On the Construction of Wavelets. on a Bounded Interval. Gerlind Plonka, Kathi Selig and Manfred Tasche. Universitat Rostock. Germany.

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1 1 On the Construction of Wavelets on a Bounded Interval Gerlind Plonka, Kathi Selig and Manfred Tasche Fachbereich Mathematik Universitat Rostock D{18051 Rostock Germany Abstract This paper presents a general approach to a multiresolution analysis and wavelet spaces on the interval [;1 1]. Our method is based on the Chebyshev transform, corresponding shifts and the discrete cosine transform (DCT). For the wavelet analysis of given functions, ecient decomposition and reconstruction algorithms are proposed using fast DCT{algorithms. As examples for scaling functions and wavelets, polynomials and transformed splines are considered. 1 Introduction Recently, several constructions of wavelets on a bounded interval have been presented. Most of these approaches are based on the theory of cardinal wavelets. The simplest construction consists in the trivial extension of functions f :[0 1]! R by setting f(x) :=0 for x 2 R n [0 1]. These functions can be analyzed by means of cardinal wavelets. But in general, this extension produces discontinuities at x = 0 as well as x = 1, which are reected by large wavelet coecients for high levels near the endpoints 0 and 1, even if f is smooth on [0 1]. Thus the regularity off is not characterized by the decay ofwavelet coecients. Another simple solution, often adapted in image analysis, consists in the even 2{periodic extension ~ f of f : [0 1]! R. If f 2 C[0 1], then ~ f 2 C(R). But in general, if f 2 C 1 [0 1], then the derivative of ~ f has discontinuities at the integers. The smoothness

2 2 of f is again not characterized by the decay ofwavelet coecients. In [8], Meyer has derived orthonormal wavelets on [0 1] by restricting Daubechies' scaling functions and wavelets to [0 1] and orthonormalizing their restrictions by the Gram{ Schmidt procedure. This idea led to numerical instabilities such that further investigations of wavelets on a bounded interval were necessary (see [4]). We are interested in wavelet methods on a bounded interval which can exactly analyze the boundary behaviour of given functions. Up to now, three methods are known to solve this problem. The often used rst method is based on special boundary and interior scaling functions as well as wavelets (see [3, 4, 13]) such that numerical problems at the boundaries can be reduced. Then the bases of sample and wavelet spaces do not consist in shifts of single functions. The second method (see [9]) works with two generalized dilation operations, since the classical dilation is not applicable for functions on a bounded interval. A third wavelet construction on the interval I := [;1 1], rst proposed in [6], is based on Chebyshev polynomials. Both scaling functions and wavelets are polynomials which satisfy certain interpolation properties. As shown in [16], this polynomial wavelet approach can be considered as generalized version of the well{known wavelet concept, which is based on shift{invariant subspaces of the weighted Hilbert space L 2 w(i) with respect to the Chebyshev shifts (see [2]), where w denotes the Chebyshev weight. The objective of this paper is a new general approach tomultiresolution of L 2 w(i) and to wavelets on the interval I, based on the ideas in [16]. As known, the Fourier transform and shift{invariant subspaces of L 2 (R) are essential tools for the construction of cardinal multiresolution and wavelets (see [5]). Analogously, the nite Fourier transform and shift{ invariant subspaces of L 2 2 lead to a unied approach to periodic wavelets (see [7, 11]). This concept can be transferred to the Hilbert space L of even periodic functions using the shift operator S a F := 1 (F ( + a)+f (;a)) (a 2 R) 2 for F 2 L 2. The isomorphism between 2 0 L2 and 2 0 L2 w (I) can be exploited in order to construct new sample and wavelet spaces in L 2 w(i). Using fast algorithms of discrete cosine transforms (DCT), ecient frequency based algorithms for decomposition and reconstruction are proposed. As special scaling functions and wavelets, we consider algebraic polynomials and transformed splines. It is remarkable that our decomposition algorithm for polynomial wavelets needs less multiplications up to a certain level than the fast decomposition algorithm for cubic spline wavelets on [0 1] proposed in [13]. The outline of our paper is as follows. In Section 2 we briey introduce the Chebyshev transform, related shifts and the DCT. In Section 3 we analyze shift{invariant subspaces of L 2 w(i). The scalar product of functions from shift{invariant subspaces can be simplied to a nite sum by means of the so{called bracket product. In Section 4 we consider a nonstationary multiresolution of L 2 w (I) consisting of shift{invariant subspaces V j (j 2 0 ) generated by shifts of scaling functions ' j. The required conditions for the multiresolution of L 2 w(i) and their consequences for the scaling functions ' j are analyzed in detail. In Section 5 we introduce the wavelet space W j (j 2 0 ) as the orthogonal complement of V j in V j+1. Then W j is a shift{invariant subspace generated by shifts of the wavelet j. Using the two{scale symbol of ' j and the bracket product of ' j and ' j+1,thewavelet j is characterized in Theorem 5.3. Section 6 provides fast, numerically stable decomposition

3 3 and reconstruction algorithms based on fast DCT{algorithms. In Section 7 we present polynomial wavelets on I (see [6, 16]). Finally in Section 8, we adapt the theory of periodic splines to the interval I with respect to the Chebyshev nodes. ote that the transformed spline wavelets are supported on small subintervals of I. The examples show that periodic multiresolutions of L 2 2 with even scaling functions ' j can be transformed into a multiresolution of L 2 w (I). 2 Chebyshev Transform and Shifts In this section, we introduce the Chebyshev transform and corresponding shifts and we examine their relations to the even shifts of periodic even functions. For more details on Chebyshev shifts we refer to [2, 16]. Throughout this paper, we consider the interval I := [;1 1] and the Chebyshev weight w(x) :=(1; x 2 ) ;1=2 for x 2 (;1 1). Let L 2 w(i) be the weighted Hilbert space of all measurable functions f : I! R with the property Z I w(y) f(y) 2 dy < 1 : For f g 2 L 2 w (I), the corresponding inner product and norm are dened by hf gi := 2 Z I w(y) f(y) g(y) dy kfk := hf fi 1=2 : Let l 2 denote the Hilbert space of all real, square summable sequences a = (a n ) 1, b =(b n ) 1 with the weighted inner product and norm given by (a b) l 2 := 1 2 a 0b n=1 a n b n kak l 2 := (a a) 1=2 l 2 : Let C(I) be the set of all continuous functions f : I! R. By n (n 2 0 )we denote the set of all real polynomials of degree at most n restricted on I. Asknown, the Chebyshev polynomials T n := cos (n arccos) 2 n (n 2 0 ) form a complete orthogonal system in L 2 w (I). ote that arccos : I! [0 ] is the inverse function of cos restricted on [0 ]. For m n 2 0 wehave 8 < 2 m = n =0 ht m T n i = 1 m = n>0 : 0 m 6= n: Further, we use the Chebyshev transform of L 2 w(i) into l 2 mapping f 2 L 2 w(i) into a[f] := (a n [f]) 1 2 l2 with the Chebyshev coecients a n [f] := hf T n i (n 2 0 ) : Then for f g 2 L 2 w (I), we have the Parseval identities hf gi =(a[f] a[g]) l 2 kfk = ka[f]k l 2 : (2.1) ote that the Chebyshev transform is a linear bijective mapping of L 2 w(i) onto l 2. For more details on the Chebyshev transform see [2, 10].

4 4 The Chebyshev transform is strongly related with the Fourier cosine transform. Let L 2 2 be the Hilbert space of all 2{periodic, square integrable functions F G : R! R with the inner product (F G) 2 := 1 2 Z ; F (s) G(s) ds: Let L be the subspace of all even functions of L 2 2. Foragiven function f 2 L 2 w(i), the cos{transformed function F := f(cos) 2 L has the Fourier expansion with the Fourier cosine coecients F = 1 2 a 0(F )+ a n (F ) := 2 Z 0 1 n=1 a n (F )cos(n ) (2.2) F (s) cos(ns) ds (n 2 0 ) : (2.3) In order to adapt the concept of shifts to the interval I, we consider the even shift S a F of F 2 L by a 2 R, which is dened as the even part of the translated function F ( ;a), i.e. S a F := 1 2 (F ( + a)+f ( ;a)) 2 L2 : (2.4) 2 0 Observe that for n 2 0 S a cos(n ) = cos(na) cos(n ) a n (S a F ) = cos(na) a n (F ) : Restricting F = f(cos) on [0 ], the arccos{transformed function F (arccos) coincides with f 2 L 2 w(i). From (2.2) { (2.3) it follows directly the Chebyshev expansion f = 1 2 a 0[f] + 1 n=1 a n [f] T n a n [f] = a n (f(cos)) (n 2 0 ) : Further, the even shift S a of F = f(cos) (a 2 R) goesinto the Chebyshev shift s h f of f with h := cos a 2 I, i.e. (s h f)(x) := 1 2 f(xh ; v(x)v(h)) + 1 f(xh + v(x)v(h)) (x 2 I) (2.5) 2 with v(x) :=(1; x 2 ) 1=2 (x 2 I). For the realization of the Chebyshev transform in nite dimensional subspaces of L 2 w (I), we will use fast algorithms of the discrete cosine transform (DCT). In the following, we briey introduce the dierent types of DCT. Let j := d 2 j, where j 2 0 stands for the level and d 2 is a constant depending on the application. Further, let k l be the Kronecker symbol and " j 0 = " j j := 2 ;1, " j k := 1 (k =1 ::: j ; 1). We introduce the matrices C j := ( cos kl j ) j k l=0 D j := diag (" j k ) j I j := ( k l ) j k l=0 ~C j := ( cos (2s+1)r j+1 ) j;1 r s=0 ~ Dj := diag (" j s ) j;1 s=0 ~ Ij := ( r s ) j;1 r s=0

5 5 which full the relations This follows from j ;1 cos ~C T j C j D j C j D j = j 2 I j (2.6) ~D j ~ Cj = ~ Cj ~ C T j " j k cos ku j = (2k +1)u j+1 = j 8 < : ~D j = j 2 ~ I j : (2.7) u 0mod j+1 0 otherwise j u 0mod j+2 ; j u j+1 mod j+2 0 otherwise (2.8) (2.9) (cf. [16]). For further development, we dene some variants of the DCT. The type I{DCT of length j + 1 (DCT{I ( j + 1)) is a mapping of R j+1 into itself dened by ^x I := C I j x (x 2 R j+1 ) (2.10) with C I j := C j D j. By j (C I j) ;1 = 2C I j, this mapping is bijective. ote that (2.6) and (2.10) imply (^x I ) T D j ^x I = x T (C I j) T D j C I j x = xt D j C j D j C j D j x = j 2 xt D j x : (2.11) The type II{DCT of length j (DCT{II ( j )) is a mapping of R j into itself dened by with C II j := ~ Cj. Then by (2.7) and (2.12), we obtain ~y II := C II j y (y 2 R j ) (2.12) (~y II ) T ~ Dj ~y II = y T ~ C T j ~D j ~ Cj y = j 2 y T y : (2.13) The type III{DCT of length j (DCT{III ( j )) is a mapping of R j into itself dened by ~y III := ~ C III j y (y 2 R j ) with C III T j := C ~ j ~D j. By (2.7), the inverse of the DCT{II ( j ) is the mapping (2= j ) DCT{III ( j ). Fast and numerically stable algorithms for the DCT{I ( j + 1), DCT{II ( j ) and DCT{III ( j ), which work in real arithmetic, are described in [1, 15]. 3 Shift{Invariant Subspaces Using the Gauss{Chebyshev nodes h j u := cos (u= j )(u 2 Z) of level j (j 2 0 ) and the Chebyshev shift (2.5), we obtain the shifts of level j j u := s hj u (u 2 Z) which possess the following properties (see [2, 16]):

6 6 Lemma 3.1 For j 2 0, u v 2 Zand f g 2 L 2 w (I) we have (i) j u+j+1 = j u = j+1 2u (ii) 2 j u j v =2 j v j u = j u+v + j u;v (iii) h j u f gi = hf j u gi (iv) j u T n =cos(nu= j ) T n a n [ j u f] = cos (nu= j ) a n [f] (n 2 0 ) (v) j u f 2 n for f 2 n (n 2 0 ). ote that j 0 f = f and j j f = f(;)forf 2 L 2 w (I). Further, for f 2 C(I) wehave ( j u f)(1) = f(h j u ) (u 2 Z) : (3.1) A linear subspace S of L 2 w (I) is called shift{invariant of level j (j 2 0), if for each f 2 S all shifted functions j l f (l =0 ::: j ) are contained in S. The shift{invariant subspace of level j S j 0 (') := span f j l ' : l =0 ::: j g is said to be of type 0 generated by' 2 L 2 w(i). The shift{invariant subspace of level j S j 1 (') := span f j+1 2l+1 ' : l =0 ::: j ; 1g is said to be of type 1 generated by' 2 L 2 w(i). It is obvious by Lemma 3.1, (i) { (ii) that S j 0 (') S j+1 0 (') ands j 1 (') =S j 0 ( j+1 1 ') S j+1 0 ('). By denition, f 2 S j+1 0 (') can be represented in the form f = j+1 " j+1 k j+1 k (f) j+1 k ' ( j+1 k (f) 2 R) : (3.2) Using Lemma 3.1, (iv) and Chebyshev transform, we obtain the Chebyshev coecients with a n [f] = ^ j+1 n (f) a n ['] (n 2 0 ) (3.3) ^ j+1 n (f) := j+1 " j+1 k j+1 k (f) cos kn j+1 : (3.4) Observe that (^ j+1 n (f)) j+1 is the DCT{I ( j+1 +1) of ( j+1 k (f)) j+1 and that the following properties of periodicity and symmetry hold for n 2 0 and k =0 ::: j+1 ; 1 ^ j+1 n (f) = ^ j+1 j+2 +n(f) ^ j+1 k (f) = ^ j j+2 ;k(f) : In particular, for f 2 S j 0 (') we get the representation (3.2) with j+1 2l+1 (f) :=0 (l =0 ::: j ; 1) : (3.5) Then it follows that the vector (^ j+1 n (f)) j with components (3.4) is the DCT{I ( j+1) of ( j+1 2l (f)) j.for f 2 S l=0 j 1(') we obtain (3.2) with j+1 2l (f) :=0 (l =0 ::: j ) (3.6)

7 7 and the corresponding vector (^ j+1 n (f)) j;1 is the DCT{II ( j )of( j+1 2l+1 (f)) j;1, l=0 i.e. ^ j+1 n (f) = j ;1 l=0 j+1 2l+1 (f) cos (2l +1)n j+1 : In the following, we derive some important properties of the subspaces S j ('). We characterize S j (') ( 2f0 1g) by the Chebyshev transform: Lemma 3.2 Let j 2 0, 2f0 1g and ' f 2 L 2 w(i) be given. (i) Then f 2 S j (') if and only if there exist ^ j+1 n (f) 2 R (n 2 0 ) with ^ j+1 n (f) = ^ j+1 j+2 +n(f) (n 2 0 ) ^ j+1 j+1 n(f) = (;1) ^ j+1 n (f) (n =0 ::: j ) (3.7) such that (3.3) is satised. (ii) Let j+1 f 2 S j ('). Then S j (f) =S j (') if and only if supp a[ j+1 f] = supp a[ j+1 '] (3.8) where supp a[f] :=fn 2 0 : a n [f] 6= 0g is the support of a[f]. Proof: As mentioned before, if f 2 S j ('), then (3.7) is satised. Since the Chebyshev transform is a linear bijective mapping, the proof of the reversed direction is straightforward. Hence (i) is valid. ow we show (ii) for = If S j 0 (f) =S j 0 ('), then ' 2 S j 0 (f). From (i) it follows that supp a['] supp a[f]. Analogously, by f 2 S j 0 (') we nd supp a[f] supp a[']. Hence we obtain (3.8). 2. Assume that (3.8) is satised. We only need to show that ' 2 S j 0 (f). Since f 2 S j 0 ('), we have (3.3) with (3.7). By supp a[f] = supp a['], we conclude that a n ['] = ^ j+1 n a n [f] (n 2 0 ) with ^ j+1 n := ^j+1 n (f) ;1 if ^ j+1 n (f) 6= 0 0 otherwise, for which (3.7) is also satised. For = 1, the assertion follows immediately from S j 1 (') =S j 0 ( j+1 1 ')ands j 1 (f) = S j 0 ( j+1 1 f). For a further analysis of the shift{invariant subspaces of L 2 w(i), we introduce the bracket product of a := (a n ) 1 and b := (b n ) 1 2 l 2. Let for k =0 ::: j+1 [a b] j k := 1 m=0 Observe that [a b] j k satises the symmetry property (a mj+1 +kb mj+1 +k + a (m+1)j+1 ;kb (m+1)j+1 ;k) : (3.9) [a b] j j+1 ;l = [a b] j l (l =0 ::: j+1 ; 1) : We extend the values [a b] j k (k =0 ::: j+1 )toan j+1 {periodic sequence, i.e. [a b] j k =[a b] j k+j+1 (k 2 0 ) :

8 8 Then the type I{bracket product of length j + 1 is dened by [a b] I j := ([a b] j k ) j 2 R j+1 and the type II{bracket product of length j by [a b] II j := ([a b] j k ) j;1 2 R j : Lemma 3.3 Let j 2 0, 2f0 1g and ' 2 L 2 w(i) be given. Further, let f 2 S j ('), g 2 S j ( ) with a n [f] = ^ j+1 n (f) a n ['] a n [g] = ^ j+1 n (g) a n [ ] (n 2 0 ) be given, where ^ j+1 n (f), ^ j+1 n (g) 2 R possess the properties (3.7). Then we have hf gi = j+1 " j+1 k ^ j+1 k (f) ^ j+1 k (g)[a['] a[ ]] j+1 k : In particular, for =, hf gi = j ; " j k ^ j+1 k (f) ^ j+1 k (g)[a['] a[ ]] j k : Using the Parseval identity (2.1), the proof follows by straightforward calculations. In particular, with f := j l ' g := j m for arbitrary ' 2 Lw 2 (I), we obtain for l m =0 ::: j the relations i.e., h j l ' j m i = " j k cos kl j cos km j [a['] a[ ]] j k ( h j l ' j m i ) j l m=0 = C j D j diag [a['] a[ ]] I j C j : (3.10) Analogously, for f := j+1 2l+1 ', g := j+1 2m+1 we have forl m =0 ::: j ; 1 h j+1 2l+1 ' j+1 2m+1 i = and thus j ;1 " j k cos k(2l +1) j+1 cos k(2m +1) j+1 [a['] a[ ]] j k (h j+1 2l+1 ' j+1 2m+1 i ) j;1 = T ~ l m=0 C j ~D j diag [a['] a[ ]] II j ~C j : (3.11) Corollary 3.4 For j 2 0 and ' 2 L 2 w(i), we have (i) S j (')? S j ( ) ( 2f0 1g) if and only if [a['] a[ ]] j k = 0 (k =0 ::: j ; ) (3.12)

9 9 (ii) S j 0 (')? S j 1 ( ) if and only if [a['] a[ ]] j+1 k = 0 (k =0 ::: j+1 ) : For ' 2 L 2 w(i), we consider the systems B j 0 (') :=f j l ' : l =0 ::: j g and B j 1 (') := f j+1 2l+1 ' : l =0 ::: j ; 1g. For B j 0 ('), we dene a special orthonormality criterion. We say that B j 0 (') isorthonormal, if the modied Gramian matrix fulls (" j m h j l ' j m 'i ) j l m=0 = I j : (3.13) The system B j 1 (') is called orthonormal, if the Gramian matrix satises (h j+1 2l+1 ' j+1 2m+1 'i) j;1 l m=0 = ~ I j : Then we obtain the following characterizations for the bases B j (') ( 2f0 1g) in terms of the bracket products. Lemma 3.5 Let 2f0 1g and j 2 0 be given. (i) The system B j (') isabasis of S j (') if and only if for all k =0 ::: j ; [a['] a[']] j k > 0 : (3.14) (ii) The system B j (') is an orthonormal basis of S j (') if and only if [a['] a[']] j k = 2= j (k =0 ::: j ; ): (3.15) (iii) If ' satises (3.14), and if '? 2 L 2 w(i) is dened by a n ['? ] := ^c j+1 n ('? ) a n ['] (n 2 0 ) with coecients ^c j+1 n ('? ) determined by(3.7) and ^c j+1 n ('? ) := (2= j ) 1=2 [a['] a[']] ;1=2 j n then B j ('? ) is an orthonormal basis of S j ('). (n =0 ::: j ; ) Proof: Let = 0. The system B j 0 (') forms a basis of S j 0 (') if and only if the Gramian matrix ( h j l ' j m 'i ) j l m=0 is regular. Since C j and D j are regular, by (3.10) this is the case if and only if diag [a['] a[']] I j is regular, i.e., if and only if (3.14) is satised. By denition, B j 0 (') is an orthonormal basis of S j 0 (') if and only if (" j m h j l ' j m 'i ) j l m=0 = (h j l' j m 'i ) j l m=0 D j = I j : By (3.10) and (2.6), this is true if and only if (3.15) holds. Finally, byverifying [a['? ] a['? ]] j k = 2= j (k =0 ::: j ) we see that by construction B j 0 ('? ) is an orthonormal basis of S j 0 ('? ). By Lemma 3.2, the denition of '? implies that S j 0 ('? )=S j 0 ('). Using (2.7) and (3.11), the assertions follow analogously for = 1. With the help of the bracket product, we are able to give a simple description of the orthogonal projectors P j ( =0 1) of L 2 w(i) onto S j (').

10 10 Lemma 3.6 Let j 2 0, 2f0 1g and let ' 2 L 2 w (I) with (3.14) be given. Then for f 2 L 2 w (I) we have a n [P j f] = ^c j+1 n (P j f) a n ['] (n 2 0 ) where the coecients ^c j+1 n (P j f) satisfy the relations (3.7) and ^c j+1 l (P j f) := [a[f] a[']] j l [a['] a[']] j l (l =0 ::: j ; ): (3.16) The projector P j is shift{invariant of level j, i.e., for all f 2 L 2 w(i) and k =0 ::: j ; j k (P j f) = P j ( j k f) : Proof: We show the assertion only for =0. For f 2 L 2 w(i), the orthogonal projection P j 0 f 2 S j 0 (') is determined by f ; P j 0 f? S j 0 ('). Then there are coecients ^c j+1 n (P j 0 f)(n 2 0 ) satisfying the properties (3.7) of symmetry and periodicity with a n [P j 0 f] = ^c j+1 n (P j 0 f) a n ['] (n 2 0 ): Using Lemma 3.3, we obtain for all l =0::: j 0 = hf ; P j 0 f j l 'i = = " j k cos kl j [a[f ; P j 0 f] a[']] j k " j k cos lk j ([a[f] a[']] j k ; ^c j+1 k (P j 0 f)[a['] a[']] j k ) : Hence the coecients ^c j+1 k (P j 0 f) satisfy (3.16). The shift{invariance of P j 0 follows from a n [ j l (P j 0 f)] = ^c j+1 n (P j 0 f) a n ['] cos ln j = ^c j+1 n (P j 0 ( j l f)) a n ['] = a n [P j 0 ( j l f)] (n 2 0 ): 4 Multiresolution of L 2 w (I) We form shift{invariant subspaces V j := S j 0 (' j ) with ' j 2 L 2 w(i) for each level j 2 0. The sequence of subspaces V j (j 2 0 ) is called a nonstationary multiresolution of L 2 w (I), if the following three conditions are satised: (M1) V j V j+1 (j 2 0 ): (M2) clos 1S j=0 V j! = L 2 w(i):

11 11 (M3) The systems B j 0 (( j =2) 1=2 ' j )(j 2 0 ) are L 2 w (I){stable, i.e., there exist positive constants independent ofj such that for all j 2 0 and for any ( j n ) j 2 R j+1, " j n 2 j n " j n j n ( j =2) 1=2 j n ' j 2 " j n 2 j n : (4.1) By (M3), B j 0 (( j =2) 1=2 ' j ) is a basis of V j. ote that dim V j = j +1. The shift{ invariant subspace V j is called sample space of level j. The function ' j of V j is said to be the scaling function of V j. If all systems B j 0 (( j =2) 1=2 ' j ) are orthonormal bases of V j (j 2 0 ) in the sense of (3.13), then we say that ( j =2) 1=2 ' j (j 2 0 )areorthonormal scaling functions. In this case the constants in condition (M3) are = = 1. Concerning (M2), we observe the following Theorem 4.1 Let fv j g 1 j=0 be a nested sequence of shift{invariant subspaces V j := S j 0 (' j ) with ' j 2 L 2 w (I), i.e., (M1) is valid. Then the condition (M2) is satised if and only if 1[ j=0 supp a[' j ] = 0 : (4.2) Proof: 1. Suppose that (4.2) is not satised. Then there is a number n n 1[ j=0 supp a[' j ] such that for the Chebyshev polynomial T n0 it holds that! 1[? clos V j : Thus, (M2) is not satised. 2. Assume that (4.2) holds. By (M1) and Lemma 3.2, we have T n0 j=0 supp a[' j ] supp a[' j+1 ] (j 2 0 ): (4.3) Suppose that there exists f 2 L 2 w(i) (f 6= 0) with f? clos 1[ j=0 V j! : (4.4) By k 0 2 0,we denote an index for which ja k0 [f]j = max fja k [f]j : k 2 0 g > 0 : By (4.2) { (4.3) we conclude that there is an index j such that k 0 2 supp a[' j0 ] and j0 k 0. Since ' j0 2 V j for all j j 0,we nd that f? S j 0 (' j0 )(j j 0 ). Hence, for j j 0,wehave by (3.12) [a[f] a[' j0 ]] j k0 = 0

12 12 i.e., a k0 [f] a k0 [' j0 ]+a j+1 ;k 0 [f] a j+1 ;k 0 [' j0 ] P + 1 (a k0 +n j+1 [f] a k0 +n j+1 [' j0 ]+a (n+1)j+1 ;k 0 [f] a (n+1)j+1 ;k 0 [' j0 ]) = 0 : n=1 (4.5) Put and choose j 1 j 0 such that 0 := ja k0 [f] a k0 [' j0 ]j > 0 n j1 ja n [f] a n [' j0 ]j 0 =2: (4.6) This choice of j 1 is possible, since by Cauchy{Schwarz inequality 1 n=1 ja n [f] a n [' j0 ]j ka[f]k l 2 ka[' j0 ]k l 2 < 1: But (4.6) contradicts equation (4.5) for j = j 1. This implies that f = 0, i.e., (M2) is satised. Theorem 4.2 The system fb j 0 (( j =2) 1=2 ' j ): j 2 0 g is L 2 w(i){stable with positive constants independent of j if and only if for all k =0 ::: j and for all j j 4 [a[' j] a[' j ]] j k : (4.7) Proof: 1. From Lemma 3.3, it follows that for j 2 0 and ( j k ) j 2 R j+1 with j By (2.10) { (2.11) we have " j k j k ( j =2) 1=2 j k ' j 2 = j 2 ^ j n := " j n ^ 2 j n [a[' j ] a[' j ]] j n " j k j k cos kn j (n 2 0 ) : " j k 2 j k = 2 j With the considerations above, (4.1) reads as follows " j n ^ 2 j n 2 j 4 " j n ^ 2 j n : " j n ^ 2 j n [a[' j ] a[' j ]] j n " j n ^ 2 j n

13 13 with arbitrary (^ j n ) j 2 R j+1 and j 2 0, which is equivalent to (4.7). In the following, we assume that (M1) { (M3) are satised. From (M1) it follows ' j 2 V j+1, i.e., there exist unique coecients j+1 k (' j ) 2 R (k =0 ::: j+1 ) such that ' j = j+1 " j+1 k j+1 k (' j ) j+1 k ' j+1 : This is the so{called two{scale relation or renement equation of ' j. transformed two{scale relation of ' j reads The Chebyshev a n [' j ] = A j+1 (n) a n [' j+1 ] (n 2 0 ) (4.8) with the two{scale symbol or renement mask of ' j A j+1 (n) := j+1 " j+1 k j+1 k (' j ) cos kn j+1 (n 2 0 ): By denition we obtain the relations of periodicity and symmetry for all n 2 0 l =0 ::: j+2 ; 1, and If a scaling function ' j A j+1 (n) = A j+1 ( j+2 + n) A j+1 ( j+2 ; l) = A j+1 (l) : (4.9) (j 2 0 ) satisfying (4.7) is given, then an orthonormal basis B j 0 (( j =2) 1=2 '? j)(j 2 0 ) can be easily obtained by Lemma 3.5, (iii). Let '? j (j 2 0 ) be dened by its Chebyshev coecients j 2 a n['? j] := ([a[' j ] a[' j ]] j n ) ;1=2 a n [' j ] (n 2 0 ): Then B j 0 (( j =2) 1=2 '? j) is an orthonormal basis of V j = S j 0 (' j ). The two{scale symbol A? j+1 satisfying a n ['? j ] = A? j+1 (n) a n['? j+1 ] (n 2 0) is connected with A j+1 by A? j+1(n) := 2 [a['j+1 1=2 ] a[' j+1 ]] j+1 n A j+1 (n) (n 2 0 ): [a[' j ] a[' j ]] j n The following connection between the bracket product [a[' j ] a[' j ]] I j and the two{scale symbol A j+1 can be observed: Lemma 4.3 For j 2 0 and k =0 ::: j, we have [a[' j ] a[' j ]] j k = A j+1 (k) 2 [a[' j+1 ] a[' j+1 ]] j+1 k + A j+1 ( j+1 ; k) 2 [a[' j+1 ] a[' j+1 ]] j+1 j+1 ;k: In particular, if ( j =2) 1=2 '? j is an orthonormal scaling function and if A? j+1 scale symbol of '? j, then is the two{ A? j+1 (k)2 + A? j+1 ( j+1 ; k) 2 = 4 (k =0 ::: j ): (4.10)

14 14 Proof: By the denition of the bracket product and by (4.8) { (4.9), we obtain for k =0 ::: j [a[' j ] a[' j ]] j k = 1 (a nj+2 +k[' j ] 2 + a (n+1)j+2 ;k[' j ] 2 + a nj+2 + j+1 +k[' j ] 2 + a nj+2 + j+1 ;k[' j ] 2 ) = A j+1 (k) 2 [a[' j+1 ] a[' j+1 ]] j+1 k + A j+1 ( j+1 ; k) 2 [a[' j+1 ] a[' j+1 ]] j+1 j+1 ;k: For orthonormal scaling functions, the assertion follows by Lemma 3.5, (ii). 5 Wavelet Spaces Let the wavelet space W j of level j (j 2 0 ) be dened as the orthogonal complement of V j in V j+1, i.e. W j := V j+1 V j (j 2 0 ): Then it follows dim W j =( j+1 +1);( j +1) = j. By denition, the wavelet spaces W j (j 2 0 ) are orthogonal. By (M1) { (M2), we obtain the orthogonal sum decomposition L 2 w(i) = V 0 Further, W j can be characterized by the orthogonal projector P j 0 of L 2 w (I)onto V j, namely by W j = ff ; P j 0 f : f 2 V j+1 g: The subspace W j is shift{invariant of level j, since by Lemma 3.6 wehave for g := f ;P j 0 f (f 2 V j+1 ), j l g = j l f ; j l (P j 0 f) = j l f ; P j 0 ( j l f) 2 W j : Assume that the shift{invariant subspace W j can be of type 1 generated by a function j 2 V j+1 such that W j = S j 1 ( j ). Further, we suppose that the set B j 1 (( j =2) 1=2 j )= f( j =2) 1=2 j+1 2l+1 j : l =0 ::: j ; 1g is L 2 w(i){stable, i.e., there are constants 0 < <1 independent ofj such that for all j 2 0 and for any ( j n ) j;1 2 R j, j ;1 2 j n j;1 1M j=0 W j : j n ( j =2) 1=2 j+1 2n+1 j 2 j ;1 2 j n : (5.1) Under these assumptions, j is called semiorthogonal wavelet. If B j 1 (( j =2) 1=2 j )(j 2 0 ) are orthonormal bases, then ( j =2) 1=2 j are called orthonormal wavelets. Obviously, for orthonormal wavelets the condition (5.1) is satised with = =1. By W j V j+1, there are unique coecients j+1 k ( j ) 2 R (k =0 ::: j+1 ) such that a two{scale relation or renement equation of j of the form j = j+1 " j+1 k j+1 k ( j ) j+1 k ' j+1

15 15 is satised. By means of the Chebyshev transform this yields a n [ j ] = B j+1 (n) a n [' j+1 ] (n 2 0 ) (5.2) a n [ j+1 1 j ] = cos n j+1 B j+1 (n) a n [' j+1 ] (n 2 0 ) with the two{scale symbol or renement mask of j B j+1 (n) := j+1 " j+1 k j+1 k ( j ) cos kn j+1 (n 2 0 ): It is clear that B j+1 satises the same properties of periodicity and symmetry as A j+1 in (4.9). As in the sample space V j, the bracket products are important for the characterization of the L 2 w(i){stability ofw j and the orthogonality W j? V j : Theorem 5.1 (i) The condition (5.1) for j 2 0 equivalent to with positive constants and independent of j is 2 j 4 [a[ j] a[ j ]] j n (n =0 ::: j ; 1) : (5.3) (ii) For j 2 0 and k =0 ::: j ; 1 we have [a[ j+1 1 j ] a[ j+1 1 j ]] j k = = cos k j+1 2 [a[ j ] a[ j ]] j k cos k j+1 2 B j+1 (k) 2 [a[' j+1 ] a[' j+1 ]] j+1 k + B j+1 ( j+1 ;k) 2 [a[' j+1 ] a[' j+1 ]] j+1 j+1 ;k : (5.4) (iii) For j 2 0, we have S j 1 ( j )? V j if and only if for all n =0 ::: j ; 1 A j+1 (n) B j+1 (n)[a[' j+1 ] a[' j+1 ]] j+1 n ; A j+1 ( j+1 ; n) B j+1 ( j+1 ; n)[a[' j+1 ] a[' j+1 ]] j+1 j+1 ;n =0: (5.5) Proof: The proofs for (i) and (ii) are similar to those of Theorem 4.2 and Lemma 4.3. In order to show (iii),we observe that V j? S j 1 ( j )=S j 0 ( j+1 1 j ) is equivalent to the equations [a[' j ] a[ j+1 1 j ]] j k = 0 for all k =0 ::: j by Corollary 3.4, (i). Inserting the two{scale relations (4.8) and (5.2), we obtain the assertion. ote that from (5.5) it follows that this equation is also valid for n = j +1 ::: j+1. We introduce the two{scale symbol matrices of level j (j 2 0 ) for n =0 ::: j ; 1by S j+1 (n) := A j+1 (n) B j+1 (n) A j+1 ( j+1 ; n) ; B j+1 ( j+1 ; n) : (5.6) As usual, these matrices will play an important role in deriving the decomposition and reconstruction algorithms. Therefore we have toinvestigate the invertibility ofs j+1 (n). Let ( =0 1) be the eigenvalues of S j+1 (n), i.e., it holds det (S j+1 (n) ; I)=0 with the unit matrix I.

16 16 Lemma 5.2 Assume that (M1) { (M3) and (5.1) hold with positive constants. Then the two{scale symbol matrices S j+1 (n) are regular for all n =0 ::: j ;1 satisfying In particular, it holds that 4 minf g j j p jdet Sj+1 (n)j 4 Furthermore, we have for n =0 ::: j ; 1 max f g ( =0 1) : (5.7) p (n =0 ::: j ; 1) : (5.8) S j+1 (n) ;1 = diag ([a[' j ] a[' j ]] ;1 j n [a[ j] a[ j ]] ;1 j n )T S j+1 (n) T diag ([a[' j+1 ] a[' j+1 ]] j+1 n [a[' j+1 ] a[' j+1 ]] j+1 j+1 ;n) T : (5.9) Proof: Using Lemma 4.3 and (5.4) { (5.5), we nd for n =0 ::: j ; 1 S j+1 (n) T diag ([a[' j+1 ] a[' j+1 ]] j+1 n [a[' j+1 ] a[' j+1 ]] j+1 j+1 ;n) T S j+1 (n) = diag ([a[' j ] a[' j ]] j n [a[ j ] a[ j ]] j n ) T : Thus (5.9) holds for n =0 ::: j ; 1. By (5.9), (4.7) and (5.3), the eigenvalues and the determinant ofs j+1 (n) can be easily estimated in terms of the constants. ow by the help of the conditions for the two{scale symbol B j+1 of j in Theorem 5.1, we obtain Theorem 5.3 Assume that (M1) { (M3) are fullled. Then for all j 2 0, B j+1 : 0! R is a two{scale symbol of a semiorthogonal wavelet j 2 L 2 w (I) if and only if for n =0 ::: j+1, the two{scale symbol B j+1 (n) is of the form B j+1 (n) = [a[' j+1] a[' j+1 ]] j+1 j+1 ;n A j+1 ( j+1 ; n) [a[' j ] a[' j ]] j n K j (n) (5.10) where B j+1 has the same properties (4.9) of periodizity and symmetry as A j+1, and where K j : 0! R satises the conditions 0 < jk j (n)j < 1 (n =0 ::: j ; 1) for some constants and. K j (n) = K j (n + j+1 ) (n 2 0 ) (5.11) K j ( j+1 ; n) = K j (n) (n =0 ::: j ; 1) Proof: 1. Let B j+1 be given in the form (5.10) with K j satisfying (5.11). Then by Theorem 5.1, (iii) the orthogonality V j? S j 1 ( j ) is satised, since for n =0 ::: j ; 1, A j+1 (n) B j+1 (n)[a[' j+1 ] a[' j+1 ]] j+1 n ; A j+1 ( j+1 ; n) B j+1 ( j+1 ; n)[a[' j+1 ] a[' j+1 ]] j+1 j+1 ;n = 0:

17 17 It follows that S j 1 ( j ) W j. It remains to show that B j 1 (( j =2) 1=2 j )(j 2 0 ) are L 2 w (I){stable. By (5.4) and Lemma 4.3, we nd for n =0 ::: j ; 1 [a[ j ] a[ j ]] j n = [a[' j+1] a[' j+1 ]] j+1 n [a[' j+1 ] a[' j+1 ]] j+1 j+1 ;n [a[' j ] a[' j ]] j n K j (n) 2 : By (4.7) and (5.11), we can estimate 0 < j 4 [a[ j] a[ j ]] j n < 1 : (5.12) 2. For each j 2 0, let B j+1 : 0! R be the two{scale symbol of a semiorthogonal wavelet j and let B j 1 (( j =2) 1=2 j )(j 2 0 )bel 2 w(i){stable. Thus B j+1 satises (5.4) and (5.5). ow, put for n =0 ::: j+1 K j (n) :=A j+1 (n) B j+1 ( j+1 ; n)+a j+1 ( j+1 ; n) B j+1 (n) : ote that K j (n) = K j ( j+1 ; n) for n = 0 ::: j ; 1. Then we continue K j on 0 by K j (n + r j+1 ) := K j (n) for all n = 0 ::: j+1 ; 1 and r 2 0. Thus K j satises the conditions (5.11). Multiplying (5.5) with A j+1 (n), by Lemma 4.3 we obtain for n =0 ::: j ; 1 and also for n = j +1 ::: j+1 0 = A j+1 (n) 2 B j+1 (n)[a[' j+1 ] a[' j+1 ]] j+1 n ; A j+1 (n) A j+1 ( j+1 ; n) B j+1 ( j+1 ; n)[a[' j+1 ] a[' j+1 ]] j+1 j+1 ;n = B j+1 (n)[a[' j ] a[' j ]] j n ; K j (n) A j+1 ( j+1 ; n)[a[' j+1 ] a[' j+1 ]] j+1 j+1 ;n: Hence, B j+1 (n) is of the form (5.10) for n =0 ::: j ; 1 j +1 ::: j+1. Dening B j+1 ( j )by (5.10) for n = j, the proof is complete. Corollary 5.4 Assume that B j 0 (( j =2) 1=2 '? j)(j 2 0 ) are orthonormal bases of V j. Let A? j+1 be the two{scale symbols of '? j 2 L 2 w(i). Then for every j 2 0, B? j+1 : 0! R is a two{scale symbol of a wavelet j? 2 L2 w (I) generating an orthonormal basis B j 1 (( j =2) 1=2 j?) of W j if and only if B? j+1 possesses the form B? (n) = j+1 A? ( j+1 j+1 ; n) (n =0 ::: j+1 ) and fulls the same properties (4.9) of periodizity and symmetry as A j+1. Proof: By Lemma 3.5, (ii) we have 2 j [a['? j ] a['? j ]] j n = 4 (n =0 ::: j ) 2 j [a[? j ] a[? j ]] j n = 4 (n =0 ::: j ; 1) for all j 2 0. Hence, = = = =1. From (5.12) it follows that = =4, and thus K j (n) = 4(n 2 0 ). Then the assertion can be obtained by application of Theorem 5.3.

18 18 6 Decomposition and Reconstruction Algorithms ow we derive ecient decomposition and reconstruction algorithms. In order to decompose agiven function f j+1 2 V j+1 (j 2 0 ) of the form f j+1 = j+1 l=0 " j+1 l j+1 l (f j+1 ) j+1 l ' j+1 (6.1) the uniquely determined functions f j 2 V j and g j 2 W j have to be found such that l=0 f j+1 = f j + g j : (6.2) Assume that the coecients j+1 l 2 R (l =0 ::: j+1 )off j+1 or their DCT{I ( j+1 +1) data j+1 ^ j+1 k := " j+1 l j+1 l (f j+1 )cos kl (k =0 ::: j+1 ) (6.3) j+1 are known. The wanted functions f j 2 V j and g j 2 W j can be uniquely represented by f j = m=0 " j m j m (f j ) j m ' j g j = j ;1 r=0 j r (g j ) j+1 2r+1 j (6.4) with unknown coecients j m (f j ) j r (g j ) 2 R. Let^ j k ~ j s 2 R denote the following DCT{I ( j + 1) and DCT{II ( j ) data ^ j k := ~ j s := m=0 j ;1 r=0 " j m j m (f j )cos km j (k =0 ::: j ) (6.5) j r (g j ) cos (2r +1)s j+1 (s =0 ::: j ; 1) : (6.6) In order to reconstruct f j+1 2 V j+1 (j 2 0 ), we have to compute the sum (6.2) with given functions f j 2 V j and g j 2 W j. Assume that j m (f j ) j r (g j ) 2 R in (6.4) or the corresponding DCT data (6.5) { (6.6) are known. Then f j+1 2 V j+1 can be uniquely represented in the form (6.1). The decomposition and reconstruction algorithms are based on the following connection between (6.3) and (6.5) { (6.6): Theorem 6.1 Assume that for j 2 0 a k [' j ] 6= 0 (k =0 ::: j ) : (6.7) For j 2 0, let f j+1 2 V j+1, f j 2 V j and g j 2 W j with (6.1) { (6.6) be given. Then we have ^ j+1 r ^ j+1 j+1 ;r = S j+1 (r) ^j r ~ j r ^ j+1 j = A j+1 ( j )^ j j : (r =0 ::: j ; 1)

19 19 Proof: From (6.4), it follows by Lemma 3.1, (iv) that for all n 2 0 a n [f j ] = ^ j n a n [' j ] with ^ j n := l=0 " j l j l (f j ) cos ln j : Analogously, by (6.1) and (6.3) { (6.6) we have for all n 2 0 a n [f j+1 ] = ^ j+1 n a n [' j+1 ] a n [g j ] = ~ j n a n [ j ] (6.8) where ^ j+1 n is dened similar to ^ j n and ~ j n := j ;1 r=0 j r (g j ) cos The relation (6.2) holds if and only if for all k 2 0 (2r +1)n j+1 : a k [f j+1 ] = a k [f j ]+a k [g j ] : Using the Chebyshev transformed two{scale relations (4.8) and (5.2), we obtain ^ j+1 k a k [' j+1 ] = ^ j k A j+1 (k) a k [' j+1 ]+ ~ j k B j+1 (k) a k [' j+1 ] : Analogously, wehave for k =0 ::: j+1 ^ j+1 j+1 ;k a k [' j+1 ] = ^ j j+1 ;k A j+1 ( j+1 ; k) a k [' j+1 ]+ ~ j j+1 ;k B j+1 ( j+1 ; k) a k [' j+1 ] : Using the assumption (6.7) and observing that ^ j j+1 ;k =^ j k, ~ j+1 ;k = ; ~ j k (k = 0 ::: j+1 ), we obtain the assertion. ote that from (4.7) and Lemma 4.3 it follows that 2 ;1 A j+1 ( j ) 2 2 ;1, i.e. A j+1 ( j ) 6= 0. We obtain the following algorithms: Algorithm 6.2 (Decomposition Algorithm) Input: j 2 0, ^ j+1 k 2 R (k =0 ::: j+1 ). Form for r =0 ::: j ; 1, ^j r ~ j r := S j+1 (r) ;1 ^ j+1 r ^ j+1 j+1 ;r Output: ^ j r (r =0 ::: j ), ~ j r (r =0 ::: j ; 1). ^ j j := A j+1 ( j ) ;1 ^ j+1 j :

20 20 Algorithm 6.3 (Reconstruction Algorithm) Input: j 2 0, ^ j r 2 R (r =0 ::: j ), ~ j r 2 R (r =0 ::: j ; 1). Form for r =0 ::: j ; 1, ^ j+1 r ^ j+1 j+1 ;r := S j+1 (r) ^j r ~ j r Output: ^ j+1 k (k =0 ::: j+1 ). 7 Polynomial Wavelets ^ j+1 j := A j+1 ( j )^ j j : As the rst example, we consider polynomial wavelets on I (see [6, 16]). Set j := 2 j (j 2 0 ). As scaling function ' j of level j we use the following function dened by its Chebyshev coecients Then it holds that j a n [' j ] := j 2 ' j = 8 < : 2 n =0 ::: j ; 1 1 n = j 0 n> j : " j k T k 2 j : By (7.1), the corresponding bracket product reads as follows (7.1) 2 j [a[' j] a[' j ]] j k = 4 k =0 ::: j ; 1 2 k = j : (7.2) Using (2.8), we obtain the following interpolation property of' j ' j (h j l ) = j l ' j (1) = 2 j " j k cos kl j = 2 0 l (l =0 ::: j ) : (7.3) By (7.1), the shifted scaling functions j k ' j (k =0 ::: j ) are contained in j.further, these functions j k ' j (k =0 ::: j ) are modied Lagrange fundamental polynomials with respect to the Gauss{Chebyshev nodes h j l (l =0 ::: j ), since for k l =0 ::: j from Lemma 3.1, (ii) and (3.1) it follows j k ' j (h j l ) = ( j l j k ' j )(1) = 1 2 ( j l+k ' j (1) + j jl;kj ' j (1)) = " ;1 j k k l : Figure 1 shows the scaling function ' 5, and Figure 2 presents the shifted function 5 16 ' 5. The function j k ' j (k =0 ::: j ) is supported on the whole interval I, and has signicant values in a small neighbourhood of h j k,ifj is large enough.

21 21 Let V j := S j 0 (' j ) be the sample space of level j. Consequently by Lemma 3.5, (i), the polynomials j k ' j (k =0 ::: j ) form a basis of V j, i.e., V j = j dim V j = j +1: ote that the operator L j : C(I)! V j dened by L j f := " j k f(h j k ) j k ' j (f 2 C(I)) is an interpolation operator, which maps C(I) onto V j with the property L j f(h j l ) = f(h j l ) (l =0 ::: j ) : All sample spaces V j (j 2 0 )formamultiresolution of L 2 w(i), where (M3) reads as follows: The systems B j 0 (( j =2) 1=2 ' j )(j 2 0 ) are L 2 w (I){stable with optimal constants =1=2 and = 1, i.e., for all j 2 0 and for any ( j k ) j 2 R j+1 wehave the sharp estimate 1 2 " j k 2 j k j " j k j k ( j =2) 1=2 j k ' j 2 Using (7.1), we nd the Chebyshev transformed two{scale relation of ' j with the corresponding two{scale symbol a n [' j ] = A j+1 (n) a n [' j+1 ] (n 2 0 ) A j+1 (n) := 8 < : 2 n =0 ::: j ; 1 1 n = j 0 n = j +1 ::: j+1 : " j k 2 j k : Let W j := V j+1 V j be the wavelet space of level j. Thus, dimw j = j. Consider the polynomials j 2 V j+1 (j 2 0 ) given by their Chebyshev coecients j a n [ j ] := 8 < : 2 n = j +1 ::: j+1 ; 1 1 n = j+1 0 otherwise : (7.4) Then the corresponding bracket product reads as follows 2 j [a[ j] a[ j ]] j+1 k = 8 < : 0 k =0 ::: j 4 k = j +1 ::: j+1 ; 1 2 k = j+1 : (7.5) The shifted polynomials j+1 2r+1 j satisfy the interpolation properties j+1 2r+1 j (h j+1 2s+1 ) = r s (r s=0 ::: j ; 1) : Figure 3 shows the wavelet 5, and Figure 4 presents the shifted wavelet The wavelet space W j is a shift{invariant subspace of L 2 w(i) oftype 1 generated by j. The

22 22 systems B j 1 (( j =2) 1=2 j )(j 2 0 ) are L 2 w (I){stable with optimal constants =1=2 and = 1, i.e., for all j 2 0 and for any ( j r ) j;1 r=0 2 R j,wehave the sharp estimate 1 2 j ;1 r=0 2 j r j;1 r=0 j r ( j =2) 1=2 j+1 2r+1 j 2 j ;1 r=0 2 j r : Using (7.1) and (7.4), we obtain the Chebyshev transformed two{scale relation of j with the corresponding two{scale symbol a n [ j ] = B j+1 (n) a n [' j+1 ] (n 2 0 ) B j+1 (n) := 0 n =0 ::: j 2 n = j +1 ::: j+1 : In the following, we compare the arithmetical complexity of our decomposition algorithm 6.2 for these polynomial wavelets on I with that of the fast decomposition algorithm for linear and cubic spline wavelets on [0 1] proposed in [13]. Let j 3. Assume that 2 j function values of f j+1 2 V j+1 are given. The decomposition algorithm for linear spline wavelets in [0 1] needs 6 2 j+1 real multiplications in order to compute all wavelet coecients of g j 2 W j. For the same problem, the decomposition algorithm for cubic spline wavelets in [13] can be implemented using 14 2 j+1 real multiplications. Compared to that, our algorithm 6.2 requires fewer real multiplications up to the level j = 14. { ow we consider the complete decomposition of f j+1 2 V j+1. Here we have to determine all coecients of the related functions in W j W j;1 ::: W 3 and V 3. Figure 5 shows the numbers of needed real multiplications (divided by 2 j+1 ) for the complete decomposition with linear spline wavelets (3), cubic spline wavelets (2) and polynomial wavelets (+). Our procedure needs fewer real multiplications than the method in [13] for cubic spline wavelets up to level j = 20. Since a level j 2f7 ::: 11g is often used in praxis, our algorithm is an interesting alternative to the method in [13]. As numerical application of the decomposition algorithm 6.2, we would like to mention that an exact detection of singularities of a given function near the boundary 1 is possible. For example, we consider a linear spline function in order to determine its spline knots. Let B 2 denote the cardinal linear B{spline. Interpolating the function f(x) :=B 2 (4x +3:96) (x 2 I) at level j = 7 and decomposing f, we can observe the singularities at ;0:99 ;0:74 and ;0:49 in the corresponding wavelet part of level j = 6 (see Figure 6). On the other hand, the decomposition of the function f(x) :=B 2 (4x +4) (x 2 I) shows that f has singularities at ;0:75 and ;0:5, but not at ;1 (see Figure 7). We can generalize this example of polynomial wavelets in a similar manner as done for periodic functions in [14]. Set j = d 2 j (j 2 0 ) with xed d 2. Further let for xed 2 0 M j := 1 j 2 j; j>

23 23 where 3 2 d is fullled. Then, j + M j j+1 ; M j+1. Let the scaling function ' j of level j be given by its Chebyshev coecients j a n [' j ] := 8 >< >: 2 0 n j ; M j j +M j ;n M j j ; M j <n< j + M j 0 n j + M j : The smaller, the better localized the scaling functions are on I. We obtain the same interpolation property of' j as in (7.3). The sample space V j = S j 0 (' j ) can be described by V j = j ;M j spann Mj o +k M j+1 T j ;k + M j;k M j+1 T j +k : k =0 ::: M j ; 1 i.e., j ;M j V j j +M j ;1. The corresponding wavelet space W j (j 2 0 )isoftype 1 generated by the polynomial j := 2 ' j+1 ; ' j 2 V j+1 such that j a n [ j ] = 8 >< >: n; j +M j M j j ; M j <n< j + M j 2 j + M j n j+1 ; M j+1 j+1 +M j+1 ;n M j+1 j+1 ; M j+1 <n< j+1 + M j+1 0 otherwise: The shifted polynomials j+1 2r+1 j also satisfy the interpolation properties j+1 2r+1 j (h j+1 2s+1 ) = r s (r s=0 ::: j ; 1) : For the Chebyshev transformed two{scale relations of ' j and j,we obtain the two{scale symbols 8 < 2 0 n j ; M j A j+1 (n) := j +M j ;n M : j j ; M j <n< j + M j 0 j + M j n j+1 and B j+1 (n) := 8 < : 0 0 n j ; M j n; j +M j M j j ; M j <n< j + M j 2 j + M j n j+1 : 8 Transformed Spline Wavelets In principle, the following is obtained by transferring the construction of [12] onto the interval. Let m 2 beaxedeven number and let M m be the centered B-spline of order m with the knots ;m=2 +k (k =0 ::: m). Set j := 2 j (j 2 0 ). Further, let Mm j ~ (j 2 0 ) be the 2{periodization of M m ( j =), i.e. 1 ~M m j := l=;1 M m ( j = ; j+1 l): From M ~ m j = M ~ m j (;) it follows that Mm j ~ 2 L The Fourier cosine coecients of ~M m j read as follows a n ( M ~ m j ) = 1 ^Mm (n= j ) = 1 sinc j j n j+1 m (n 2 0 ):

24 24 ow restricting Mm j ~ on [0 ], we choose as scaling function ' j := Mm j ~ (arccos), that is, ' j has the Chebyshev coecients j a n [' j ] = ^Mm (n= j ) = sinc For the two{scale symbol of ' j we nd A j+1 (n) = 2 n j+1 m (n 2 0 ): cos n j+2 m (n 2 0 ): Let the sample spaces V j be generated by ' j, i.e. V j := S j 0 (' j )(j 2 0 ). Then by 1[ j=0 supp a[' j ] = 0 it follows that the condition (M2) is satised. For the bracket product we obtain by Poisson summation formula 2 j [a[' j] a[' j ]] j n = and hence = = 1 1 l=0 1 l=;1 k=;1 ^M m (j+1 l + n) j ^M 2m (2l + n= j ) M 2m (k) e ;ink= j 2 j [a[' j ] a[' j ]] j n = 2m (e ;in= j ) with the well{known Euler{Frobenius polynomial 2m (z) := 1 k=;1 2 (j+1(l 2! +1); n) + ^Mm j M 2m (k) z k (z 2 C jzj =1): The systems B j 0 (( j =2) 1=2 ' j )(j 2 0 )arel 2 w(i){stable, since we have with 2 j 4 [a[' j] a[' j ]] j n (j 2 0 ) 4 = 2m (;1) = 22m (2 2m ; 1) (2m)! jb 2m j 4 = 2m (1) = 1 where B 2m denotes the 2m{th Bernoulli number. Observe that we have found the same constants as in the case of the multiresolution generated by cardinal splines of order m (see [12]). ote that dierent scaling factors of scaling functions are used in [12]. Let the wavelet j be dened by its Chebyshev coecients a n [ j ] := 2 sin n j+2 m 2m (;e ;in= j+1 ) a n [' j+1 ] (n 2 0 )

25 25 i.e., j possesses the two{scale symbol B j+1 (n) = 2 sin n j+2 m 2m (;e ;in= j+1 ) (n 2 0 ): By denition it is clear that j 2 V j+1. In order to show thatw j = S j 1 ( j ), we have to check the orthogonality V j? S j 1 ( j ) and the L 2 w(i){stability ofb j 1 (( j =2) 1=2 j ) (j 2 0 ). Since m is even, we easily observe that (5.5) is satised for the two{scale symbols A j+1 (n) B j+1 (n) above. Furthermore, inserting B j+1 (n) into (5.4) we obtain for n =0 ::: j ; 1by Lemma j [a[ j ] a[ j ]] j n = 2m (e ;in= j ) 2m (e ;in= j+1 ) 2m (;e ;in= j+1 ) : In particular, we obtainl 2 w (I){stability ofb j 1(( j =2) 1=2 j ) with the constants 4 = min f 2m (z) 2m (;z) 2m (z 2 ): z 2 C jzj =1g = 2m (;1) 2m (;i) 2m (i) > 0 4 = max f 2m (z) 2m (;z) 2m (z 2 ): z 2 C jzj =1g < 1 : Observe that these constants are the same as the constants found for the well{known cardinal Chui{Wang wavelet (cf. [12]). ote that dierent scaling factors of wavelets are used in [12]. In contrast with polynomial wavelets, the shifted scaling functions and wavelets are supported on small subintervals of I. Finally, we will consider the connection of the wavelet j above with the cardinal Chui{ Wang wavelet w m given by its Fourier transform ^w m (u) := (;e ;iu=2 ) m;1 1 ; e ;iu=2 m 2m(;e ;iu=2 ) 1 ; e ;iu=2 2 iu=2 m (u 2 R nf0g): Let ~ j be the 2{periodization of ~w m ( j ) with i.e., ~w m := (;1) m=2+1 (w m (= + m)+w m (;= + m)) ~ j := 1 l=;1 ~w m ( j ; j+1 l) : Then ~ j 2 L The Fourier cosine coecients read a n ( ~ j) = 2 j cos n j+1 ( sin n j+2 ) m 2m (;e ;in= j+1 ) (sinc = 2 cos n j+1 ( sin n j+2 ) m 2m (;e ;in= j+1 ) a n [' j+1 ] : n j+2 ) m Comparing with the Chebyshev coecients of j we nd for the restriction of ~ j on [0 ]: j+1 1 j = ~ j(arccos) :

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