. Introduction Let M be an s s dilation matrix, that is, an s s integer matrix whose eigenvalues lie outside the closed unit disk. The compactly suppo

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1 Multivariate Compactly Supported Fundamental Renable Functions, Duals and Biorthogonal Wavelets y Hui Ji, Sherman D. Riemenschneider and Zuowei Shen Abstract: In areas of geometric modeling and wavelets, one often needs to construct a compactly supported renable function with sucient regularity which is fundamental for interpolation (that means, () = and () = for all Z s nfg). Low regularity examples of such functions have been obtained numerically by several authors and a more general numerical scheme was given in [RiS]. This paper presents several schemes to construct compactly supported fundamental renable functions with higher regularity directly from a given continuous compactly supported renable fundamental function. Asymptotic regularity analyses of the functions generated by the constructions are given. The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces. A very important consequence of the constructions is a natural formation of pairs of dual renable functions, a necessary element in the construction of biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual renable functions given in [RiS], we are thus able to obtain symmetric compactly supported multivariate biorthogonal wavelets with arbitrarily high regularity. Several examples are computed. AMS Subject Classication: Primary A5, A5, A5 A Secondary 9B6, A8 Keywords: Fundamental functions, renable functions, dual functions, interpolatory subdivision scheme, biorthogonal wavelets. y Research supported in part by the Strategic Wavelet Program at the National University of Singapore and NSERC Canada under Grant # A7687.

2 . Introduction Let M be an s s dilation matrix, that is, an s s integer matrix whose eigenvalues lie outside the closed unit disk. The compactly supported distribution is M-renable with nite mask sequence a, when satises the renement equation (:) = Z s j det Mja()(M ) for a nitely supported sequence a. When M = I ss, we simply say is renable. In most of the applications, should be a function with some regularity (smoothness). If is continuous and equal to at the origin and at the other integers: ; if = ; (:) () = := ; if Z s nfg, then is called fundamental (because any sequence of data can be interpolated at the lattice by an innite linear combination of the shifts of with the data as coecients). All the essential information about the M-renable function is carried in the renement mask a. The construction of compactly supported fundamental M-renable functions with desirable regularity properties is motivated from needs arising in geometric modeling and the application of wavelet theory to signal processing (as well as in many other applications of wavelets and biorthogonal wavelets). It is also the logical starting point for the construction of biorthogonal wavelets. The generator for the wavelets and its dual function is in some sense a \splitting" of a fundamental M-renable function with the regularity of the resulting pair at least partially determined by the original function. Examples in the univariate case were rst given by [Du]. Daubechies in [D] obtained a general construction and also provided a good asymptotic regularity analysis. For higher dimensions this analysis has been limited to a few cases that can be reduced to a univariate setting. The goal of the present paper is to carry out the following program: (i) \Iterate" a fundamental M-renable function of known smoothness to obtain a smoother fundamental M-renable function. An asymptotic analysis of the regularity of the new function can be given in terms of the regularity of the original function and the number of steps taken. The latter quantity can be related in turn to other quantities, such as the length of the renement mask. (ii) \Split" the new function into a pair of dual functions, for purposes of generating biorthogonal wavelets in such a way that the original function is one of the pair. This provides compactly supported M-renable functions of arbitrary regularity which are dual to a given compactly supported fundamental M-renable function. Combined with the biorthogonal wavelet construction for a pair of dual renable functions given in [RiS], we are able to obtain symmetric compactly supported multivariate biorthogonal wavelets with arbitrarily high regularity.

3 We next motivate the construction of such functions by a brief description of how they arise in geometric modeling and signal processing. After that we provide the basic notation and Fourier analysis formulations that will be used in the sequel. The remainder of the introductory section will discuss more fully earlier work and the connections between fundamental M-renable functions and biorthogonal wavelets. In x, we will give the implementable constructions alluded to above, while the asymptotic analysis of regularity for these constructions is derived in x. How the constructions can be used for the formation of dual pairs of functions is the topic of x. Finally, in x5 we will illustrate our methods by examples. In geometric modeling, subdivision algorithms are dened by renement masks. For simplicity, assume that the dilation matrix is M = I. Subdivision algorithms begin with some initial set of discrete data, c := fc : Z s g, called control points, which can be visualized as the vertices of a given polyhedral surface. The subdivision operator S is dened by the mask a as (:) S(c) := Z s a( )c ; Z s : This gives s sets of new control points c + = Z s a( + )c ; f; g s Thus, the new control point sequence c is determined linearly from c by s dierent convolution rules, and the sequence c consists of s dierent copies of the original control point sequence c which are averaged by dierent dyadic cosets of the mask. With the scaling factor, the new control polygon is parameterized so that the points c correspond to the ner grid Z s. Continuing this process, we get control point sequences c n = S n c corresponding to the grids n Z s. If the mask a is well chosen, these data sets will approach some continuous limiting surface in a computationally stable manner. The subdivision operator can be visualized as an operator which smoothes the corners of a given polyhedral surface. If the mask a comes from a fundamental renable function, then from (.) and (.) we nd a() =. Thus, c = c(), and the new control point sequence interpolates the previous one on the grid Z s. More generally, the sequences c n = S n c corresponding to the grids n Z s interpolate the previous control points c n on the grids n Z s. This iterative process is called an interpolatory subdivision algorithm. The limiting surface can be written as c()( ); Z s and thus has the same regularity as the function. That is, smoother surfaces require smoother fundamental renable functions. We refer the reader to [CDM] for the details. As mentioned earlier, the construction of interpolatory subdivision algorithms is directly connected to construction of orthogonal and biorthogonal wavelets which provides the second motivation described next. We refer to [St], [M] and [RiS] for the more details.

4 Biorthogonal wavelets start with the construction of a dual pair of compactly supported renable functions, and d in L (R s ) such that (:) h; d ( )i = ; Z s : To construct orthogonal wavelets, should be equal to d. The functions and d are then used to construct biorthogonal wavelets (cf. x). If and d are compactly supported, then the wavelet families they generate will also be compactly supported and will have the same regularity as their respective generator. When the wavelet transform is used to analyze signals, the compactness gives localization in the time domain, while the regularity provides localization in the frequency domain. Ideal wavelets would possess high regularity and small support. However, the Heisenberg uncertainty principle asserts the contrary; higher regularity leads to larger support. A balance between the time-frequency localization requirements is often needed. By constructing wavelets with larger support to increase the regularity, we lose accuracy in time domain to gain accuracy in frequency domain. Given one of the dual renable functions or d, we may wish to choose the approximate regularity of the other, which is precisely what our method allows. The construction of dual renable functions, or what is the same, the choice of two masks a and a d, can be used to design a pair of (biorthogonal) low pass lters. The corresponding construction of the wavelet masks from a and a d given in x can be used to construct the biorthogonal high pass lters. The lters constructed by the methods in this paper are linear phase (symmetric) lters. Wavelet construction and lter bank design in signal processing are interrelated subjects; see [SN] and [VH]. We now turn to the Fourier analysis formulations needed in the paper. A compactly supported continuous function is fundamental if and only if (:5) Z s b (! + ) = : Applying the Fourier transform to the renement equation (.) give the relation (:6) b (!) = ^a(m t!)b (M t!); where ^a(!) = P Z s a() exp( i!) and M t is the inverse of the transpose M t of the matrix M. We also call ^a the mask (sometimes it is called the symbol of the mask) and often nd it convenient to write it as a Laurent polynomial A(z) = Z s a()z : When ^a() =, then there is a unique compactly supported distributional solution of renement equation (.) with b () =. The Fourier transform of the solution of the M-renement equation (.) can be obtained as the innite product (:7) b (!) := j= ^a(m t j!):

5 Let be a compactly supported fundamental M-renable function with the mask ^a. Condition (.5) has consequences for the mask ^a which result from applying (.6). Before describing this result, we need some notation. The integers Z s can be decomposed into the disjoint sets (cosets) f + M t Z s g, Z s M t, where Z s M t = Z s = M t Z s. Combining (.6) and (.5) using this decomposition of the sum, we nd that if the M-renable compactly supported continuous function is fundamental, then its mask ^a satises (:8) where Z s Mt ^a! + M t Z s Mt = ;! R s ; or, equivalently, A( z) = ; jzj = ; := exp im t : Condition (.8) is called the interpolatory condition for the mask on Z s M t. It was shown in [LLS] that a continuous renable function is fundamental if and only if is stable and its mask, ^a, satises the interpolatory condition (.8). Here, we recall that is stable, when L (R s ) and its shifts form a Riesz basis of S(), where S() is the shift invariant subspace in L (R s ) generated by. A function L (R s ) is stable if and only if the inequality (:9) c Z s jb (! + )j C; a:e:! R s holds for some constants < c C. pre-stable, if the inequality c jb (! + )j Z A compactly supported distribution is holds for some constant < c, or equivalently, if b has no -periodic zeros. If a compactly supported pre-stable function is in L (R s ), then it is stable. The construction of compactly supported fundamental functions starts with the construction of masks that satisfy the interpolatory condition (.8). Since the masks considered here are Laurent polynomials, the problem of constructing masks to satisfy the interpolatory condition (.8) can be reduced to a problem of solving a system of equations. However, this is not enough for the corresponding renable function to be fundamental as the following examples will show that the interpolatory condition (.8) is only a necessary condition. The rst simple example shows that the corresponding renable function can be stable and have a mask that satises the interpolatory condition (.8), but may not be continuous. Example.. Let ^a(!) = (+exp( i!)). Then ^a satises the interpolatory condition (.8) and the corresponding renable function is the characteristic function on [; ], which is not continuous. The next example shows that the mask can satisfy the interpolatory condition and the corresponding renable function can be very smooth, however, it may not be fundamental.

6 Example.. Let ^a := cos (!=). Then ^a satises the interpolatory condition (.8). Since the corresponding renable function is the autocorrelation function of the characteristic function of [; ], it is not stable, hence not fundamental. Further, let where ^a ^a k := ^a k ( ^a k ); k = ^a. Then, by Theorem.5 of [JiS], ^a k satises the interpolatory condition (.8) and the regularity of the corresponding renable function k of the mask ^a k can be made as high as we wish by taking k suciently large. However, the function k cannot be fundamental, since b k has b = b as a factor which implies that k is not stable. The procedure commonly used in the literature to construct examples of compactly supported fundamental renable functions is: (i) Solve a system of equations derived from the interpolatory condition (.8) to obtain a mask satisfying (.8). (ii) Check the stability and calculate the regularity of the corresponding renable function via the transition operator. This procedure faces the following diculties: First, except in very few cases, the masks obtained from the system of equations are numerical in nature and have no explicit closed form. Secondly, after the numerical solutions are obtained, one still needs to check whether the corresponding renable functions are fundamental and to determine their regularity. A general method of construction should not only build the masks to satisfy the interpolatory condition (.8), but also should provide the analysis of (a) whether the resulting functions are fundamental and (b) the asymptotic regularity. In this regard, there are few such constructions available. Daubechies in [D] obtained such a general construction for the univariate case where each mask is explicitly given as the convolution of the mask of a B-spline with a mask of a renable distribution. It is this construction that leads to the general construction of compactly supported orthogonal wavelets, and later to the construction of compactly supported biorthogonal wavelets (see [CDF]). This construction also is the basis of the bivariate constructions in [CD] and [HL], where the bivariate problem is reduced to a univariate problem to which the Daubechies' construction can be applied directly. Strictly bivariate examples of continuous and continuously dierentiable compactly supported bivariate fundamental renable functions were obtained by [DDD] and [DGL] respectively. A set of examples of compactly supported bivariate fundamental renable functions with increasing regularity was provided in [RiS] from masks formed by the convolution of box spline masks with masks of renable distributions. A construction was given in [HJ] for examples of continuous fundamental renable functions with various optimal properties. Several iterative methods for the construction of compactly supported fundamental renable functions were obtained in [JiS] in their construction of univariate biorthogonal wavelets from a multiresolution generated by fundamental renable function. In x, the ideas of [JiS] are more fully developed to obtain methods for the construction of compactly supported fundamental M-renable functions for any dilation matrix and in any number of variables starting from a given compactly supported M-renable fundamental function. The examples mentioned in the previous paragraph provide a sucient number of starting points to justify our methods. 5

7 For dual pairs of M-renable functions, and d in L (R s ), we require that the functions be stable and satisfy the biorthgonality relation (.). Then the function d in L (R s ) is called a dual function to. Often, one of these two functions is given. For a given stable function L (R s ), it is possible to nd noncompactly supported dual renable functions d, but the construction of compactly supported dual renable functions requires more as proved in [BR]: A compactly supported function has a compactly supported (not necessary renable) dual function if and only if its shifts are linearly independent, that is, if and only if for an arbitrary sequence a, the equality a()( ) = Z s implies that a =. The linear independence of the function L (R s ) and its shifts is a stronger requirement than the stability of. One method for the construction of a dual pair of renable functions is as follows: First, nd a nitely supported mask ^a whose corresponding compactly supported renable function ' is fundamental. To obtain a dual pair, one factors the mask ^a in an appropriate way and then separates the factors into two masks. In the univariate case, when ' is fundamental, the mask ^a always has the mask of a B-spline as a factor and this method leads to the construction of the biorthogonal wavelets in [CDF]. Further, in the univariate case, when ^a = ^a( ) and is non-negative, one can separate the mask into identical pairs of factors to obtain a renable function with orthonormal shifts. That is the case in Daubechies' construction of compactly supported orthonormal wavelets. For the multivariate case, we do not have factorization of the mask any more. In [CS], an example of biorthogonal bivariate wavelets with dilation matrix M = I was constructed where one basis is continuous piecewise linear polynomials while its dual basis is a compactly supported L function found by solving linear equations. In [RiS], pairs of dual renable functions are obtained by separating the masks from [RiS] where ^a are the products of box splines masks with trigonometric polynomials. Thus, one set of wavelets consisted of nite linear combinations of box splines while various dual functions of diering smoothness were obtained. In some cases, the construction of compactly supported M- renable dual functions can be reduced to the problem of the construction of compactly supported fundamental M-renable functions as follows: If the renable M-function has linearly independent shifts, then one nds a compactly supported M-renable function d L (R s ), so that the function ' = ( ) d is fundamental. Then d is a dual function of. In this regard, our particular methods will lead naturally to the construction of dual functions, as we will show in x.. Methods for Constructions of Masks Suppose we have a fundamental M-renable function in hand, there are two questions we wish to address here. () If the smoothness of the given renable function is not high enough, can we \iterate" in some fashion to obtain a smoother one from it while still preserving the interpolatory property for the new mask? One likely way to obtain higher 6

8 smoothness is to obtain higher powers of the given mask as a factor of the mask obtained through the iterative procedure. This would require controlling the remaining factor so as to not detract too much from the advantage gained. If such is the case, then there is hope that after removing one power of the original mask from the new mask, what remains will be a mask which answers the second question: () Can we nd a dual function with smoothness as high as we wish? The constructions of new compactly supported M-renable fundamental functions from a given one and the construction of a dual renable function given below are based on the following simple idea: Let P be the mask symbol ^a of the given fundamental M-renable function. Dene (:) P (z) := P ( z); Z s Mt; jzj = : Then the interpolatory condition (.8) becomes (:) Z s Mt Z s Mt P (z) = ; mn P (z) = jj=mn jzj = ; which implies C mn Z s Mt P (z)! = ; jzj = for any integer N and m. In particular, we take m = j det Mj. Since we mainly consider this polynomial on the torus, we shall abuse notation and write P (!) instead of P (exp( i!)). Theorem.. Let P = P be a Laurent polynomial satisfying (.) (i.e. (.8)) for a dilation matrix M with m = j det Mj. Dene o G := n IN m :jj = mn; > N and > ; Z s M n tnfg G j := IN m :jj = mn; > N and ; Z s M tnfg; with exactly j equalities o ; j = ; : : : ; m : and dene H := m j= j + C mn P G j P Z s Mt nfg! + C (N;:::;N) mn Z s Mt P N ; where C mn are the multinomial coecients. Then, as a symbol, the Laurent polynomial P H also satises (.8). Remark. It should be noted that some of the sets G j, j = ; : : : ; m, may be empty for particular choices of m and N. In this case, the corresponding terms in the denition of H are zero. 7

9 Proof. Let Q (z) := (P H)( z), Z s M t. Observe that the mapping 7! + is one-to-one and onto Z s M t for each xed Z s M t. Hence, when z is replaced by z in the sequence P Z P s only a permutation of the sequence is obtained. For the last term Mt of Z s (P H), we nd Mt C (N;:::;N) mn Z s Mt P N+ 6= P N = C (N;:::;N) mn = C (N;:::;N) mn Z s Mt Z s Mt P N P N! Z s Mt P by (.8) for P. The term in P H corresponding to j, j m with G j 6= ;, is (:) j + C mn P G j P Z s Mt nfg When z is replaced by z, we obtain a similar sum with G j replaced by n G j () := IN m :jj = mn; > N and ; Z s M tnfg; with exactly j equalities o :! : Summing over Z s M t, we obtain j + Z s G Mt j () C mn Z s Mt P! = G j C mn Z s Mt P! ; where G j := n IN m : jj = mn; with exactly j + of the equal to jj o ; (because G j (), Z s M t covers G j Z s Mt Q = m j= G j = jj=mn exactly j + times). Thus for jzj =, C mn Z s Mt C mn Z s Mt P P!! = + C (N;:::;N) mn Z s Mt Z s Mt P N P! mn = : 8

10 Here are a few cases that will be used in later examples: (:5) m = ; N = : m = ; N = ; m = ; N = ; m = ; N = ; m = ; N = ; m = ; N = ; m = ; N = ; H = P + P P = P + P : H = P (P + P + 6P ): H = P (P + 6P P + 5P + P ): H = P (P 7 + 8P 6 P + 8P 5 P + 56P P + 7P P ): H = P P + (P + P ) + 6P P : H = P P + 6P (P + P ) + 5P (P + P ) + 6(P P + P P ) + (P + P ) + 9P P : H = P P + P (P + P + P ) + (P P + P P + P P ) + (P + P + P ) + P P P : Of course, to make use of these equations, we must assign some ordering to the coset representers, Z s M t nfg, say ; : : : ; m, and set P j = P j, j = ; : : : ; m. The Theorem will be used in the following ways: Since P H satises the interpolatory condition for a mask, P H is a candidate as a mask for a new fundamental function with hopefully higher smoothness, while H would then be the mask for a dual function. The trick of placing the extra power of P in the last term in H means that H has the factorization P N T. With good initial choice of P = P, the power of P will add to the smoothness of the functions generated from the masks P H and H, provided that T can be bounded suitably. The next result addresses the bounds on T. Lemma.6. On jzj =, if P = P is non-negative, then the function H of Theorem. has the form H = P N T where T (!) C N+ mn + C(N;:::;N) mn (m ) (m )N ; with the zero set of T being a subset of the zero set of P. Therefore, (:7) N;m := max! T (!) C(N; m) m Nm+= p N(m ) N(m ) = where C(N; m) :95 if N = ; m and C(N; m) 5: if N ; m. 9

11 Proof. Since P + : : : + P m = and P is non-negative, we conclude that P. The conclusion ensures that T is the sum of non-negative terms hence, the zero set of T is a subset of P, since the term with highest power of P has only P as a factor. Note further that P H contains only \monomials" in the expansion P Z s Mt P with P N but perhaps with smaller coecients (even zero in some cases). Since P = Z s Mt nfg P, we may use instead the expansion of (P + ( P )) mn to obtain (:8) T mn j=n+ mn j=n+ C j mn P j N ( P ) mn j + C (N;:::;N) mn C j mn P j N ( P ) mn j + C (N;:::;N) mn P Z s Mt nfg P m (m )N ; Q where the last inequality comes from the fact that the maximum of Z s Mt nfg P subject to the constraints P and Z s Mt nfg P = const; (namely, the constant P,) occurs when all the P are equal. Since the derivative with respect to P of the rst term on the right hand side of (.8) is (after a regrouping) mn N+ h C k+ mn mn (k N)P k N ( P ) mn k C k mn (mn k)p k N ( P ) mn k i mn = ( N ) CmN k+ P k N ( P ) (mn k ) < ; N+ the right hand side is decreasing on P. Therefore, the right hand side of (.8) has its maximum when P =. That gives the desired bound. The bound (.7) follows from the strong form of Stirling's formula (x + ) x x e xp x < Indeed, applying this formula, we nd that C(N; m) 8 >< >: + p m e p (m ) + p N + p (N+) + p (m ) + p m e mp m p Nm p N(m ) + p Nm p x ; x : p Nm ; when N = and m ; p N pn m ; if N and m.

12 Another consideration is whether good properties of P are passed on to the newly dened P H and H. The next lemma will be useful in establishing the stability of the functions generated by masks P H and H Lemma.9. Suppose the M-renable functions and have Fourier transforms which are continuous and non-vanishing at. Suppose further that the zero set of the mask ^a contains the zero set of the mask ^a. If is pre-stable, then is pre-stable. Proof. The pre-stability of will follow from the fact that any zero for b is a zero for b. Suppose b (! ) =. Then for k large enough, with b M t k! b (! ) = k j= ^a M t j!! b M t k! 6= by the nonvanishing and continuity the origin and the fact that M t k!! as k!. Hence, M t j! is a zero of ^a for some j. But then the same M t j! is a zero of ^a as well, which implies! is a zero of b. Corollary.. If P = ^a is the non-negative mask of a continuous compactly supported fundamental M-renable function, then the M-renable functions generated by the masks P H and H in Theorem. are pre-stable. Proof. We have noted already that such a is stable. Clearly b is continuous since is compactly supported. Since P is non-negative, Lemma.6 implies that the zero sets of P, P H and H coincide. Hence, the result follows from Lemma.9. We have clearly established that Theorem. provides a family (indexed by N) of functions H so that P H and H preserve many of the good properties inherent in P ; positivity, interpolatory condition (for P H) and pre-stability. Further, we have the following result: Corollary.. Let P = ^a be the non-negative mask of a continuous compactly supported fundamental M-renable function. If the M-renable function corresponding to the mask P H is continuous, then it is fundamental. If the M-renable function corresponding to the mask H is in L (R s ), then it is stable and dual to. Further, since both P H and H are real, we have the following corollary: Corollary.. Let P = ^a be the non-negative mask of a continuous compactly supported fundamental M-renable function. Then the M-renable functions corresponding to the mask P H and to the mask H are symmetric to the origin. In the next section, we will discuss the question of gain in smoothness by this procedure. Since larger N gives higher powers of P in P H, this may already be a method of building smoothness and indeed, it is! But, a quick glance at the examples (.5) shows that with larger m or N, the complexity of H increases rapidly. Hence, it may be better to obtain the higher smoothness through iteration.

13 The Iteration Algorithm.. Given a mask a for an M-renable continuous function with a mask ^a which satises the interpolatory condition (.5). Fix an N and set P = ^a, k =. DO: Step. Set k = k +, P = P, and compute P, Z s M t nfg as in (.). Step. Form H according to Theorem. (as in (.5)). Step. Set P = P H and/or when k >, set H d = P H=^a. Step. Dene b N;k (!) := b d;n;k (!) := j= j= P (M t j!) and/or b d N;k (!) := H d (M t j!): j= H(M t j!); and/or Step 5. STOP if the smoothness of the desired function(s) ( N;k and/or d N;k and/or d;n;k ) is reached, ELSE repeat Steps {5. A word about the choices of output. The function N;k will be a new, smoother function with a non-negative mask which satises the interpolatory condition (.5). The function d is a candidate for a dual function to N;k N;k since the product of the mask of N;k (P in Step ) and the mask of d N;k (H in Step ) satises the interpolatory condition (.5) (we elaborate on this in Section ). For a similar reason, the functions d;n;k are candidates (with smoothness increasing in k) for a dual function to the original. The rst three steps in Algorithm. are easy to implement in symbolic software such as MAPLE. Step 5 can also be carried out with only the information provided by the new masks through the procedure described in the next section. Remark.. The bivariate fundamental renable functions constructed by [RiS] provide us many initial functions for the construction here. In general, one can use box spline as suggested by [RiS] to generate examples of the fundamental renable function with low regularity numerically by solving a system of equations, or one can always obtain examples by using tensor product of fundamental renable functions with lower regularity. Then apply the constructions in this section on those fundamental renable functions to build nonseparable fundamental renable functions with higher regularity.. Regularity We rst show that provided the initial function with mask ^a = P is suitably smooth, then the regularity of P H will increase with N or with each iteration in Algorithm.. A function belongs to C for n < < n +, provided that C n and (:) jd (x + t) D (x)j const jtj n ; for all jj = n and jtj

14 for some constant independent of x. The number is related to weighted L exponents dened as (:) sup := supf : Z R s ( + j!j) jb (!)j d! < g: The relation is given by the inequality sup sup. Therefore, an increase in the decay rate of the Fourier transform at innity will mean that the corresponding function will have increased smoothness. We say b has decay rate if jb (!)j C( + j!j) : Our object is to show that the constructions in the last section lead to smoother functions provided the original functions have sucient smoothness. We will show that the Fourier transforms of the constructed functions have increasing decay rate provided the Fourier transform of the initial function has sucient rate of decay. The analysis depends on the factorization of H as P N T, the estimates on T as given in Lemma.6, and on the characteristics of the dilation matrix M t. For simplicity, we require that M has a complete orthonormal set of eigenvectors. Then there is an equivalent norm on R s for which j min j jj!jj jjm t!jj j max j jj!jj; where j min j (respectively, j max j) is the minimum (respectively, maximum) modulus of the eigenvalues of M t. Hence, if B := B(R s ) is the closed unit ball in R s, then (:)! M tk+ Bn M t K B =) j min j K jj!jj j max j K+ : The matrix M will remain xed, but we must label the various elements of the construction in the last section to identify how they arise. We do this with a subscript N denoting the choice of N and a subscript k to denote the iteration step k. Thus, we use H N;k with a factorization P N;k N TN;k to obtain the function b N;k, while b d N;k is the function obtained from the mask P N;k = P N;k H N;k. We make use of the bound on T N;k found in Lemma.6. We follow the well-known analysis of [D] for the univariate case. We begin with the observation that P N;k () = P N;k () = implies that T N;k () =. This in turn implies that T N;k (!) + Cj!j, and consequently allows the estimate, (:) sup j!j j= T N;k (M t j!) sup j!j j= Therefore, when! M tk+ Bn M t K B, we have that exp(jcm t j!j) C: T N;k (M t j K!) = T N;k (M t j!) T N;k (M t j K!) C K N;m j= j= j= = Cj min j K log N;m = log j min j C( + j!j) log N;m = log j min j ;

15 by (.) and (.). For brevity in the ensuing expressions, we set (:5) N := log N;m )= log(j min j : If we assume that b N;k has rate of decay N;k (with N; :=, the decay rate of the original ), then since b N;k (!) = N+TN;k P N;k M t j! M t j! j= = bn;k (!) N+ N b d N;k (!) = bn;k (!) T N;k M t j! ; j= T N;k M t j! j= and, similarly we nd that (:6) b N;k (!) C( + j!j) (N+) N;k + N ; b d N;k (!) C( + j!j) N N;k + N : and It is easy to show inductively from the estimate (.6) that the following formulas for decay rates hold for b N;k and b d N;k respectively: (:7) N;k = (N + ) k N =N + N =N d N;k = N(N + ) k N =N : and For the decay rate of b d;n;k, we take b d;n; = b d N; and observe via induction that b d;n;k (!) = b d;n;k (!)b d N;k (!) = Hence, from (.7), we obtain the decay rate of b d;n;k as k j= b d N;j(!): d;n;k = (N + ) k N =N for k >. It follows from (.7) that N =N is bounded by a constant dependent only on the matrix M. Hence, we have established Theorem.8. Let be an M-renable function with non-negative mask P = ^a. If the decay rate of b is suciently large, then the decay rate of b N;k, b d N;k and b d;n;k can be made arbitrarily large by increasing N or k. In particular, a decay rate of > log N;m =N log jmin j ;

16 will suce, where N;m can be bounded as in (.7). Remark. We note that with k = and N increasing, both the diameter of the support of the mask and the rate of decay of the Fourier transform are increasing linearly in N. For a xed N and increasing k, both the diameter of the support and the rate of decay of the Fourier transform are increasing geometrically with k. Therefore, to increase regularity eciently while maintaining control on the size of the support, it is better to increase N. On the other hand iteration with low values of N is easier to implement by (e.g.) MAPLE, because of the complexity of H for large N. Example..9 Let be the the simplest fundamental function constructed in [RiS] which is obtained by convolving the mask of a three direction box spline of equal multiplicity in each direction with a distribution (see x5). The mask ^a of is non-negative. By choosing a suciently large N, or a suciently large iteration k, we get fundamental renable functions with high order box spline factor. Further, the regularity of the renable function grows linearly with N (geometrically with k). Hence, the constructions given in x together with the constructions given in [RiS] provide several methods for the construction of fundamental renable functions with high order box spline factor. The regularity of the renable function increases as the order of the box spline factor does. The crude bounds on T N served their purpose to establish Theorem.8, but they may mislead one to believe that the starting point needs to be unrealistically high. However, in practice things are better since the smoothness can be assessed much more accurately. Here we summarize the approach to be used in our examples at the end. By now it has received treatment at many levels, by several people [CGV], [CD], [E], [H], [J], [RiS], [RS] and [V]. The criterion to be used to bound sup from below is contained in the following statement: For an integer r, let V r := v `(Z s ) : Z s p()v() = ; 8 p r ; where r denotes the polynomials of total degree r. Assume M is a dilation matrix with a complete set of orthonormal eigenvectors. If the mask ^a for the stable fundamental M- renable function satises (:) ^a ; ^a() = ; and D ^a M t = for jj r and Z s M t nfg; then for a suitable choice of with supp a, V r is invariant under the matrix h i IH := a(m ) : Let IHjV r ; be the spectral radius of IHj V r. Then the exponent sup satises log IHjV r (:) sup log j max j : 5

17 The proof of this statement can be obtained by modifying the proof in [RiS] or from [CGV], [J] and [RS]. The invariant set was dened in [LLS] and [LLS] and [HJ]. An explicit formula for the invariant set was given in [HJ, Theorem.]: (:) := j= M j supp a:. Dual Functions and Biorthogonal Wavelets In this section we combine the construction of biorthogonal wavelets in our previous paper [RiS] with the constructions of fundamental renable functions in section to construct dual functions of arbitrary smoothness. Suppose that is a continuous, compactly supported, M-renable function on R s, and the set of functions f( )g Z s are linearly independent. We want to construct a stable compactly supported M-renable function d in L (R s ) so that the set of functions f d ( )g Z s forms a Riesz basis of S( d ) and (:) h; d ( )i = ; Z s : The latter equations hold (see e.g. [RS]) if and only if the Fourier transform for and d respectively satisfy (:) Z s b (! + ) c d (! + ) = ;! TT s : Therefore, when (.) holds, the masks ^a and ^a d must satisfy the following necessary condition: (:) Z s Mt ^a( + M t )^a d ( + M t ) = : This can be used to advantage when compared to the interpolatory condition (.8), because if the mask ^h for a fundamental M-renable function factors into a product of two masks ^h = ^a^a d, then ^a and ^a d can be considered as candidates for the masks of an M-renable function and its dual function. In our case, we will deal only with real masks and therefore can dispense with the conjugation. It was shown in [S] that and d are a dual pair if and only if and d are stable and their masks satisfy (.). Therefore, if masks ^a and ^a d satisfy (.), then one needs show that the renement equations of both masks ^a and ^a d have compactly supported solutions (, d, respectively) in L (R s ), and that the functions and d are stable. 6

18 These verications are much easier through the constructions and results of x. We begin with a continuous, compactly supported fundamental M-renable function with mask ^a = P. An application of the constructions provide new mask P H which also satises the interpolatory condition (.5), and therefore, we have (.) with ^a := P, our original mask, and ^a d := H. The existence and stability for are given while the existence in L (R s ) of the solution d is assured if is smooth enough, and the stability of d is assured if the mask ^a in non-negative (by Corollary.). Thus, if we begin with a continuous, compactly supported fundamental M-renable function with mask ^a, then the existence of L (R s ) solutions and the stability of d are essentially automatic (one may have to iterate a few times or choose a larger N to ensure that d is in L (R s ), but as Theorem.8 shows, we may then achieve any regularity we wish for the dual). Here is a summary of two constructions for dual functions and biorthogonal wavelets using the results of x. The two constructions dier only in how the ^a and ^a d are chosen in Step of Algorithm.5 to commence the actual construction. The rst assumes that we want to nd a dual function for the given fundamental function (which may have been arrived at after an iterative procedure to gain smoothness) with the smoothness of the dual function to be determined solely by the choice of N, namely, the case k = and N xed. In this case, we take ^a = P and ^a d = H. In the second construction, we want to nd a dual function for the given fundamental function with the smoothness of the dual enhanced by repeated iterations. In this case, after a suitable number of iterations to obtain the smoothness of the candidate for the dual function, we take ^a and set the dual mask ^a d = P H d =^a, since ^a is a factor of the iterations. Let and d be a dual pair. Then, it was shown in [BDR] (also see [JS]) that the sequence of subspaces dened by and the sequence of subspaces dened by S k () := ff(m k ) : f S()g; k Z; S k ( d ) := ff(m k ) : f S( d )g; k Z; form a `dual' pair of multiresolutions of L (R s ). Here we recall that a sequence S k () forms a multiresolution, when the following conditions are satised: (i) S k () S k+ (); (ii) [ kz S k () = L (R s ) and \ kz S k () = fg; (iii) and its shifts form a Riesz basis of S(). Once a dual pair of the multiresolutions are available, the construction of the biorthogonal wavelets from the pair of multiresolutions is equivalent to a problem of the matrix extension. The algorithm presented here for biorthogonal wavelet construction makes use of an algorithm for design of matrix pairs provided in [RiS]: Algorithm for Matrix Pairs.. For given m vectors of Laurent polynomials P = [P ; P ; : : : ; P m ] and Q with P (z)q T (z) = for all z fc nfgg s, DO: Step. Extend the row P to an m m matrix K in P, where P is the set of all nite order matrices with entries being Laurent polynomials in (C nfg) s. 7

19 Step. Alter the last m rows of K to be orthogonal to Q: Let K j, j = ; : : : ; m, be the last m rows of the matrix K and dene G j := K j (K j Q T )P; j = ; : : : m: Step. Dene := P T ; G T ; : : : ; G T m T. Then P and is nonsingular on (C nfg) s. Step. Find := F T ; F T ; : : : ; F T m and set T := Q T ; F T ; : : : ; F T m Then P and (z) T (z) = I m ; z (C nfg) s : Step is denitely trivial here, since is fundamental, for then substituting () = () into the renement equation (.) shows that a(m ) = ()=jdet(m)j. Hence, Z s a(m)z M = =jdet(m)j: Thus, if = is the rst element in our ordering of Z s M t, then the matrix can be extended simply by placing =j det(m)j's in the diagonal. Once the step one is ready, the other steps can be implemented easily. Remark. If each entry of the polynomial vectors P and Q T are real on jzj =, and if P is a real constant (as in the case just described), then each entry of is real on jzj = and each entry of is real on jzj = except that the rows ; : : : ; m may be multiplied by j det j which is a monomial, that is, a real constant times an exponential exp( i!) on jzj =. Therefore, if we start with a fundamental M-renable function and a dual function both symmetric to the origin, then the wavelets and dual wavelets constructed in the next algorithm will also be symmetric. Algorithm for Biorthogonal Wavelets from Interpolatory Subdivision.5. For a given continuous, compactly supported fundamental M-renable function with mask ^a. DO: Step. Apply Algorithm. to a desired level k to obtain H or H d as required. We take ^a and ^a d = H for k = or ^a d = H d for k >. Let b d (!) := ^a d M t j! : Step. Dene polynomials corresponding to the masks restricted to the cosets of Z s =MZ s by A ; (z) := Z s a( + M)z M ; Z s =MZ s ; j= and A d ;(z) := Z s a d ( + M)z M ; Z s =MZ s : Step. Apply Algorithm. to P (z) := A (z) := (A ; (z)) Z s =MZ s, and Q(z) := Ad := (A d ;(z)) Z s =MZs to obtain matrices and. 8

20 Step. Label the rows and columns of and by Z s =MZ s with the rst row labeled by and the remaining rows in some xed order (e.g. by the lexicographic order). For the -th rows A = (A ) Z s =MZ s and Ad := (A d ) Z s =MZs respectively of and, dene and Step 5. Dene two sets of functions A (!) := exp( i!)a (exp( im t!)); Z s =MZ s A d (!) := exp( i!)a d (exp( im t!)): Z s =MZ s (:6) b (M t!) := A (!)b (!); and bd (M t!) := A d (!)b d (!); Z s =MZ s : Then (i) = and d = d. (ii) The functions, Z s =MZ s nfg, are called the wavelets for the renable function. (iii) The functions d, Z s =MZ s nfg, are called the dual wavelets, that is, the wavelets of the dual function d. Finally, as was noted in [RiS] and in a more general setting for dilation matrices M as discussed here in [RS], the systems (:7) and n (M k ) : Z s Mnfg; k Z; Z so (:8) n d (M k ) : Z s Mnfg; k Z; Z so ; are the biorthogonal wavelet systems from the above construction, i.e. h (M k ); d (M k )i = ; k ;k ; ; Z s ; k ; k Z: Further, they form a biorthogonal Riesz basis of L (R s ) if they are Bessel systems (see [RS]. Let := : Z s M nfg and d := d : Z s Mnfg. It was shown in [RS],[RS] that the dilations and shifts of functions in form a Bessel system if the functions (:9) R E := max jb ( + )j; Z s and (:) R D := jb (M tk )j ;kzz 9

21 are in L. This will be true provided the functions have certain decay rates at innity and a certain order of the zeros at the origin. Remark. Let be an arbitrary given compactly supported fundamental M-renable function with mask ^a non-negative. By the methods given in this paper, we are able to construct a compactly supported dual renable function with any desired regularity and the corresponding biorthogonal wavelet systems. Further, since we can construct the renable fundamental function to have arbitrary regularity in any number of variables by Remark., we are able to construct multivariate biorthogonal wavelets with arbitrary regularity by the methods given here. Remark. In the bivariate case, for an arbitrary given three direction mesh box spline one can construct dual renable function with arbitrary regularity by Example.. Therefore, biorthogonal wavelet systems with arbitrary regularity such that one of the bases is formed by piecewise polynomials can be constructed. Remark. Let be a given M-renable function with mask ^a that is not fundamental. Assume that and its shifts are linearly independent. One may nd a renable dual function d with mask ^a d that has a lower regularity numerically by solving a system of equations. If the mask ^a^a d is non-negative, then we can use the constructions of x to nd a mask P with ^a as a factor and for which the compactly supported M-renable function corresponding to P=^a has high regularity. Hence, smoother biorthogonal wavelets can be obtained. When is a linearly independent box spline in R s, we may get a compactly supported dual renable function with any desired regularity. Therefore, the problem of the construction of a compactly supported dual renable function with any desired regularity for a given linearly independent renable function is reduced to the problem of either nding numerically a distributional dual function or nding numerically a dual function with a lower regularity. 5. Examples In this section we apply the techniques developed in the paper to illustrate the theory in concrete cases. We have chosen bivariate examples with dierent dilation matrices. The ndings are summarized in Table. Bivariate example with dilation matrix M = I. For the rst example we use the simplest of the fundamental functions constructed in [RiS]. It was found by multiplying the mask of a three direction box spline of equal multiplicity in each direction by a suitable factor to give a mask satisfying (.5) with smallest support and with the same symmetries as the box spline. The mask for the fundamental function, call

22 (,) (,) (,) (,) 6 Figure. The interpolatory renable function ;; derived in Algorithm (.5). and the wavelets from it as it ;;, is reproduced below: (5:) a ;; = The mask ^a ;; is cos! cos! ! +! cos 5 cos(! ) cos(! ) cos(! +! ) =; 7 5 :

23 which is clearly non-negative. The support of a is in [ ; ] and it has maximum value at the origin. In [RiS], the corresponding function was shown to have smoothness C " for any " > (,) 6.5 (,) (,) (,) and the wavelets from it as derived in Algo- Figure. The dual function d ;; rithm (.5). Hence Algorithm. can be applied to ^a with N = to obtain a mask H supported on [ ; ]. The mask H satises (.) for r =, hence, V is invariant under the matrix IH for the mask H. Applying the criterion at the end of x, we nd that the corresponding function d ;; is in smoothness class C, where : Therefore, d ;; is stable and dual to ;; by Corollary.. Since we have a reasonable dual function to ;;, we can apply Algorithm.5 to construct biorthogonal wavelets for the pair ;; and d ;;. Figure shows the renable function ;; and the corresponding wavelets, while Figure shows the dual function d ;; and the dual wavelets.

24 ten d ten Figure. The function ten and the dual function d ten found by Algorithm. applied to a simple tensor product. Nonseparable bivariate example from univariate hat functions with dilation matrix M = I. As mentioned earlier, the initial fundamental function may be the tensor product of simple univariate fundamental functions. The application of the constructions in x will lead to nonseparable fundamental functions. Here, we take the tensor product, ten, of the univariate hat function which has the mask a ten = =6 =8 =6 =8 = =8 =6 =8 =6 Clearly, the mask ^a ten will be non-negative since it is the product of the univariate mask in each variable. Applying Algorithm (.) with N =, we nd a dual function in L (R ) (but not continuous) and a new nonseparable fundamental function ten which belongs to C ", for any " >. Iterating once more (k = ) we nd a dual, d ten, for ten that has smoothness. In Table, we have labeled this next iteration of ten as the rst iteration of ten to give its dual function and to better show the comparison with ;;. Figure shows the functions ten and d ten. The supports of the functions derived using ;; are smaller. Examples with dilation matrix M = : Here we take a quick look at an example which has already been well studied in the literature (see [DD] and [CD]). For the dilation matrix M = and the mask (5:) a = =8 =8 = =85 ; =8 5 :

25 we have that ^a = + cos(! ) + cos(! ). Therefore, the methods of x apply to build smoother fundamental functions and for those smoother functions we can try to nd dual functions as well. Let M be the fundamental function corresponding to the mask (5.). Since ^a is non-negative, we may apply the smoothness criterion given at the end of x. For the matrix M, j min j = j max j = p and therefore, N;k is in C ", where := log IHjV r = log() and where Vr is to be determined for each choice of N and k. Table summarizes our calculations for N = ; ; ; with k = and N = ; k =. Consulting Table we see quite clearly the increasing smoothness with N. An iterate, (k = for N = ), also shows the increasing smoothness with the iteration. For the dual functions however, one must progress further to obtain smoothness. With N =, we do not get an L (R ) function for d ;, but with increasing N we do get existence in L (R ) for the dual functions d ;, d ;, and a continuous dual function in d ;. Function N; k r in V r N;k C " with d;n;k ;; N = ; k = r = = : : : : N = ; k = r = 7 = :659 : : : d;; C :858 ten N = ; k = r = = : : : : N = ; k = r = = : : : : d;; L (R s ) ten N = ; k = r = = : : : : N = ; k = r = 7 = :68 : : : d;; C :8578 M N = ; k = r = = :65 : : : N = ; k = r = = :58 : : : d;; 6 L (R ) N = ; k = r = 5 = :998 : : : d;; L (R ) N = ; k = r = 7 = :57 : : : d;; L (R ) N = ; k = r = 9 = :59 : : : d;; C :6 N = ; k = r = 7 = :687 : : : d;; L (R ) Table. Results of applying the constructions in x to the Examples of this section References [BR] A. Ben-Artzi and A. Ron, On the integer translates of a compactly supported function: dual bases and linear projectors, SIAM J. Math. Anal. (99),

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