Table Of Content

ACCURACY OF SEVERAL MULTIDIMENSIONAL CRYSTAL-REFINABLE FUNCTIONS 7 1 URSULA MOLTER, MARÍA DEL CARMEN MOURE, 0 AND ALEJANDRO QUINTERO 2 n a Abstract. LetΓbeacrystalgroupinRd. Afunctionf :Rd −→ J Cr issaidtobeΓ−refinableifitisalinearcombinationsoffinitely 8 many of the rescaled and translated functions f(γ−1(Ax)), where 2 thetranslationsγaretakenonacrystalgroupΓ,andAisanexpan- ] sive dilation matrix such that AΓA−1 ⊂ Γ. Γ−refinable functions A satisfy a refinement equation f(x)= d f(γ−1(Ax)) with d γ∈Γ γ γ C being r×r matrices,and f(x)=(f1(Px),...,fr(x))T. The accuracy . of f is the highest degree p such that all multivariate polynomials h t q(x) of degree(q)<p are exactly reproduced fromlinear combina- a tionsoftranslatesoff alongthecrystalgroupΓ.Inthispaper,we m determine the accuracy p from the matrices d . Moreover, we de- γ [ duce from our conditions, a characterization of accuracy for some 1 lattice refinable vector function F, which simplifies the classical v conditions. 6 2 2 8 0 . 1 1. Introduction 0 7 Crystal groups (Crystallographic groups or space groups) Γ, are 1 groups of isometries of Rd that generalize the notion of translations : v on a lattice, allowing to move using different (rigid) movements in Rd i X following a bounded pattern that is repeated until it fills up space. r Precisely (see [7]): a Definition 1.1. A crystal group is a discrete subgroup Γ ⊂ Isom(Rd) such that Isom(Rd)/Γ is compact, where Isom(Rd) is endowed with the topology of pointwise convergence. Or equivalently, one can define a crystal group to be a discrete subgroup Γ ⊂ Isom(Rd) such that there exists a compact fundamental domain P for Γ. 2010 Mathematics Subject Classification. Primary 42C40, Secondary 52C22. Key words and phrases. Crystallographic groups, Accuracy, Refinement equation, Composite dilations. 1 2 U.MOLTER, M. MOURE, ANDA. QUINTERO Intuitively, a crystal should have a bounded pattern that is repeated until it fills up space, i.e. there exists a bounded closed set P such that γ(P) = Rd and γ(P◦)∩γ′(P◦) 6= ∅ then γ = γ′, γ∈Γ [ where P◦ is the interior of P. This set is called fundamental domain, which corresponds to the fundamental domain for lattices, only that here its shape can be much more general. Note that a particular case of crystal group is the group of translations on a lattice. In this paper we are interested in systems of functions, generated by dilations and movements along a crystal group. Let us start recalling the necessary definitions. Definition 1.2. Let Γ be a crystal group. We will say that A is a Γ−admissible matrix, if A is an expanding affine map and AΓA−1 ⊂ Γ. It is easy to see that if A is a Γ−admissible matrix, then m = |detA| is an integer. Therefore, the quotient group Γ/AΓA−1 is of order m. A function f : Rd −→ Cr is Γ-refinable with respect to A and Γ if it is a linear combinations of the rescaled and ‘translated’ functions f(γ−1Ax), where the ‘translates’ γ ∈ Γ are movements on Γ. Precisely, f(x) = (f (x),...,f (x))T satisfies a refinement equation, dilation 1 r equation or two-scale difference equation of the form (1) f(x) = d f(γ−1Ax) γ γ∈Γ′ X for some finite Γ′ ⊂ Γ and some r×r matrices d . These matrices are γ called coefficients of the refinement equation. Refinable functions with respect to A and Γ are related to Crystal Wavelets and Wavelets with composite dilations [8], [9], [14]. In this paper we study the general multidimensional, multifunction case (d ≥ 1,r ≥ 1) with a Γ−admissible matrix A. We seek to determine one fundamental property of the space spanned by a Γ−refinable function f based on the coefficients d . That property is the accuracy γ of f: Definition 1.3. Let f : Rd → Cr, the accuracy of f is the largest integer p such that all multivariate polynomials q(x) = q(x ,...,x ) of 1 d deg(q) < p lie in the space k (2) S(f) = span d f(γ x) : d ∈ C1×r , γi i γi ( ) i=1 X ACCURACY OF CRYSTAL-REFINABLE FUNCTIONS 3 which is the closure of all finite linear combinations of Γ−translates of f. As usual, equality of functions is interpreted as holding almost everywhere (a.e.). Note that in fact, the accuracy is a property of the space S(f), but since the space is generated by Γ−translates of the function f, we will talk in-distinctively about the accuracy of f, or S(f). Accuracy hasplayed animportantroleinbothapproximationtheory and in wavelet theory. In approximation theory, it is closely related to the approximation properties of shift invariant spaces. In wavelet theory, one of the most successful and systematic ways of constructing smooth, compactly supported, orthonormal wavelet bases for L2(R) is based on the factorization of a symbol which determines a scaling function [6]. This factorization of the symbol is closely related to the accuracy of the scaling function. If the scaling function has accuracy p, then the corresponding wavelet will have p zero moments. Accuracy is necessary for a refinable function to be smooth, although it is not sufficient. General results of accuracy can be found in [3, 4, 5, 11] and references therein. Our goal in this paper is to obtain necessary and/or sufficient conditions for a vectorial crystal refinable function f to have accuracy p. In this direction, our first result establishes necessary conditions for S(f) (2) with f an arbitrary function, to have accuracy p. For the case that f is a Γ−refinable function, we will give necessary and sufficient conditions to ensure that f has accuracy p. In the case in which f satisfies a refinement equation (1) with f : Rd → C, we can associate a vector function (f ,...,f ) on Rd, that satisfies a refinement equation (using 1 r translations on a lattice). Using our approach, accuracy conditions on the coefficients of the refinement equation are much simpler than in the general case (see Theorem 4.8). Following the ideas in [3] we consider together the monomials xα = xα1...xαd of a given degree in a single array. With this key observation, 1 d it is surprisingly simpler than expected to move from one to several variables. 2. Crystal Groups Definition 2.1. A crystal group is a discrete subgroup Γ ⊂ Isom(Rd) such that Isom(Rd)/Γ is compact, where Isom(Rd) is endowed with the topology of pointwise convergence. Or equivalently, one can define a crystal group to be a discrete subgroup Γ ⊂ Isom(Rd) such that there existens a compact fundamental domain P for Γ. 4 U.MOLTER, M. MOURE, ANDA. QUINTERO The theorem of Bieberbach [1] yields the following: Theorem 2.2 (Bieberbach). Let γ be a crystal subgroup of Isom(Rd). Then (1) Λ = Γ∩Trans(Rd) is a finitely generated abelian group of rank d which spans Trans(Rd), and ∼ (2) the linear parts of the symmetries ad(Γ) = Γ/Λ, the point group of Γ, is finite. (See also [12], IV-4). Here Trans(Rd) stands for translations of Rd. We will denote the point groupofΓ by G. andcall (Γ,G,Λ)acrystal triple. Remark 2.3. • Note that the set Λ is not empty by Bierberach’s theorem [1]. Moreover, Λ consists of translations on a lattice L which is isomorphic to Zd. We will denote by L and L∗ the fundamental domains of the lattices L and its dual, L∗ respectively. Here L = R(Zd) with R an invertible d×d matrix and hence L∗ = (R∗)−1(Zd). • The Point Group G of Γ is a finite subgroup of O(d), the or- thogonal group of Rd, that preserves the lattice of translations, i.e. GL = L. General results on crystal groups, can be found for example in [10], [17], [13], [1], [2]. Note that the simplest example of a crystal group is the group of translations on a lattice L, i.e. Γ = {τ : k ∈ L}, where τ (x) = x+k. k k One very important class of crystal groups, are the splitting crystal groups: Definition 2.4. Γ is called a splitting crystal group if it is the semidi- rect product of the subgroups Λ and G. In this case Γ = G ⋉ Λ and for each γ,γ ∈ Γ, with γ = (g ,τ ) and γ = (g ,τ ), we have i k j l γ ·γ = (gjgi,τkg−1(l)) where gi,gj ∈ G, τk,τl ∈ Λ and γ(x) = gi(x+k). i Every crystal geroup is naturally embedded in ea splitting group, and veery often arguments for general groups can be relatively easy reduced to the splitting case and then be proved for that simpler case. This justifies, thatfromnowonwewillonlyconsidersplittingcrystalgroups. For simplicty of notation, for each γ ∈ Γ we will use the notation γ = (g ,k) in stead of (g ,τ ). If γ = (g ,k) and γ = (g ,l), then i i k i j γ ·γ = (g g ,k +g−1(l)). j i i Example 2.5. Consider the vectors u = (0,1) and ve= (1,0) and let eS be the symmetry with respect to the X-axis (i.e S(x,y) = (x,−y)). ACCURACY OF CRYSTAL-REFINABLE FUNCTIONS 5 Let Γ be the group generated by {τ ,τ ,S}. Then Λ = {τ : l ∈ L} u v l where L = {αu+ βv : α,β ∈ Z} and G = {Id,S}. The fundamental domain P is the rectangle of vertices {(0,0);(1,0);(0,1/2);(1,1/2)} 11 22 P 1 S(P) -1 2 Figure 1. P ∪S(P) is the fundamental domain for Λ. 3. Notation The notation of this paper is complicated due to the dimension and the multiplicity of functions. We use the standard multi-index notation xα = xα1...xαd, where 1 d x = (x ,...,x )T ∈ Rd and α = (α ,...,α ) with each α a nonnegative 1 d 1 d i integer. The degree of α is |α| = α +... +α . The number of multi- 1 d s+d−1 indices α of degree s is d = . We write β ≤ α if β ≤ α s d−1 i i (cid:18) (cid:19) for i = 1,...,d. For each integer s ≥ 0 we define the vector-valued function X : [s] Rd → Cds by X (x) = [xα] ,; x ∈ Rs. [s] |α|=s For our purposes we need define two special matrices, A and Q [s] [s,t] for integers s,t ≥ 0. Given a matrix A, we define the matrices A and [s] Q by [s,t] X (Ax) = A X (x), [s] [s] [s] s X (x−y) = Q (y)X . [s] [s,t] [t] t=0 X These matrices have three properties that will be of great impor- tance. Lemma 3.1. Let A ∈ Rd×d be a matrix, and L be the lattice associated to the crystal group Γ (see Remark 2.3). Then: (1) If A is an expansive matrix then A is an expansive matrix for [s] each s ≥ 0. (2) If A is an invertible matrix then Q (Az) = A Q (z)A−1. [s,t] [s] [s,t] [t] 6 U.MOLTER, M. MOURE, ANDA. QUINTERO (3) Let B ∈ Cdt×r be given matrices, for 0 ≤ t ≤ s. If t s Q (Al)B = 0 for each l ∈ L, then B = 0 for 0 ≤ t ≤ s. t=0 [s,t] t t The pProof of the previous lemma as well as the explicit form and properties of these matrices can be seen in [3]. From the matrices A and Q we obtain the following definition. [s] [s,t] Definition 3.2. Let (Γ,G,Λ) be a splitting crystal triple. Let γ ∈ Γ, γ = (b,l) then we define the matrices Q by [s,t] Q (γ) = Q (l)b−1. [s,t] [s,et] [t] Lemma 3.3. Let (Γ,G,Λ) be a splitting crystal triple, and A an in- e vertible matrix such that AΓA−1 ⊂ Γ. We then have: (1) Q (τ ) = Id. [s,s] l (2) Q (γ) = (−1)sX (l) for each γ = (b,l) ∈ Γ. [s,0] [s] s (3) Q (γ γ ) = Q (γ )Q (γ ). [s,t] 1 2 u=t [s,u] 2 [u,t] 1 (4) Q (AγA−1) = A Q (γ)A−1. [s,t] P [s] [s,t] [t] The proof the previous lemma is immediate from Lemma 3.1 and Lemma 4.1 of [3]. Given a collection {v = (v ,...,v ) ∈ C1×r : 0 ≤ |α| < p}, α α,1 α,r of row vectors of length r, we shall associate special matrices and functions, which play an important role in our analysis of accuracy. We use the following notation throughout the paper. We group the v by degree to form d ×1 column vectors v with α s [s] block entries that are the 1×r row vectors v . Specifically, we set α v = [v ] , 0 ≤ s < p. [s] α |α|=s Note that, when α = 0 then v = [v ] = v . [0] 0 0 s (3) y (γ) = Q (γ)v , [s] [s,t] [t] t=0 X where γ = (b,l) and b is the matrix that satisfies X (bx) = b X (x). [t] [t] [t] [t] Finally, we define the infinite row vector (4) Y = (y (γ)) . [s] [s] γ∈Γ The functions y have the following properties. [s] ACCURACY OF CRYSTAL-REFINABLE FUNCTIONS 7 Lemma 3.4. Let {v ∈ C1×r : 0 ≤ |α| < p} be a collection of vectors α s and let y be the functions y (γ) = Q (γ)v . Let γ and γ in [s] [s] t=0 [s,t] [t] 1 2 Γ, then s P e y (γ γ ) = Q (γ )y (γ ). [s] 1 2 [s,t] 2 [t] 1 t=0 X If γ = (Id,l ) = τ , then the previous equality yields 2 2 l2 s y (γ τ ) = Q (l )y (γ ). [s] 1 l2 [s,t] 2 [t] 1 t=0 X Proof. For the proof we use Lemmas 4.1, 4.2 and 4.3 of [3]. By definition s s s y (γ γ ) = Q (γ γ )v = Q (γ )Q (γ )v [s] 1 2 [s,t] 1 2 [t] [s,u] 2 [u,t] 1 [t] t=0 t=0 u=t X XX s u s = Q (γ ) Q (γ )v = Q (γ )y (γ ). [s,u] 2 [u,t] 1 [t] [s,t] 2 [u] 1 u=0 t=0 u=0 X X X (cid:3) We will say that translates of f along Γ are Γ−independent if for every choice of row vectors b ∈ C1×r, γ b f(γx) = 0 if, and only if, b = 0 for every γ. γ γ γ∈Γ X Equivalently, for every choice of an infinite row vector b = (b ) with γ γ∈Γ block entries b ∈ C1×r, γ bF(x) = 0 if, and only if, b = 0. Here F(x) is the infinite column vector with block entries f(γ(x)), i.e. (5) F(x) = [f(γ(x))] . γ∈Γ 4. Characterization of Accuracy 4.1. Necessary conditions for arbitrary functions. In this section, we will present necessary conditions for an arbitrary (not neces- sarily Γ−refinable) function f : Rd −→ Cr with Γ−independent translates, to have accuracy p. Theorem 4.1. Assume that f : Rd → Cr is compactly supported, and that translates of f are Γ−independent. If f has accuracy p then there exists a collection {v ∈ C1×r : 0 ≤ |α| < p} α 8 U.MOLTER, M. MOURE, ANDA. QUINTERO of row vectors such that i): v 6= 0. 0 ii): X (x) = y (γ)f(γ(x)) = Y F(x) for 0 ≤ s < p. [s] [s] [s] γ∈Γ Where Y =Py (γ) as in (3) and (4). [s] [s] γ∈Γ Proof. Since f has ac(cid:0)curacy(cid:1)p, there exist row vectors w ∈ C1×r such α,γ that every polynomial xα of degree α, 0 ≤ |α| < p can be written as a finite linear combination of Γ−translates of f. xα = w f(γ(x)) a.e. α,γ γ∈Γ X For each γ ∈ Γ, group the vectors w by degree to form the column α,γ vectors w (γ) = [w ] . [s] α,γ |α|=s For each σ ∈ Γ define the infinite row vector W (σ) = w (γσ) . [s] [s] γ∈Γ Next, set v = w (where I is th(cid:0)e identity(cid:1)of Γ) and define the vectors α α,I v and the matrices of polynomials y by [s] [s] s v = [v ] and y (γ) = Q (γ)v . [s] α |α|=s [s] [s,t] [t] t=0 X Then, grouping the polynomials xα by degree, we have for 0 ≤ s < p, that X (x) = [xα] = w f(γ(x)) [s] |α|=s α,γ " # Xγ∈Γ |α|=s = w (γ)f(γ(x)) = W (I)F(x). [s] [s] γ∈Γ X Therefore, for each σ = (g,h) ∈ Γ W (σ)F(x) = W (I)F(σ−1(x)) = X (g−1(x)−h) [s] [s] [s] s s = Q (h)g−1X (x) = Q (σ)X (x) [s,t] [t] [t] [s,t] [t] t=0 t=0 X X s = Q (σ)W (I) F(x). [s,t] [t] ! t=0 X ACCURACY OF CRYSTAL-REFINABLE FUNCTIONS 9 Considering our assumption that translates of f are Γ−independent, this implies that s W (σ) = Q (σ)W (I), [s] [s,t] [t] t=0 X and therefore for each γ ∈ Γ we have that s w (γσ) = Q (σ)w (γ) . [s] γ∈Γ [s,t] [t] ! (cid:0) (cid:1) Xt=0 γ∈Γ In particular, taking γ = I we obtain s s w (σ) = Q (σ)w (I) = Q (σ)v = y (σ). [s] [s,t] [t] [s,t] [t] [s] t=0 t=0 X X Thus X (x) = y (γ)f(γ(x)) = Y F(x). [s] [s] [s] γ∈Γ X Consider now the case s = 0. Since y (γ) = v for every γ ∈ Γ we [0] 0 have 1 = x0 = X (x) = y (γ)f(γ(x)) = v f(γ(x)). [0] [0] 0 γ∈Γ γ∈Γ X X Therefore v 6= 0. (cid:3) 0 4.2. Accuracy for Γ−refinable functions. In this section we will obtainnecessary and/orsufficient conditions fora Γ−refinable function to have accuracy p. First, we rewrite equation (1) in matrix form. Let (Γ,G,Λ) be a splitting crystal triple and A a Γ−admissible matrix. Remember that a function f is Γ−refinable if satisfies f(x) = d f(γ−1Ax). γ γ∈Γ X We consider F(x) the infinite column vector defined by equation (5), i.e. F(x) = [f(γ(x))] . Note that for a given x, only finitely many γ∈Γ entries f(γ(x)) of F(x) are non zero since f has compact support. Lemma 4.2. Let f : Rn → Cd, A ∈ Cr×r a Γ−admissible matrix and F the function defined by F(x) = [f(γ(x))] (see (5)). Then, the γ∈Γ function f is Γ−refinable if and only if LF(AX) = F(X) a.e., where L is the Γ ×Γ matrix given by L = [dAγA−1σ−1]γ,σ∈Γ, with r ×r block entries dAγA−1σ−1. 10 U.MOLTER, M. MOURE, ANDA. QUINTERO Proof. Suppose that f satisfies the Γ−refinement equation (1). We consider L = [dAγA−1σ−1]γ,σ∈Γ and we know that F(x) = [f(γ(x))] = d f(σ−1(Aγx)) by (1). γ∈Γ γ " # Xσ∈Γ γ∈Γ Hence LF(Ax) = [dAγA−1σ−1]γ,σ∈Γ[f(σ(Ax))]σ∈Γ = dAγA−1σ−1f(σ(Ax)) " # Xσ∈Γ γ∈Γ = d f(α−1AγA−1Ax) = [f(γ(x))] = F(x). α γ∈Γ " # Xα∈Γ γ∈Γ Conversely, let us suppose that LF(Ax) = F(x). Then for each γ ∈ Γ we have dAγA−1σ−1f(σ(Ax)) = f(γ(x)). σ∈Γ X In particular, if we consider γ = Id, then dσ−1f(σ(Ax)) = dσf(σ−1(Ax)) = f(x), σ∈Γ σ∈Γ X X therefore f is a Γ−refinable function. (cid:3) The following result characterizes the accuracy of Γ−refinable functions. Theorem 4.3. Assume that f : Rd → Cr is integrable, compactly supported and satisfies the refinement equation (1). Consider the following statements I) f has accuracy p. II) There exist a collection of row vectors {v ∈ C1×r : 0 ≤ |α| < p} α such that ˆ (i) v f(0) 6= 0 and 0 (ii) Y = A Y L for 0 ≤ s < p where Y = (y (γ)) as in [s] [s] [s] [s] [s] γ∈Γ (3) and (4). Then we have the following: a) If the transaltes of f along Γ are independent, then (I) implies (II) b) (II) implies (I). In this case, if we scale all the vectors v by α C = (v fˆ(0))−1|P| then 0 X (x) = y (γ)f(γ(x)) = Y F(x), 0 ≤ s < p. [s] [s] [s] γ∈Γ X