The Journal of Fourier Analysis and Applications Volume 3, Number 5, 1997 Tight Frames of Multidimensional Wavelets Marcin Bownik ABSTRACZ In this paper we deal with multidimensional wavelets arising from a multiresolution analysis with an arbitrary dilation matrix A, namely we have scaling equations ~°S(x)= Z hSk Iv/~ttal~°l(Ax-k) fors= 1..... q, k~Z n whereto I is a scaling fimction for this multiresolution and ~o2 ..... ~oq ( q = [ det A [) are wavelets. O rthog- onality conditions for ~oI ..... ~oq naturally impose constraints on the scaling coefficients {hSk}~=L,~..,q, which are then called the wavelet matrix. We show how to reconstruct functions satisfying the scaling equations above and show that ~o2 ..... ~oq always constitute a tight frame with constant 1. Furthermore. we generalize the sufficient and necessary conditions of orthogonality given by Lawton and Cohen to the case of several dimensions and arbitrary dilation matrix A. 1. Preliminaries In this section we fix some definitions and notations and we present theorems we use later. We use the following definition of the Fourier transform in R n. .T f(x) = f(x) = ~, f(y)e-Z~i(x'y)dy . (1.1) This is well defined for integrable functions f. Nevertheless, .F can be defined on L2(]~n), and then :" : L2(R n) ----> L2(~ n) is unitary (Plancherel theorem). Let II" 112 be the norm in L2(Rn). Let us recall some useful properties of the Fourier transform. Let us denote by 7.y the operator of translation by y, Uyf(x) = f(x - y) and by HA the scaling operator by a non-degenerate matrix A E Mn(R),HAf(x) = f(Ax). Then .T'Ty f ( x ) = e-2~ri(x'Y)3~(x), (1.2) .T'LIA f (x ) - A-----~idet f (()-,)A T x. (1.3) Definition. A family of vectors (vj)j~s in Hilbert space 7-/ is called a frame if there are A > 0, B < ~, such that for all v 6 7-/, AIIvll 2 _< ~ I(v, vj)l z _< nllvll z • (1.4) jEJ Math Subject Classifications. 42C 15. Keywords and Phrases. Tight frame, wavelets, multiresolution analysis. ~D 1997 CRC Pre~.sLLC ISSN 1069-5869 526 Ma~'inBownik A and B are called frame constants. If A = B, then (vj)j~j is called a tight frame. [] General information about frames can be found in [3]. For a tight frame with frame constant 1 we need only the simple fact that if [Ivjll = 1 for all j 6 J, then (vj)j~j is an orthonormal basis of 7-/. To see this, note that linear combinations of vj are dense in 7-( and [I vjo[I 2 = )--~4eJ I(vjo' vj)12 = Ilvjo 114 + Y~j~J.j#Jo ]{uJo' vJ )12; hence, (vjo, vj) = 0forj # j0. Definition. For any integer m > 0 we introduce the Sobolev space (with exponent 2) by wm(~,n)= {f ~ L2(IR"): Daf ~ LZ(R n) for lot I < m} . (~.5) with norm [Ifllw .... __ , (l.6) [al<m 0 where c~ = (ot~ ..... ~,,) is a multi index and by Dct -- ,3~ ~...~x~''~I,,I is the distributional derivative. W m (N n) equipped with the norm (1.6) is Hilbert space. [] We will use the Sobolev lemma and a simple lemma about Sobolev spaces (see[ 17]). Lemma 1. (Sobolev). lf f ~ W 'n (R n) and m > n/2, then (eventually after a change of values on a set of measure O) • f ~ cr(Rn)forr < m - n/2, • the derivatives Daffor lct[ < r satisfy the inequality IID=flt~ ~ cllfllwm, (1.7) with constant c > 0 independent of f . Lemma 2. Let h ~ wrn(~,n) and g ~ Cc¢(R n) have compact support. Then the sequence ak IIhT"kgl]w,,, k ~ Z", belongs to 12(Z'7). Proof. lakl z = IIh~gllZm = ~ IID~(h~g)lt~ <_ C ~ [IDUhDl~(~g)ll22 lal<m lal+l~l_<m < C sup ItDt~gll~ ~ fs IOC~h(x)12dx" I~l<m tal_<m uppT"-kg Because suppg is bounded, fs tDah(x)12dx 5_ c" fR IDah(x)12dx kEZZ n uppT-kg . which finishes the proof. [] 2. Introduction Assume we have some matrix A ~ Mn(N) acting on a lattice F, (F = PZ n for some non- degenerate matrix P 6 Mn (]R)), such that: Tight Frames of Multidimensional Wavelets 527 • A is a dilation matrix, i.e., all eigenvatues ), of A satisfy [)-I > I • 1" is invariant for A, that is AF C 1-'. p- 1A P is a matrix with integer entries hence q = ]det A I = Idet P- 1A P] is an integer greater than 1. Definition. By a multiresolution analysis associated with (A, F) we mean a sequence of closed subspaces (Vi)iez C L2(Rn), satisfying following conditions: i. Vi C Vi+l for/ E Z. ii. giEz Vi is dense in LZ(IRn). iii. AiEz V/ = {0}. iv. f(x) ~ V~ iff f(A-ix) ~ Vo. v. There exists ~o called a scaling function such that {~o(x - Y)}vsr is an orthonormat basis of VO. [] Remark. Conditions (iv) and (v) can be expressed by saying that for each i {~o(Aix - Y)}v~r is an orthonormal basis of Vi. Condition (i) then implies that a scaling equation is satisfied ~o(x) = Z hy~o(Ax - y) , (2.1) y~F for some coefficients (hy)yer'. Thus, the main ingredient of a multiresolution analysis is scaling function ~o satisfying (2.1). [] If we have a multiresolution analysis with scaling function ~o, then one can show there are numbers/hst,,×Jy~r ls=2"-"q so that the q - I functions q)2, .. ~oq, called wavelets, generated from ~o by the formula q¢'(x) = Z hSr"~9)(Ax - Y) for s = 2 ..... q, (2.2) y~F have the property that {rpS(x - •,,aff=2,...,q" JJyEF is an orthonormal basis of W0 = V1 • V0 (the orthogonal compliment of V0 in V 1 ), see [19]. Equations (2.1) and (2.2) can be expressed jointly by ~Os( X ) = v--.,_,----------2,h~,/]detAl~ol(A x _ g) fors = 1 ..... q, (2.3) y~F where ~o I = ~o is the scaling function and h~, = h×, g c P. Therefore I s=2....,q {]det A[J/aps(AJx - Y)/×sr.jsZ (2.4) is an orthonormal basis of L2(~n). In order to simplify many calculations, we will deal with a multiresolution analysis asso- ciated with (A,Zn), where A is some dilation matrix with integer entries. One should stress that this is not an essential restriction. For any multiresolution analysis (Vi)i~z, associated with (A, F) with scaling function ¢p, we can consider another multiresolution analysis associated with (P-lAP, zn), where P has the same meaning as above. Since A and p-lAp have the same char- acteristic polynomials P-lAP is also a dilation matrix. Consider the unitary operator Up given by Uef(x) = t~/~Plf(Px). Because Up preserves scalar product in L2(~ n) (Up Vi)i~ Z is a muttiresolution analysis with scaling function Up~p. Scalings and translates of Up~o2 ..... Upq)q form an orthonormal basis. The operator Up preserves other properties, such as tightness of the 528 Marcin Bownik frame, smoothness, vanishing in infinity, compact support, etc. This is why we will deal only with dilation matrices A acting invariantly on F = Z n. Self similar tilings of R n arise naturally when one considers a multiresolution analysis for which the scaling function is the indicator function of some measurable set. This was first noticed in the paper [6]. Many other authors have worked on related subjects, see [8, 12] and [5]. The following fact which can be extracted from [9] is of great use. Fact 1. Let A be a dilation matrix and 79 = {k! ..... kq} be q = Idet A[ representatives of different cosets ofZn/AZ n and Q = Q(A, 73) = {x 6 IR" • x = ~--~-~=t A-i6i' Ei ~ 79}. lff 6 L loc"1 t]Rn,~ J (locally integrable on R n) is Z n-periodic then fQf(X)dx = JQJ f[o.,l f(x)dx " 3. Solution of the Sealing Equation Suppose we have some scaling coefficients {h E }~cz" q and using them we try to reconstruct the wavelets appearing in (2.3). Naturally, we should add some extra conditions on these coefficients. The orthonormality of translations of the scaling function and of the wavelets is the motivation for the following definition. Definition. A sequence of vectors (h i , h 2 ..... h q) ~ (l I (Zn)) q is called a wavelet matrix, if ~--] h k+Ams h s'k+Am' -~- ~s.s'~m.rn' (3.1) kEZ n for every s, s' = 1 ..... q; m, rn' 6 Z n and him = ~. (3.2) mET, n The first vector is called the scaling vector, the others are called wavelet vectors. [] This definition in the case of one dimension appeared in [7], where the reader can find various examples of wavelet matrices. The simplest example of a wavelet matrix for the general dilation A is obtained by taking a unitary q x q matrix U = ,,t'U'uJi=l'~J=l .......... qq with a constant first row, that is u U = 1/q/~, j = 1 ..... q and defining s ifk = for some i = 1, , q , • Ui ki .,. hE = 0 otherwise, where {kl ..... kq} are representatives of different cosets of Zn/AZ n. Not much is known to the author about the existence, for a given dilation matrix, of wavelet matrices with coefficients of compact support or with strong decay at infinity.