Estimating the support of a scaling vector

Transcription

1 Estimating the support of a scaling vector Wasin So Jianzhong Wang October, 00 Abstract An estimate is given for the support of each component function of a compactly supported scaling vector satisfying a matrix refinement equation with finite number of terms The estimate is based on the highest lowest degree of each polynomial in the corresponding matrix symbol Only basic techniques from matrix theory are involved in the derivation Introduction In this paper we are interested in measurable functions from the reals R to the complex C; two functions are equal if they are identical almost everywhere Let r be a positive integer F [f f r ] T be a complex vector-valued function on R, where T denotes the transpose of a matrix A point t R is called a support point of F if the measure of the intersection x : F (x) 0 (t ɛ, t + ɛ) is not zero for any ɛ > 0 The support of F, denoted by supp(f ), is defined as the convex hull of the set of support points of F Hence equal functions have same supports; the support of a nonzero function is always a close interval with positve length Note that, in the literature of wavelet theory with r, the support of a scaling function is always taken to be a closed interval because of the result in [5] (also see [, pg 5]) Recent interests in multiwavelets lead to the study of scaling vector Φ [φ φ r ] T which is a vector-valued function satisfying a matrix refinement Department of Mathematical Information Sciences, Sam Houston State University, Huntsville, TX 7734 Both authors are partially supported by the National Science Foundation grant DMS-95038

2 equation (MRE) with finite number of terms Φ(x) N C k Φ(x k) () k0 where C k s are r r matrices In applications, shortly supported multiwavelets are always desired Support of multiwavelets can be obtained easily from the support of the corresponding scaling vectors Hence it is useful to estimate the support of scaling vectors from the defining MRE However the determination of the support of a scaling vector is not straightforward In [3], Heil Colella observed that supp(φ) [0, N] if Φ is compactly supported But this estimate is too crude as the following example, due to Geronimo, Hardin, Massopust [], shows Example Let Φ [φ φ ] T be a scaling vector satisfying the MRE () with matrix coefficients: C 0 [ ] 6 0, C [ ] , 0 C [ ], C 3 0 [ Note that supp(φ ) [0, ], supp(φ ) [0, ] so supp(φ) [0, ] [0, 3] An explanation is the existence of nilpotent matrices Note that C 3 in the above example is nilpotent In [6], Massopust, Ruch Van Fleet showed that supp(φ) [0, N r ] if C N is nilpotent, supp(φ) [ r, N] if C 0 is nilpotent However such improved estimates are still not good enough to explain the above example In this paper, we give an estimate for each componentwise support supp(φ i ) hence the global support supp(φ) Sufficient conditions are given for these estimates to be acheived The rest of the paper is organized as follows Our main results are stated in with an illustration Proofs are given in 3 4 is devoted to the study of the global support of a scaling vector ]

3 Componentwise support of a scaling vector For the rest of the paper, let Φ [φ φ r ] T be a compactly supported scaling vector satisfying the MRE () In this section we are interested in estimating the support supp(φ i ) for i r To this end, we define the associated matrix symbol by P (z) N C k z k, k0 which is a r r matrix with polynomial entries Let h(i, j) (resp l(i, j)) be the highest (resp lowest) degree of the (i, j)-entry of P (z) We adopt the convention that the highest (resp lowest) degree of the zero polynomial is (resp ) I k denotes the k k identity matrix e k denotes the k-th column of the identity matrix whose dimension is determined from the context For positive integers a, b, E ab denotes the matrix e a e T b Let J be the set of all integer sequences J (j,, j r ) where j,, j r r For each J (j,, j r ) J, define E J I r E j E rjr, h J [h(, j ) h(r, j r )] T l J [l(, j ) l(r, j r )] T Note that E J is always invertible (see Lemma 3) Theorem For i r, the support of φ i is a finite closed interval [L i, R i ] where R i max e T i EJ h J : J J L i min e T i EJ l J : J J In Theorem, both maximization minimization are with respect to the set J which has r r elements In order to reduce the complexity we introduce the following concepts For each J (j,, j r ) J i r, define a new integer sequence γ (γ 0, γ,, γ t ) satisfying the following conditions: t r, γ 0 i, 3

4 3 γ k j γ(k ) for k,, t, 4 γ 0,, γ t are distinct, 5 γ t γ s for some s t The existence of γ, s t is clear they are uniquely determined by the sequence J (j,, j r ) the integer i As examples, take r 4 If J (3,, 4, 3) i, then γ (, 3, 4, 3) t 3, s If J (3,, 4, 3) i, then γ (, ), t, s For fixed i, let Γ i be the collection of all such γ s Let s t be the numbers corresponding to a given γ Γ i Define a t t matrix by A γ I t E E 3 E (t )t E ts Note that A γ E J for J (, 3,, t, s) so A γ is invertible (see Lemma 3) Define h γ [h(γ 0, γ ) h(γ, γ ) h(γ t, γ t )] T l γ [l(γ 0, γ ) l(γ, γ ) l(γ t, γ t )] T Theorem For i r, the support of φ i is a finite closed interval [L i, R i ] where R i max e T Aγ h γ : γ Γ i L i min e T A γ l γ : γ Γ i In Theorem, both maximization minimization are with respect to the set Γ i The number of elements in Γ i is ( ) r r k0 (k + )! k which can be proved to be equal to the integral part of the positive number (r )!(r )e +, where e is the base of natural logarithm Hence the complexity of the optimization is reduced to (r )!(r )e + from r r in Theorem Using the classical adjoint formula for matrix inverse [4, pg 0], it is not hard to see that the first row of A γ is [ e T A t γ s ( t s ) t s ( t s ) ] t Therefore Theorem can be restated explicitly as follows 4

5 Theorem 3 For i r, the support of φ i is a finite closed interval [L i, R i ] where s R i max γ Γ i L i min γ Γ i k s k k h(γ (k ), γ k ) + k l(γ (k ), γ k ) + t t s t t s ks ks k h(γ (k ), γ k ), k l(γ (k ), γ k ) A family f i of functions on R is locally linearly independent if i c if i (x) 0 on any nontrivial interval (a, b) then c i 0 for those supp(f i ) (a, b) Φ [φ,, φ r ] T is called a locally linearly independent scaling vector if the family φ j (x k) : j r, k Z is locally linearly independent In this case, the family φ j (x k) : j r, k Z is also locally linearly independent This fact will be used in Lemma 34 Theorem 4 If Φ is a locally linearly independent scaling vector then all inequalities become equalities in Theorems,, 3 Choosing r in Theorem 3, it yields R max h(, ), R max h(, ), L min l(, ), L min l(, ), 3 h(, ) + 3 h(, ), h(, ) + h(, ), 3 h(, ) + 3 h(, ), h(, ) + h(, ), 3 l(, ) + 3 l(, ), l(, ) + l(, ), 3 l(, ) + 3 l(, ), l(, ) + l(, ) As an illustration, we use these formulas to estimate the support of the scaling vector mentioned in the example [ of ] The highest [ ] lowest degree matrices are respectively h l Hence L R 0 L R Furthermore, Φ is known to be locally linearly independent [] so we have supp(φ ) [L, R ] [0, ] supp(φ ) [L, R ] [0, ] 5

6 3 Proofs We need two lemmas for the proof of Theorem Lemma 3 Let f i be a family of functions on R Then ( ) supp c i f i conv ( i supp(f i )) where conv denotes the convex hull of a set i Lemma 3 For J J, the matrix E J is invertible its inverse has nonnegative entries Proof Let E E jr + + E rjr Note that E where is the maximum row sum norm Then E J I r E is invertible actually E J k0 k+ Ek which has nonnegative entries because E has nonnegative entries We are ready to prove Theorem Proof of Theorem For each i r, using the MRE (), we have φ i (x) N r k0 j r N j k0 r h(i,j) j kl(i,j) where C k (i, j) is the (i, j)-entry of the matrix C k Since φ i has compact support, we let supp(φ i ) [L i, R i ] By Lemma 3, we have [L i, R i ] conv r jsupp h(i,j) kl(i,j) ( [ conv r j (L j + l(i, j)), ]) (R j + h(i, j)) 6

7 Hence we have R i max R j + h(i, j) : j r, L i min L j + l(i, j) : j r For each i r, there exist integers j,, j r r such that In matrix form, E J R R r (I R i R ji + h(i, j i ) R r E tjt ) t R r h(, j ) h(r, j r ) h J where J (j,, j r ) By Lemma 3, EJ is nonnegative matrix so Hence R R r E J h J R i e T i EJ h J max J J Similarly, the lower bound for L i is obtained et i E J h J Lemma 33 Let p be a permutation on,, r P be the r r matrix associated with p Then P P T, e T k P et p (k), P T E ab P E p (a)p (b), [v v v r ]P [v p() v p() v p(r) ] Finally we give the proof of Theorem Proof of Theorem Given J J i r, let γ, s, t be the corresponding sequence numbers defined in It suffices to prove that e T i EJ h J e T A γ h γ Take a permutation p on,, n such that p(k) γ (k ) for k,, t Such permutation exists because the integers γ 0,, γ t are distinct Using Lemma 33, we have P T e γ(k ) e p (γ (k ) ) e k for 7

8 k,, t It follows that e T i P et p (i) et because p() γ 0 i, [ ] P T hγ h J, Finally, P T E J P P T (I r I r P T I r ) r E kjk P k k E γ(k ) γ k + k γ E p (γ k )p (γ k ) k k γ I r E k (k+) k k>t [ ] Aγ 0 E kjk E kjk P e T i EJ h J (e T i P ) (P T EJ P ) (P T h J ) (e T i P ) (P T E J P ) (P T h J ) [ ] [ ] e T Aγ 0 hγ e T A γ h γ E p (k)p (j k ) Lemma 34 Let f, f n be a family of locally linearly independent functions on R such that supp(f i ) [a i, b i ] where a i < b i Then ( n ) supp c i f i [a, b] i where a min a i : c i 0 b max b i : c i 0 Proof Let a l min a i : c i 0 b h max b i : c i 0 By Lemma 3, n i c i f i is compactly supported ( n ) supp c i f i [a, b] [a l, b h ] conv ( i supp(f i )) i 8

9 It remains to show that a a l b b h Assume the contrary that b < b h Then n i c i f i (x) 0 on (b h ɛ, b h ) for 0 < ɛ < min bi a i i Note that [a h, b h ] (b h ɛ, b h ) By the local linear independence of f i, c h 0 which is impossible by the definition of b h The argument for a a l is similar Proof of Theorem 4 It suffices to give the proof involving Theorem For each i r, using the MRE (), we have φ i (x) N r k0 j r N j k0 r h(i,j) j kl(i,j) where C k (i, j) is the (i, j)-entry of the matrix C k Since φ i has compact support, we let supp(φ i ) [L i, R i ] By Lemma 34, we have [L i, R i ] conv r jsupp h(i,j) kl(i,j) ( [ conv r j (L j + l(i, j)), ]) (R j + h(i, j)) The rest of the proof is exactly the same as the proof of Theorem with the modification that all inequalities are changed to equalities 4 Global support of a scaling vector In this section we are interested in the global support supp(φ) of Φ satisfying the MRE () From the last section, we know that supp(φ i ) [L i, R i ] for i r Hence supp(φ) [L, R] where R maxr i : i r L minl i : i r Theorem 3 gives the estimates as R max i r max γ Γ i s k k h(γ k, γ k ) + 9 t t s ks k h(γ k, γ k ),

10 L min i r min γ Γ i s k k l(γ k, γ k ) + t t s ks k l(γ k, γ k ) Theorem 4 (i) If C N is a nilpotent matrix of index m ie (C N ) m 0, then R N m (ii) If C 0 is a nilpotent matrix of index m ie (C 0 ) m 0, then L m Proof Using Lemma 3, it is not hard to see that supp(φ) supp(aφ) for any invertible matrix A (i) Without loss of generality, we can assume that C N is reduced to the Jordan form J m (0) where J m (0) is a lower triangular Jordan block with largest size Hence the highest degree matrix satisfies h (N )One(r) + C N where One(r) is a r r matrix with all entries equal to Now it is not hard to see that the maximum is attained at i m γ (m, m,,, m) Actually, the maximum is equal to m m m k k h(γ k, γ k ) N m (ii) Without loss of generality, we can C 0 is reduced to the Jordan form J m (0) where J m (0) is a lower triangular Jordan block with largest size Hence the lowest degree matrix satisfies l One(r) C 0 Now it is not hard to see that the minimum is attained at i m γ (m, m,,, m) Actually, the minimum is equal to m m m k k l(γ k, γ k ) m Setting m r, we obtain the result of Massopust, Ruch Van Fleet mentioned in the introduction Corollary 4 (i) If C N is nilpotent, then supp(φ) [0, N r ] (ii) If C 0 is nilpotent, then supp(φ) [ r, N] 0

11 References [] I Daubechies, Ten Lectures on Wavelets, SIAM Press, Philadelphia, 99 [] JS Geronimo, DP Hardin, PR Massopust, Fractal functions wavelet expansions based on several scaling functions, J Approximation Theory, 78 (994) [3] C Heil D Colella, Matrix refinement equations: existence uniqueness, Preprint, 994 [4] R Horn C Johnson, Matrix Analysis, Cambridge, New York, 989 [5] PG Lemarie G Malgouyres, Support des fonctions de base daus une analyse multiresolution Comptes Rendus del l Acad Sci Paris, 33 (99) [6] P Massopust, D Ruch, P Van Fleet, On the support properties of scaling vectors, Preprint, 994