi_{1}}(0)\,-\,v_{\varphi_{1}}(0),\,d\in D(M)$. This can be done by Algorithm 1 from subsection 3.2. Then $\varphi_{1}\in C({\mathbb{R}}^{s})$ if and only if the joint spectral radius of the matrices $\lambda T_{d},\,d\in D(M),$ restricted to the subspace $U_{\lambda,1}$ is smaller than one. The Hölder regularity of $\varphi_{1}$ is computed by formula (15) for the matrices $A_{d}=\lambda T_{d}|_{U_{\lambda,1}}$. Similarly, we solve the other equations of the system (26) successively for $i=2,\ldots,\ell$. **

5 Existence and smoothness in $\mathbf{L_{p}({\mathbb{R}}^{s}),\ 1\leq p<\infty}$

In this section, see Theorem 4, we characterize the existence of refinable functions in $\varphi\in L_{p}({\mathbb{R}}^{s})$, $1\leq p<\infty$, and provide a formula for the Hölder exponent of such $\varphi$, see Theorem 5, in terms of the $p$-radius ($p$-norm joint spectral radius [36, 51]) of a set of transition matrices.

Definition 8**.**

For $1\leq p<\infty$, the $p$-radius ($p$-norm joint spectral radius) of a finite family of linear operators ${\mathcal{A}}\,=\,\{A_{0},\ldots,A_{m-1}\}$ is defined by

[TABLE]

Note that, for $\varphi\in L_{p}({\mathbb{R}}^{s})$, the difference space $U$ is defined similarly to (12) by

[TABLE]

Although the estimates in Theorem 5 look similar to the ones from section 3, the corresponding proofs require totally different techniques.

The Hölder exponent of a function $\varphi\in L_{p}({\mathbb{R}}^{s})$ is defined by

[TABLE]

We use the notation $\|\cdot\|_{L_{p}({\mathbb{R}}^{s})}\,=\,\|\cdot\|_{p}$. To determine the influence of the dilation matrix $M$ on the Hölder exponent of $\varphi\in L_{p}({\mathbb{R}}^{s})$, in Theorem 5, we consider the Hölder exponents of $\varphi$ along the subspaces determined by the Jordan basis of $M$. The Hölder exponent of $\varphi$ along a subspace $J\subset{\mathbb{R}}^{s}$ is defined by

[TABLE]

In the proofs of Theorems 4 and 5 we use the following auxiliary results.

5.1 Auxiliary results

The following analogues of Theorems A1 and A2 from section 3 were proved in [55].

Theorem A3. Let $1\leq p<\infty$. For a finite family ${\mathcal{A}}$ of $m$ operators acting in ${\mathbb{R}}^{n}$ and for any $\varepsilon>0$, there exists a norm $\|\cdot\|_{\varepsilon}$ in ${\mathbb{R}}^{n}$ such that

[TABLE]

For $1\leq p<\infty$ and for a finite family ${\mathcal{A}}$ of $m$ operators acting in ${\mathbb{R}}^{n}$, we denote

[TABLE]

Since each norm $\|\cdot\|$ in ${\mathbb{R}}^{n}$ is equivalent to the norm $\|\cdot\|_{\varepsilon}$, Theorem A3 yields the following result.

Corollary 6**.**

Let $1\leq p<\infty$. For every $\varepsilon>0$, there exists a constant $C(\varepsilon)>0$ such that ${\mathcal{F}}_{p}(k,u)\,\leq\,C(\varepsilon)(\rho_{p}+\varepsilon)^{k}\|u\|$ for all $u\in{\mathbb{R}}^{n}$ and $k\in{\mathbb{N}}$.

Theorem A4. Let $1\leq p<\infty$ and ${\mathcal{A}}$ be a finite family of $m$ linear operators in ${\mathbb{R}}^{n}$. Then for every $u\in{\mathbb{R}}^{n}$ that does not belong to any common invariant linear subspace of ${\mathcal{A}}$, there exists a constant $C(u)>0$ such that

[TABLE]

In Lemma 5 we relax assumptions of Theorem A4. We show that (28) holds for all points $u$ apart from the ones in a proper linear subspace of ${\mathbb{R}}^{n}$.

Lemma 5**.**

Let $1\leq p<\infty$. Every finite family ${\mathcal{A}}$ of $m$ linear operators possesses a common invariant linear subspace ${\mathcal{L}}\subset{\mathbb{R}}^{n}$, ${\mathcal{L}}\not={\mathbb{R}}^{n}$, (possibly ${\mathcal{L}}=\{0\}$) such that for every $u\notin{\mathcal{L}}$ there exists a constant $C(u)>0$ for which (28) holds.

Proof.

Without loss of generality, after a suitable normalization, it can be assumed that $\rho_{p}=1$. Let ${\mathcal{L}}$ be the biggest by inclusion common invariant subspace of ${\mathcal{A}}$ such that $\rho_{p}({\mathcal{A}}|_{{\mathcal{L}}})<1$. Note that ${\mathcal{L}}$ is a proper subspace of ${\mathbb{R}}^{n}$, otherwise we get a contradiction to $\rho_{p}=1$. Hence, ${\rm dim}\,{\mathcal{L}}\,\leq\,n-1$. Take arbitrary $u\notin{\mathcal{L}}$ and denote by ${\mathcal{L}}_{u}$ the minimal common invariant subspace of ${\mathcal{A}}$ that contains $u$. If $\rho_{p}({\mathcal{A}}|_{{\mathcal{L}}_{u}})<1$, then the $p$-joint spectral radius of ${\mathcal{A}}$ on the linear span of ${\mathcal{L}}$ and ${\mathcal{L}}_{u}$ is equal to $\max\,\{\rho_{p}({\mathcal{A}}|_{{\mathcal{L}}}),\rho_{p}({\mathcal{A}}|_{{\mathcal{L}}_{u}})\}<1$, which contradicts the maximality of ${\mathcal{L}}$. Hence, $\rho_{p}({\mathcal{A}}|_{{\mathcal{L}}_{u}})=1$. Since $u$ does not belong to any common invariant subspace of the finite family ${\mathcal{A}}|_{{\mathcal{L}}_{u}}$, by Theorem A4, there exists a constant $C(u)>0$ such that ${\mathcal{F}}_{p}(k,u)\geq C(u)\rho({\mathcal{A}}|_{{\mathcal{L}}_{u}})^{k},\,k\in{\mathbb{N}}$. On the other hand, $\rho_{p}=\rho({\mathcal{A}}|_{{\mathcal{L}}_{u}})=1$, and, hence, the claim follows. ∎

For $1\leq p<\infty$, in the rest of this section, we denote by $C(u)>0$ the largest possible constant in inequality (28), i.e.,

[TABLE]

This function is upper semi-continuous and, therefore, is measurable.

5.1.1 Properties of the space $U$

Note that, by (11), the vector $\displaystyle z=\int_{G}v(x)\,dx\in V$ is an eigenvector of the operator $\displaystyle T\,=\,\frac{1}{m}\,\sum_{d\in D(M)}\,T_{d}$ associated with the eigenvalue $1$. The following result is an analogue of Proposition 1.

Proposition 6**.**

If $\varphi\in L_{p}({\mathbb{R}}^{s})$, $1\leq p<\infty$, then the subspace $U$ in (27) coincides with the smallest by inclusion common invariant subspace of the matrices $T_{d},\,d\in D(M)$, and contains the $m$ vectors $\,T_{d}z-z,\,d\in D(M)$, where the nonzero $z\in V$ satisfies $Tz=z$.

The proof of Proposition 6 is similar to the one of Proposition 1. Note that, since $\displaystyle\sum_{k\in{\mathbb{Z}}^{s}}c_{k}=m$, the column sums of the matrix $T$ are equal to one. Hence, $T$ has at least one eigenvalue one. This eigenvalue does not have to be simple. Nevertheless, Proposition 7 guarantees that $U$ is well-defined. The proof of Proposition 7 is similar to the one of Proposition 3.

Proposition 7**.**

There exists at most one eigenvector $z\in V$ of $\,T$ associated to the eigenvalue $1$ such that $\rho_{p}({\mathcal{A}})<1$.

5.2 $\mathbf{L_{p}}$-solutions of refinement equations

Theorem 4**.**

A refinable function $\varphi$ belongs to $L_{p}({\mathbb{R}}^{s})$, $1\leq p<\infty$, if and only if $\rho_{p}({\mathcal{A}})<1$.

Proof.

Assume first that $\rho_{p}=\rho_{p}({\mathcal{A}})<1$. Choose $\varepsilon\in(0,1-\rho_{p})$ and consider the norm $\|\cdot\|_{\varepsilon}$ in $U$ as in Theorem A4. Define the function space

[TABLE]

with the norm \|f\|\,=\,\bigl{(}\,\int_{G}\|v_{f}(x)\|^{p}_{\varepsilon}\,dx\,\bigr{)}^{1/p}. The space $V_{U,p}$ is nonempty because it at least contains a piecewise constant function $f$ such that $v_{f}\equiv z$ a.e., where $z\in V$ is the eigenvector of the operator $\displaystyle\frac{1}{m}\sum_{d\in D(M)}\,T_{d}$ associated with the eigenvalue one. Note that, for $f_{1},f_{2}\in V_{U,p}$, we have

[TABLE]

Therefore, due to $\rho_{p}+\varepsilon<1$, ${\boldsymbol{T}}$ is a contraction on $V_{U,p}$, and, hence, it has a unique fixed point $\varphi$, which is the solution of the refinement equation $\varphi={\boldsymbol{T}}\varphi$.

Assume next that $\varphi\in L_{p}({\mathbb{R}}^{s})$. By Lemma 5, there exists a proper subspace ${\mathcal{L}}\subset U$ (due to $\hbox{dim}U=n$, we associate ${\mathbb{R}}^{n}$ with $U$) invariant under ${\mathcal{A}}$ such that ${\mathcal{F}}_{p}(k,u)\geq C(u)\,\rho_{p}^{\,k},\ k\in{\mathbb{N}}$, whenever $u\notin{\mathcal{L}}$. Since $U$ is the smallest by inclusion subspace of ${\mathbb{R}}^{N}$ invariant under ${\mathcal{A}}$ and containing the differences $v(x_{2})-v(x_{1})$ for almost all $x_{1},x_{2}\in G$, the set $\{(x_{1},x_{2})\in G^{2}\ :\ v(x_{2})-v(x_{1})\notin{\mathcal{L}}\}$ has a positive Lebesque measure $\mu$ in ${\mathbb{R}}^{s}\times{\mathbb{R}}^{s}$. Hence, the set

[TABLE]

has a positive Lebesque measure. By the Fubini theorem, the set in (30) has sections of positive Lebesque measure. Thus, there exists $h\in{\mathbb{R}}^{s}$ such that

[TABLE]

Therefore, there exist $\delta>0$ and a set $H\subset G$ of positive Lebesque measure such that the function in (29) satisfies C\bigl{(}v(x+h)-v(x)\bigr{)}>\delta for almost all $x\in H$. Thus,

[TABLE]

Denote $h_{k}=M^{-k}h$, $k\in{\mathbb{N}}$, then, by (11), we get

[TABLE]

Since $y\in H$, by (31), we obtain

[TABLE]

Thus,

[TABLE]

Since $v\in L_{p}({\mathbb{R}}^{s},{\mathbb{R}}^{N})$ and $h_{k}$ goes to [math] as $k\to\infty$, we get $\rho_{p}<1$. ∎

Remark 5**.**

The proof of Theorem 4 is much simpler than that of Theorem 1. Indeed, the argument with a contraction operator ${\boldsymbol{T}}$ on the affine subspace $V(U,p)$ cannot be directly extended to prove the continuity of $\varphi$ due to the following reason: the piecewise constant function $f$ for which $v_{f}\equiv z$ is not continuous. Thus, it is not clear how to show that $V(U,p)$ is nonempty. We are not aware of any simple proof of this fact in the multivariate case. **

5.3 Hölder regularity in $\mathbf{L_{p}({\mathbb{R}}^{s})}$

To be able to determine the exact Hölder regularity of a refinable function $\varphi\in L_{p}({\mathbb{R}}^{s})$, $1\leq p<\infty$, we need to adjust the definitions of the transition matrices $T_{d}$, $d\in D(M)$. To do so we replace the set $\Omega$ in Definition 2 by the set $\tilde{\Omega}$ in (34), the latter contains $\Omega$ and is determined by a certain admissible absorbing set $\Delta$.

Definition 9**.**

Let $1\leq p<\infty$. A set $\Delta\subset{\mathbb{R}}^{s}$ is called absorbing if, for all $f\in L_{p}({\mathbb{R}}^{s})$, the Hölder exponent of $f$ along $\Delta$ satisfies $\alpha_{f,\Delta,p}\,=\,\alpha_{f,p}$.

Remark 6**.**

An arbitrary set that contains some neighborhood of the origin is absorbing. It is also easy to show that any convex body (convex set with a nonempty interior) that contains the origin is absorbing.**

For the sake of simplicity, we choose $\Delta$ to be an arbitrary simplex with one of the vertices at the origin and such that its interior intersects all the spaces $J_{i},i=1,\ldots,q(M)$. In this case $\Delta$ is absorbing, and the sets $\Delta\cap J_{i}$, $i=1,\ldots,q(M)$, are absorbing in the corresponding subspaces $J_{i}$. We call such a simplex $\Delta$ admissible. Note that for each $t>0$, the set $t\,\Delta$ is also an admissible simplex.

Define $\tilde{\Omega}\subset{\mathbb{Z}}^{s}$ to be the minimal set such that

[TABLE]

Such a set $\,\tilde{\Omega}\,$ always exists, due to $\displaystyle\cup_{k\in{\mathbb{Z}}^{s}}(k+G)\,=\,{\mathbb{R}}^{s}$. Note that $\Omega\subseteq\tilde{\Omega}$. In many cases $\tilde{\Omega}=\Omega$, but not always, see examples 4 and 5. Thus, $\,{\rm supp}\,\varphi\,+\,\Delta\ \subset\ \tilde{\Omega}+G\,.$ Using $\tilde{\Omega}$ we redefine

[TABLE]

and

[TABLE]

are now of size $\tilde{N}=|\tilde{\Omega}|$. This leads to the appropriate modification

[TABLE]

of the finite set ${\mathcal{A}}$ in (13). The modified subspaces $\tilde{V}$, $\tilde{U}$, $\tilde{U_{i}}$, $i=1,\ldots,q(M)$, and $\tilde{W}$ differ from the subspaces $V,U,U_{i}$ and $W$, respectively, only by the lengths of their corresponding elements. We are now ready to formulate the main result of this subsection.

Theorem 5**.**

For a refinable function $\varphi\in L_{p}({\mathbb{R}}^{s})$, $1\leq p<\infty$,

[TABLE]

and, consequently,

[TABLE]

Proof.

We first show that $\alpha_{\varphi,J_{i},p}\geq\log_{\,1/r_{i}}\,\rho_{p}(\tilde{{\mathcal{A}}}|_{\tilde{U}_{i}})$. Set $\rho_{i,p}=\rho_{p}(\tilde{{\mathcal{A}}}|_{\tilde{U}_{i}})$. Choose an arbitrary $\tilde{h}\in J_{i}\cap\Delta$ such that $\|\tilde{h}\|<\delta$, $\delta\in(0,1)$. Then the function $\psi=\varphi(\cdot+\tilde{h})-\varphi$ is supported on $K_{\Gamma}+\Delta$. Hence, the vector-valued function $\tilde{v}_{\psi}$ is well defined on $G$. Thus, for arbitrary $k\in{\mathbb{N}}$, we have

[TABLE]

By Corollary 6, for every $\varepsilon>0$, there exists a constant $C(\varepsilon)>0$ such that

[TABLE]

Thus, we obtain

[TABLE]

where $\tilde{C}(\varepsilon)>0$ is independent of either $k$ or $\tilde{h}$. Choose an arbitrary $h\in J_{i}\cap\Delta,\,\|h\|<\delta$, and let $k\in{\mathbb{N}}$ be the largest integer such that $\|M^{k}h\|<\delta$. From $\|M^{k+1}h\|\geq\delta$ and substituting $\tilde{h}=M^{k}h$ in (37), we obtain

[TABLE]

for some constant $C_{1}>0$. Combining these estimates, we obtain $\alpha_{\varphi,J_{i},p}\,\geq\,\log_{\,1/r_{i}}\,(\rho_{i,p}+\varepsilon)$. Taking the limit for $\varepsilon\to 0$, we obtain the claim.

To establish the reverse inequality $\alpha_{\varphi,J_{i},p}\leq\log_{\,1/r_{i}}\,\rho_{i,p}$, we argue as in the second part of the proof of Theorem 4. We show the existence of a vector $h\in J_{i}\cap\Delta$ and of a subset $H\subset J_{i}$ of positive Lebesque measure (on the space $J_{i}$) for which inequality (33) holds (with $\rho_{p}$ replaced by $\rho_{i,p}$). Taking a limit in that inequality as $k\to\infty$ and using the fact that $k\,\leq\,C_{2}+\log_{\,r_{i}}\,\frac{\|h\|}{\|h_{k}\|}$, where $C_{2}>0$ independent of $k$, we complete the proof. ∎

5.4 Examples

The following examples illustrate the need for the modifications of the set $\Omega$ in subsection 5.3.

Example 4**.**

The solution of the univariate refinement equation

[TABLE]

is the characteristic function $\varphi=\chi_{[0,1)}$. The $L_{p}$-regularity of $\varphi$ is $\alpha_{\varphi,p}=\frac{1}{p}$. In this case, $M=2$ and, for the standard set of dyadic digits $D(M)=\{0,1\}$, we have $G=[0,1]$ and $\Omega=\{0\}$. Hence, $N=1$ and we get $T_{0}=T_{1}=1$. The common invariant subspace of $T_{0}$ and $T_{1}$ is trivial $U=\{0\}$, hence, by definition, $\rho_{p}({\mathcal{A}})=0$. Thus, $\alpha_{\varphi,p}=\frac{1}{p}$, while $\log_{1/2}\rho_{p}({\mathcal{A}})\,=\,+\infty$. We see that $\alpha_{\varphi,p}\neq\log_{1/2}\rho_{p}({\mathcal{A}})$.

On the other hand, for $\tilde{\Omega}=\{0,1\}$, we get

[TABLE]

The corresponding common invariant subspace $\tilde{U}=\tilde{W}=\{u\in{\mathbb{R}}^{2}\ :\ u_{1}+u_{2}=0\}$ is one dimensional, and $\tilde{A}_{0}=\tilde{A}_{1}=1$. Clearly, $\rho_{p}(\{\tilde{A}_{0},\tilde{A}_{1}\})=2^{-1/p}$. By Theorem 5, we obtain the correct Hölder exponent $\alpha_{\varphi,p}=\frac{1}{p}$. **

The next example shows that in some cases $\Omega=\tilde{\Omega}$.

Example 5**.**

The solution of the univariate refinement equation

[TABLE]

is $\varphi=\chi_{G}$, where $G$ is the tile in ${\mathbb{R}}$ corresponding to the dilation $M=3$ and to the digit set $D(M)=\Gamma=\{0,1,5\}$. Thus, $\hbox{supp}(\varphi)\subseteq[0,\frac{5}{2}]$.

For the standard set of triadic digits $D(M)=\{0,1,2\}$, we have $G=[0,1]$ and $\Omega=\{0,1,2\}$, i.e. $N=3$. In this case, $K+\Delta\subseteq[0,3]$. Hence, one can take the complemented set $\tilde{\Omega}=\Omega=\{0,1,2\}$.

Remark 7**.**

It is well known that, if $p$ is an even integer, then $\rho_{p}({\mathcal{A}})$ can be efficiently computed as an eigenvalue of a certain matrix derived from the matrices in ${\mathcal{A}}$ [40, 51]. Hence, Theorem 5 allows us to find the Hölder $L_{p}$-regularity at least for even integers $p$, in particular, for $p=2$, see Example 6. **

Example 6**.**

For the refinement equation from subsection 3.6, we have $\tilde{\Omega}=\Omega$ and $U=W$. Therefore, Theorem 5 yields $\alpha_{\varphi,p}\,=\,\log_{1/\rho(M)}\rho_{p}({\mathcal{T}}|_{W})$. Furthermore, $\rho(M)=\frac{1+\sqrt{13}}{2}\,=\,2.30277\ldots$ and $\rho_{2}({\mathcal{T}}|_{W})\,=\,0.79736\ldots$. Hence, $\alpha_{\varphi,2}\,=\,\log_{1/\rho(M)}\rho_{2}({\mathcal{T}}|_{W})\,=\,0.27148\ldots$. Recall that the Hölder exponent in $C({\mathbb{R}}^{s})$ of $\varphi$ is $\alpha_{\varphi}\,=\,0.07652\ldots$. **

Acknowledgements. The authors are grateful to N. Guglielmi and T. Mejstrik who kindly spent their valuable time to help us with computation issues.