On the Construction of Wavelets on a Bounded ... - Semantic Scholar
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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 re ected by large wavelet coecients for high levels near the endpoints 0 and 1, even if f is smooth on [0; 1]. Thus the regularity of f is not characterized by the decay of wavelet 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 [0; 1], then the derivative of f~ has discontinuities at the integers. The smoothness 1
2 of f is again not characterized by the decay of wavelet 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 Lw (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 to multiresolution of Lw (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 (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 lead to a uni ed 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 Sa F := 21 (F ( + a) + F ( ? a)) (a 2 R) for F 2 L ; . The isomorphism between L ; and Lw (I ) can be exploited in order to construct new sample and wavelet spaces in Lw (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 brie y introduce the Chebyshev transform, related shifts and the DCT. In Section 3 we analyze shift{invariant subspaces of Lw (I ). The scalar product of functions from shift{invariant subspaces can be simpli ed to a nite sum by means of the so{called bracket product. In Section 4 we consider a nonstationary multiresolution of Lw (I ) consisting of shift{invariant subspaces Vj (j 2 N ) generated by shifts of scaling functions 'j . The required conditions for the multiresolution of Lw (I ) and their consequences for the scaling functions 'j are analyzed in detail. In Section 5 we introduce the wavelet space Wj (j 2 N ) as the orthogonal complement of Vj in Vj . Then Wj 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 , the wavelet j is characterized in Theorem 5.3. Section 6 provides fast, numerically stable decomposition 2
2
2
2 2
2 2
2 2
2 2
0
0
0
2
2
2
2
0
2
0
+1
+1
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. Note that the transformed spline wavelets are supported on small subintervals of I . The examples show that periodic multiresolutions of L with even scaling functions 'j can be transformed into a multiresolution of Lw (I ). 2 2
2
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 )? = for x 2 (?1; 1). Let Lw (I ) be the weighted Hilbert space of all measurable functions f : I ! R with the property 2
Z
1 2
2
w(y) f (y) dy < 1 : 2
I
For f; g 2 Lw (I ), the corresponding inner product and norm are de ned by Z 2 hf; gi := w(y) f (y) g(y) dy ; kf k := hf; f i = : 2
1 2
I
Let l denote the Hilbert space of all real, square summable sequences a = (an)1n , b = (bn)1n with the weighted inner product and norm given by 2
=0
=0
1 X (a; b)l2 := 21 a b + anbn ; n
kakl2 := (a; a)l2= : 1 2
0 0
=1
Let C (I ) be the set of all continuous functions f : I ! R. By n (n 2 N ) we denote the set of all real polynomials of degree at most n restricted on I . As known, the Chebyshev polynomials Tn := cos (n arccos) 2 n (n 2 N ) form a complete orthogonal system in Lw (I ). Note that arccos : I ! [0; ] is the inverse function of cos restricted on [0; ]. For m; n 2 N we have 8 < 2 m = n = 0; hTm; Tni = : 1 m = n > 0; 0 m 6= n: Further, we use the Chebyshev transform of Lw (I ) into l mapping f 2 Lw (I ) into a[f ] := (an[f ])1 n 2 l with the Chebyshev coecients an[f ] := hf; Tni (n 2 N ) : Then for f; g 2 Lw (I ), we have the Parseval identities (2.1) hf; gi = (a[f ]; a[g])l2 ; kf k = ka[f ]kl2 : Note that the Chebyshev transform is a linear bijective mapping of Lw (I ) onto l . For more details on the Chebyshev transform see [2, 10]. 0
0
2
0
2
=0
2
2
2
0
2
2
2
4 The Chebyshev transform is strongly related with the Fourier cosine transform. Let L be the Hilbert space of all 2{periodic, square integrable functions F; G : R ! R with the inner product Z 1 (F; G) := 2 F (s) G(s) ds : 2 2
2
?
Let L ; be the subspace of all even functions of L . For a given function f 2 Lw (I ), the cos{transformed function F := f (cos) 2 L ; has the Fourier expansion 2 2
0
2 2
2 2
2
0
1 X F = 21 a (F ) + an (F ) cos(n ) n
(2.2)
0
=1
with the Fourier cosine coecients
Z 2 an(F ) := F (s) cos(ns) ds (n 2 N ) :
(2.3)
0
0
In order to adapt the concept of shifts to the interval I , we consider the even shift SaF of F 2 L ; by a 2 R, which is de ned as the even part of the translated function F ( ? a), i.e. SaF := 12 (F ( + a) + F ( ? a)) 2 L ; : (2.4) Observe that for n 2 N Sa cos(n ) = cos(na) cos(n ) ; an(SaF ) = cos(na) an(F ) : Restricting F = f (cos) on [0; ], the arccos{transformed function F (arccos) coincides with f 2 Lw (I ). From (2.2) { (2.3) it follows directly the Chebyshev expansion 2 2
0
2 2
0
0
2
1 X 1 f = 2 a [f ] + an[f ] Tn ; n 0
an[f ] = an(f (cos))
(n 2 N ) : 0
=1
Further, the even shift Sa of F = f (cos) (a 2 R) goes into the Chebyshev shift shf of f with h := cos a 2 I , i.e. (2.5) (sh f )(x) := 21 f (xh ? v(x)v(h)) + 12 f (xh + v(x)v(h)) (x 2 I ) with v(x) := (1 ? x ) = (x 2 I ). For the realization of the Chebyshev transform in nite dimensional subspaces of Lw (I ), we will use fast algorithms of the discrete cosine transform (DCT). In the following, we brie y introduce the di erent types of DCT. Let Nj := d 2j , where j 2 N stands for the level and d 2 N is a constant depending on the application. Further, let k;l be the Kronecker symbol and "j; = "j;N := 2? , "j;k := 1 (k = 1; : : : ; Nj ? 1). We introduce the matrices 2 1 2
2
0
1
0
j
N Dj := diag ("j;k )Nk ; I j := (k;l)Nk;l ; C j := ( cos kl N )k;l ; C~ j := ( cos Ns +1r )Nr;s ? ; D~ j := diag ("j;s)Ns ? ; I~j := (r;s)Nr;s ? ; j
=0
=0
=0
(2 +1)
j
j
j
j
1 =0
j
j
=0
1
1 =0
j
5 which ful l the relations
This follows from N X
C j Dj C j Dj = N2j I j ; C~ Tj D~ j C~ j = C~ j C~ Tj D~ j = N2j I~j :
(2.6) (2.7)
Nj u 0 mod Nj ; (2.8) 0 otherwise ; k 8 NX ? u 0 mod Nj ; < Nj (2 k + 1) u cos N = : ?Nj u Nj mod Nj ; (2.9) j k 0 otherwise (cf. [16]). For further development, we de ne some variants of the DCT. The type I{DCT of length Nj + 1 (DCT{I (Nj + 1)) is a mapping of RN into itself de ned by x^I := C Ij x (x 2 RN ) (2.10) with C Ij := C j Dj . By Nj (C Ij )? = 2 C Ij , this mapping is bijective. Note that (2.6) and (2.10) imply (x^I)TDj x^I = xT(C Ij )TDj C Ij x = xTDj C j Dj C j Dj x = N2j xTDj x : (2.11) The type II{DCT of length Nj (DCT{II (Nj )) is a mapping of RN into itself de ned by y~II := C IIj y (y 2 RN ) (2.12) j
=0
j
"j;k cos ku Nj =
+1
1
=0
+2
+1
+1
+2
j+1
j +1
1
j
j
with C IIj := C~ j . Then by (2.7) and (2.12), we obtain
(2.13) (y~II)T D~ j y~II = y T C~ Tj D~ j C~ j y = N2j y T y : The type III{DCT of length Nj (DCT{III (Nj )) is a mapping of RN into itself de ned by j
y~III := C~ IIIj y
(y 2 RN ) j
with C IIIj := C~ Tj D~ j . By (2.7), the inverse of the DCT{II (Nj ) is the mapping (2=Nj ) DCT{III (Nj ). Fast and numerically stable algorithms for the DCT{I (Nj + 1), DCT{II (Nj ) and DCT{III (Nj ), which work in real arithmetic, are described in [1, 15].
3 Shift{Invariant Subspaces
Using the Gauss{Chebyshev nodes hj;u := cos (u=Nj ) (u 2 Z) of level j (j 2 N ) and the Chebyshev shift (2.5), we obtain the shifts of level j j;u := sh (u 2 Z) ; which possess the following properties (see [2, 16]): 0
j;u
6
Lemma 3.1 For j 2 N , u; v 2 Z and f; g 2 Lw (I ) we have 2
0
(i) j;u N +1 = j;u = j ; u ; (ii) 2 j;u j;v = 2 j;v j;u = j;u v + j;u?v ; (iii) hj;uf; gi = hf; j;ugi ; (iv) j;u Tn = cos (nu=Nj ) Tn ; an[j;uf ] = cos (nu=Nj ) an[f ] (n 2 N ) ; (v) j;uf 2 n for f 2 n (n 2 N ) . Note that j; f = f and j;N f = f (? ) for f 2 Lw (I ). Further, for f 2 C (I ) we have (j;u f )(1) = f (hj;u ) (u 2 Z) : (3.1) A linear subspace S of Lw (I ) is called shift{invariant of level j (j 2 N ), if for each f 2 S all shifted functions j;l f (l = 0; : : : ; Nj ) are contained in S . The shift{invariant subspace of level j Sj; (') := span fj;l' : l = 0; : : :; Nj g is said to be of type 0 generated by ' 2 Lw (I ). The shift{invariant subspace of level j Sj; (') := span fj ; l ' : l = 0; : : : ; Nj ? 1g is said to be of type 1 generated by ' 2 Lw (I ). It is obvious by Lemma 3.1, (i) { (ii) that Sj; (') Sj ; (') and Sj; (') = Sj; (j ; ') Sj ; ('). By de nition, f 2 Sj ; (') can be represented in the form +
+1 2
j
+
0
0
2
0
j
2
0
0
2
1
+1 2 +1 2
0
+1 0
1
f =
NX +1 j
k=0
0
"j
+1 1
+1 0
;k j +1;k (f ) j +1;k '
+1
( j
+1 0
;k (f ) 2 R) :
(3.2)
+1
Using Lemma 3.1, (iv) and Chebyshev transform, we obtain the Chebyshev coecients an[f ] = ^j ;n (f ) an ['] (n 2 N ) (3.3) with NX +1 "j ;k j ;k (f ) cos Nkn : ^j ;n(f ) := (3.4) +1
0
j
+1
k=0
+1
+1
j +1
Observe that (^ j ;n (f ))nN +1 is the DCT{I (Nj + 1) of ( j ;k (f ))kN +1 and that the following properties of periodicity and symmetry hold for n 2 N and k = 0; : : : ; Nj ? 1 ^j ;n(f ) = ^j ;N +2 n(f ) ; ^j ;k (f ) = ^j;N +2 ?k (f ) : In particular, for f 2 Sj; (') we get the representation (3.2) with j ; l (f ) := 0 (l = 0; : : :; Nj ? 1) : (3.5) j
+1
+1
=0
j
+1
=0
0
+1
+1
j
+
+1
+1
j
0
+1 2 +1
Then it follows that the vector (^ j ;n(f ))Nn with components (3.4) is the DCT{I (Nj +1) of ( j ; l(f ))Nl . For f 2 Sj; (') we obtain (3.2) with j ; l(f ) := 0 (l = 0; : : :; Nj ) ; (3.6) +1
+1 2
j
=0
1
+1 2
j
=0
7 and the corresponding vector (^ j ;n (f ))Nn ? is the DCT{II (Nj ) of ( j i.e. NX ? j ; l (f ) cos (2lN+ 1)n : ^j ;n(f ) = =0
j
+1
1
j
+1
;l
+1 2 +1
(f ))Nl ? , j
1
=0
1
+1 2 +1
l=0
j +1
In the following, we derive some important properties of the subspaces Sj; ('). We characterize Sj; (') ( 2 f0; 1g) by the Chebyshev transform:
Lemma 3.2 Let j 2 N , 2 f0; 1g and '; f 2 Lw (I ) be given. (i) Then f 2 Sj; (') if and only if there exist ^j ;n (f ) 2 R (n 2 N ) with ^j ;n(f ) = ^ j ;N +2 n (f ) (n 2 N ) ; ^ j ;N +1n (f ) = (?1) ^j ;n (f ) (n = 0; : : : ; Nj ) ; 2
0
+1
+1
+1
+1
0
+
j
(3.7)
0
+1
j
such that (3.3) is satis ed. (ii) Let j+1; f 2 Sj; ('). Then Sj; (f ) = Sj; (') if and only if
supp a[j ; f ] = supp a[j ; '] ; where supp a[f ] := fn 2 N : an [f ] 6= 0g is the support of a[f ]. +1
(3.8)
+1
0
Proof: As mentioned before, if f 2 Sj; ('), then (3.7) is satis ed. Since the Chebyshev transform is a linear bijective mapping, the proof of the reversed direction is straightforward. Hence (i) is valid. Now we show (ii) for = 0. 1. If Sj; (f ) = Sj; ('), then ' 2 Sj; (f ). From (i) it follows that supp a['] supp a[f ]. Analogously, by f 2 Sj; (') we nd supp a[f ] supp a[']. Hence we obtain (3.8). 2. Assume that (3.8) is satis ed. We only need to show that ' 2 Sj; (f ). Since f 2 Sj; ('), we have (3.3) with (3.7). By supp a[f ] = supp a['], we conclude that an['] = ^j ;n an[f ] (n 2 N ) with ? if ^ j ;n (f ) 6= 0 ; ^j ;n := 0^j ;n (f ) otherwise , 0
0
0
0
0
0
+1
0
1
+1
+1
+1
for which (3.7) is also satis ed. For = 1, the assertion follows immediately from Sj; (') = Sj; (j ; ') and Sj; (f ) = Sj; (j ; f ). For a further analysis of the shift{invariant subspaces of Lw (I ), we introduce the bracket 1 product of a := (an)1 n and b := (bn )n 2 l . Let for k = 0; : : : ; Nj 1
0
0
+1 1
1
+1 1
2
=0
[a; b]j;k :=
2
=0
1 X m=0
(amN +1 k bmN +1 j
+
j
+1
+ am
k
+
(
N +1 ?k b(m+1)N +1 ?k ) :
+1)
j
j
Observe that [a; b]j;k satis es the symmetry property [a; b]j;N +1 ?l = [a; b]j;l (l = 0; : : : ; Nj ? 1) : We extend the values [a; b]j;k (k = 0; : : : ; Nj ) to an Nj {periodic sequence, i.e. [a; b]j;k = [a; b]j;k N +1 (k 2 N ) : +1
j
+1 +
+1
j
0
(3.9)
8 Then the type I{bracket product of length Nj + 1 is de ned by [a; b]Ij := ([a; b]j;k )Nk 2 j
=0
RN +1 ; j
and the type II{bracket product of length Nj by [a; b]IIj := ([a; b]j;k )Nk ? 2 j
1
=0
RN
j
:
Lemma 3.3 Let j 2 N , ; 2 f0; 1g and '; 2 Lw (I ) be given. Further, let f 2 Sj; ('), g 2 Sj;( ) with an[f ] = ^j ;n (f ) an['] ; an[g] = ^j ;n(g) an [ ] (n 2 N ) be given, where ^ j ;n (f ), ^j ;n (g) 2 R possess the properties (3.7). Then we have 2
0
+1
+1
+1
0
+1
hf; gi =
NX +1 j
k=0
"j
^ j+1;k (f ) ^j+1;k (g) [a[']; a[ ;k
] ]j
+1
;k :
+1
In particular, for = ,
hf; gi =
NX ? j
k=0
"j;k ^j
;k (f ) ^j +1;k (g ) [a[']; a[
+1
] ]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 (I ), we obtain for l; m = 0; : : : ; Nj the relations 2
N X j
hj;l'; j;m i =
k=0
i.e.,
km [a[']; a[ ] ] ; "j;k cos kl cos j;k Nj Nj
(3.10) ( hj;l'; j;m i )Nl;m = C j Dj diag [a[']; a[ ] ]Ij C j : Analogously, for f := j ; l ', g := j ; m we have for l; m = 0; : : : ; Nj ? 1 j
=0
+1 2 +1
hj
;l
+1 2 +1
'; j
; m+1
+1 2
i=
+1 2
NX ?1 j
k=0
+1
+ 1) [a[']; a[ ] ] ; "j;k cos k(2Nl + 1) cos k(2m j;k N j +1
j +1
and thus
i )Nl;m? = C~ Tj D~ j diag [a[']; a[ ] ]IIj C~ j : Corollary 3.4 For j 2 N and '; 2 Lw (I ), we have (i) Sj; (') ? Sj; ( ) ( 2 f0; 1g) if and only if [a[']; a[ ] ]j;k = 0 (k = 0; : : :; Nj ? ) ; (hj
;l
+1 2 +1
'; j 0
; m+1
+1 2
j
1 =0
(3.11)
2
(3.12)
9 (ii) Sj; (') ? Sj; ( ) if and only if [a[']; a[ ] ]j 0
1
;k
+1
(k = 0; : : :; Nj ) :
= 0
+1
For ' 2 Lw (I ), we consider the systems Bj; (') := fj;l' : l = 0; : : : ; Nj g and Bj; (') := fj ; l ' : l = 0; : : :; Nj ? 1g. For Bj; ('), we de ne a special orthonormality criterion. We say that Bj; (') is orthonormal, if the modi ed Gramian matrix ful ls (3.13) ("j;m hj;l'; j;m'i )Nl;m = I j : The system Bj; (') is called orthonormal, if the Gramian matrix satis es N? = I~ j : (hj ; l '; j ; m 'i)l;m Then we obtain the following characterizations for the bases Bj; (') ( 2 f0; 1g) in terms of the bracket products. Lemma 3.5 Let 2 f0; 1g and j 2 N be given. (i) The system Bj; (') is a basis of Sj; (') if and only if for all k = 0; : : : ; Nj ? [a[']; a['] ]j;k > 0 : (3.14) 2
0
+1 2 +1
1
0
0
j
=0
1
+1 2 +1
+1 2
j
+1
1 =0
0
(ii) The system Bj; (') is an orthonormal basis of Sj; (') if and only if [a[']; a[']]j;k = 2=Nj (k = 0; : : : ; Nj ? ):
(3.15)
(iii) If ' satis es (3.14), and if '? 2 Lw (I ) is de ned by an['?] := c^j ;n ('?) an ['] (n 2 N ) with coecients c^j ;n ('?) determined by (3.7) and (n = 0; : : : ; Nj ? ); c^j ;n ('?) := (2=Nj ) = [a[']; a[']]?j;n= then Bj; ('?) is an orthonormal basis of Sj; ('). Proof: Let = 0. The system Bj; (') forms a basis of Sj; (') if and only if the Gramian matrix ( hj;l'; j;m'i )Nl;m is regular. Since C j and Dj are regular, by (3.10) this is the case if and only if diag [a[']; a['] ]Ij is regular, i.e., if and only if (3.14) is satis ed. By de nition, Bj; (') is an orthonormal basis of Sj; (') if and only if ("j;m hj;l'; j;m'i )Nl;m = ( hj;l'; j;m'i )Nl;m Dj = I j : By (3.10) and (2.6), this is true if and only if (3.15) holds. Finally, by verifying [a['?]; a['?]]j;k = 2=Nj (k = 0; : : : ; Nj ) ; we see that by construction Bj; ('?) is an orthonormal basis of Sj; ('?). By Lemma 3.2, the de nition of '? implies that Sj; ('?) = Sj; ('). 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 Pj; ( = 0; 1) of Lw (I ) onto Sj; ('). 2
+1
0
+1
1 2
1 2
+1
0
0
j
=0
0
0
j
j
=0
=0
0
0
0
0
2
10
Lemma 3.6 Let j 2 N , 2 f0; 1g and let ' 2 Lw (I ) with (3.14) be given. Then for f 2 Lw (I ) we have an[Pj; f ] = c^j ;n(Pj; f ) an ['] (n 2 N ) ; 2
0
2
+1
0
where the coecients c^j+1;n (Pj; f ) satisfy the relations (3.7) and
c^j ;l(Pj; f ) := [[aa[['f ]];; aa[['']]]]j;l (l = 0; : : : ; Nj ? ): (3.16) j;l The projector Pj; is shift{invariant of level j , i.e., for all f 2 Lw (I ) and k = 0; : : : ; Nj ? +1
2
j;k (Pj; f ) = Pj; (j;k f ) :
Proof: We show the assertion only for = 0. For f 2 Lw (I ), the orthogonal projection Pj; f 2 Sj; (') is determined by f ? Pj; f ? Sj; ('). Then there are coecients c^j ;n (Pj; f ) (n 2 N ) satisfying the properties (3.7) of symmetry and periodicity with an[Pj; f ] = c^j ;n(Pj; f ) an ['] (n 2 N ): 2
0
+1
0
0
0
0
0
0
+1
0
0
Using Lemma 3.3, we obtain for all l = 0 : : : ; Nj 0 = hf ? Pj; f; j;l'i N X "j;k cos kl [a[f ? Pj; f ]; a[']]j;k = N j k 0
j
0
=0
=
N X j
k=0
"j;k cos lk N ([a[f ]; a[']]j;k ? c^j
Hence the coecients c^j
j
;k (Pj;0 f ) [a['];
+1
;k (Pj;0 f ) satisfy (3.16).
a[']]j;k) :
The shift{invariance of Pj; follows from an[j;l(Pj; f )] = c^j ;n(Pj; f ) an ['] cos ln Nj = c^j ;n(Pj; (j;lf )) an ['] = an[Pj; (j;lf )] (n 2 N ): +1
0
+1
0
+1
0
0
0
0
4 Multiresolution of L2w (I )
We form shift{invariant subspaces Vj := Sj; ('j ) with 'j 2 Lw (I ) for each level j 2 N . The sequence of subspaces Vj (j 2 N ) is called a nonstationary multiresolution of Lw (I ), if the following three conditions are satis ed: (M1) Vj Vj (j 2 N ): 0
0
+1
(M2) clos
1 S j =0
!
0
Vj = Lw (I ): 2
2
2
0
11 (M3) The systems Bj; ((Nj =2) = 'j ) (j 2 N ) are Lw (I ){stable, i.e., there exist positive constants ; independent of j such that for all j 2 N and for any ( j;n)Nn 2 RN , 1 2
0
2
0
j
0
j+1
N X j
n=0
"j;n j;n 2
N
2
X
"j;n j;n (Nj =2)1=2 j;n'j
j
n=0
=0
N X j
n=0
"j;n j;n:
(4.1)
2
By (M3), Bj; ((Nj =2) = 'j ) is a basis of Vj . Note that dim Vj = Nj + 1. The shift{ invariant subspace Vj is called sample space of level j . The function 'j of Vj is said to be the scaling function of Vj . If all systems Bj; ((Nj =2) = 'j ) are orthonormal bases of Vj (j 2 N ) in the sense of (3.13), then we say that (Nj =2) = 'j (j 2 N ) are orthonormal scaling functions. In this case the constants in condition (M3) are = = 1. Concerning (M2), we observe the following Theorem 4.1 Let fVj g1j be a nested sequence of shift{invariant subspaces Vj := Sj; ('j ) with 'j 2 Lw (I ), i.e., (M1) is valid. Then the condition (M2) is satis ed if and only if 0
1 2
1 2
0
1 2
0
2
0
0
=0
1 [
j =0
supp a['j ] =
N0 :
(4.2)
Proof: 1. Suppose that (4.2) is not satis ed. Then there is a number n 2 0
N0
n
1 [ j =0
supp a['j ]
such that for the Chebyshev polynomial Tn0 it holds that
Tn0 ? clos
1 [ j =0
!
Vj :
Thus, (M2) is not satis ed. 2. Assume that (4.2) holds. By (M1) and Lemma 3.2, we have supp a['j ] supp a['j ] (j 2 N ): Suppose that there exists f 2 Lw (I ) (f 6= 0) with +1
(4.3)
0
2
f ? clos
1 [ j =0
!
Vj :
(4.4)
By k 2 N , we denote an index for which jak0 [f ]j = max fjak [f ]j : k 2 N g > 0 : By (4.2) { (4.3) we conclude that there is an index j 2 N such that k 2 supp a['j0 ] and Nj0 k . Since 'j0 2 Vj for all j j , we nd that f ? Sj; ('j0 ) (j j ). Hence, for j j , we have by (3.12) [a[f ]; a['j0 ]]j;k0 = 0 ; 0
0
0
0
0
0
0
0
0
0
0
12 i.e.,
ak0 [f ] ak0 ['j0 ] + aN +1 ?k0 [f ] aN +1?k0 ['j0 ] j
+
1 P n=1
(ak0
j
nN +1 [f ] ak0
+
j
nN +1 ['j0 ] + a n
+
and choose j j such that 1
( +1)
j
Put
N +1 ?k0 [f ] a n
N +1 ?k0 ['j0 ]) = 0 :
( +1)
j
(4.5)
j
:= jak0 [f ] ak0 ['j0 ]j > 0 ; 0
0
X nN 1
jan[f ] an['j0 ]j =2:
(4.6)
0
j
This choice of j is possible, since by Cauchy{Schwarz inequality 1
1 X n=1
jan[f ] an['j0 ]j ka[f ]kl2 ka['j0 ]kl2 < 1:
But (4.6) contradicts equation (4.5) for j = j . This implies that f = 0, i.e., (M2) is satis ed. 1
Theorem 4.2 The system fBj; ((Nj =2) = 'j ) : j 2 N g is Lw (I ){stable with positive constants ; independent of j if and only if for all k = 0; : : : ; Nj and for all j 2 N 1 2
0
2
0
0
N 4j [a['j ]; a['j ]]j;k : 2
(4.7)
Proof: 1. From Lemma 3.3, it follows that for j 2 N and ( j;k )Nk 2 RN j
0
N
2
X
1=2 "j;k j;k (Nj =2) j;k 'j
j
k=0
with
N X j
^j;n :=
k=0
By (2.10) { (2.11) we have
N X j
k=0
N X
= N2j
j
n=0
"j;k j;k cos kn N j
"j;k j;k 2
=0
j +1
"j;n ^ j;n [a['j ]; a['j ]]j;n 2
(n 2 N ) : 0
N X 2 "j;n ^ j;n : = N jn j
2
=0
With the considerations above, (4.1) reads as follows
N X j
n=0
"j;n ^j;n 2
N 4j
2
N X j
n=0
"j;n ^j;n [a['j ]; a['j ]]j;n 2
N X j
n=0
"j;n ^j;n ; 2
13 with arbitrary (^ j;n )Nn 2 RN
and j 2 N , which is equivalent to (4.7).
j +1
j
=0
0
In the following, we assume that (M1) { (M3) are satis ed. From (M1) it follows 'j 2 Vj , i.e., there exist unique coecients j ;k ('j ) 2 R (k = 0; : : : ; Nj ) such that +1
+1
NX +1 j
'j =
k=0
"j
+1
;k j +1;k ('j ) j +1;k 'j +1 :
+1
This is the so{called two{scale relation or re nement equation of 'j . The Chebyshev transformed two{scale relation of 'j reads an['j ] = Aj (n) an ['j ] (n 2 N ) (4.8) with the two{scale symbol or re nement mask of 'j +1
Aj (n) := +1
NX +1 j
k=0
"j
+1
0
cos kn Nj
;k j +1;k ('j )
+1
(n 2 N ): 0
+1
By de nition we obtain the relations of periodicity and symmetry for all n 2 N and l = 0; : : : ; Nj ? 1, Aj (n) = Aj (Nj + n) ; Aj (Nj ? l) = Aj (l) : (4.9) If a scaling function 'j (j 2 N ) satisfying (4.7) is given, then an orthonormal basis Bj; ((Nj =2) = '?j) (j 2 N ) can be easily obtained by Lemma 3.5, (iii). Let '?j (j 2 N ) be de ned by its Chebyshev coecients Nj a ['?] := ([a[' ]; a[' ]] )? = a [' ] (n 2 N ): j j j;n n j 2 n j Then Bj; ((Nj =2) = '?j) is an orthonormal basis of Vj = Sj; ('j ). The two{scale symbol A?j satisfying an['?j] = A?j (n) an['?j ] (n 2 N ) is connected with Aj by 0
+2
+1
+1
+1
+2
+1
0
1 2
0
+2
0
0
1 2
0
1 2
0
0
+1
+1
0
+1
+1
A?j+1(n)
:= 2 [a['[aj ['];]a; a['['j ]]]]j j j j;n +1
+1
;n
+1
1=2
(n 2 N ):
Aj (n) +1
0
The following connection between the bracket product [a['j ]; a['j ]]Ij and the two{scale symbol Aj can be observed: Lemma 4.3 For j 2 N and k = 0; : : : ; Nj , we have [a['j ]; a['j ]]j;k = Aj (k) [a['j ]; a['j ]]j ;k + Aj (Nj ? k) [a['j ]; a['j ]]j ;N +1?k : In particular, if (Nj =2) = '?j is an orthonormal scaling function and if A?j is the two{ scale symbol of '?j, then (4.10) A?j (k) + A?j (Nj ? k) = 4 (k = 0; : : : ; Nj ): +1
0
2
+1
+1
+1
+1
1 2
+1
2
+1
2
+1
+1
+1
+1
j
+1
+1
+1
2
14
Proof: By the de nition of the bracket product and by (4.8) { (4.9), we obtain for k = 0; : : :; Nj
[a['j ]; a['j ]]j;k 1 X
(anN +2 k ['j ] + a n
=
j
2
+
n=0 Aj+1(k)2 [a['j+1];
=
N +2 ?k ['j ]
( +1)
j
a['j ]]j +1
2
+ anN +2 j
N +1 +k ['j ]
+
2
j
;k + Aj +1 (Nj +1 ? k )
+1
2
+ anN +2 j
N +1 ?k ['j ]
[a['j ]; a['j ]]j +1
2
+
+1
j
)
;N +1 ?k :
+1
j
For orthonormal scaling functions, the assertion follows by Lemma 3.5, (ii).
5 Wavelet Spaces
Let the wavelet space Wj of level j (j 2 N ) be de ned as the orthogonal complement of Vj in Vj , i.e. Wj := Vj Vj (j 2 N ): Then it follows dim Wj = (Nj +1) ? (Nj +1) = Nj . By de nition, the wavelet spaces Wj (j 2 N ) are orthogonal. By (M1) { (M2), we obtain the orthogonal sum decomposition 0
+1
+1
0
+1
0
Lw (I ) = V 2
0
1 M j =0
Wj :
Further, Wj can be characterized by the orthogonal projector Pj; of Lw (I ) onto Vj , namely by Wj = ff ? Pj; f : f 2 Vj g: The subspace Wj is shift{invariant of level j , since by Lemma 3.6 we have for g := f ?Pj; f (f 2 Vj ), j;lg = j;lf ? j;l(Pj; f ) = j;lf ? Pj; (j;lf ) 2 Wj : Assume that the shift{invariant subspace Wj can be of type 1 generated by a function j 2 Vj such that Wj = Sj; ( j ). Further, we suppose that the set Bj; ((Nj =2) = j ) = f(Nj =2) = j ; l j : l = 0; : : : ; Nj ? 1g is Lw (I ){stable, i.e., there are constants 0
Nj : Then it holds that N Nj ' = X "j;k Tk 2 N : 2 j 0
j
j
k=0
By (7.1), the corresponding bracket product reads as follows 0; : : :; Nj ? 1 ; Nj [a['j ]; a['j ]]j;k = 42 kk = = Nj : Using (2.8), we obtain the following interpolation property of 'j
(7.2)
2
N X 2 "j;k cos kl 'j (hj;l ) = j;l'j (1) = N Nj = 2 ;l (l = 0; : : : ; Nj ) : j k j
(7.3)
0
=0
By (7.1), the shifted scaling functions j;k 'j (k = 0; : : : ; Nj ) are contained in N . Further, these functions j;k 'j (k = 0; : : : ; Nj ) are modi ed Lagrange fundamental polynomials with respect to the Gauss{Chebyshev nodes hj;l (l = 0; : : : ; Nj ), since for k; l = 0; : : :; Nj from Lemma 3.1, (ii) and (3.1) it follows j
j;k 'j (hj;l) = (j;l j;k 'j )(1) = (j;l k 'j (1) + j;jl?kj 'j (1)) = "?j;k k;l : Figure 1 shows the scaling function ' , and Figure 2 presents the shifted function ; ' . The function j;k 'j (k = 0; : : : ; Nj ) is supported on the whole interval I , and has signi cant values in a small neighbourhood of hj;k , if j is large enough. 1 2
1
5
+
5 16
5
21 2
1
0.8 1.5
0.6 1 0.4
0.5 0.2
-1
00
-0.5
1
0.5 x
-1
00
-0.5
1
0.5 x
-0.2
Fig. 1. Scaling function ' .
Fig. 2. Shifted scaling function ; ' .
5
5 16
5
Let Vj := Sj; ('j ) be the sample space of level j . Consequently by Lemma 3.5, (i), the polynomials j;k 'j (k = 0; : : : ; Nj ) form a basis of Vj , i.e., Vj = N ; dim Vj = Nj + 1 : Note that the operator Lj : C (I ) ! Vj de ned by 0
j
N X j
Lj f :=
k=0
(f 2 C (I ))
"j;k f (hj;k ) j;k 'j
is an interpolation operator, which maps C (I ) onto Vj with the property Lj f (hj;l ) = f (hj;l) (l = 0; : : : ; Nj ) : All sample spaces Vj (j 2 N ) form a multiresolution of Lw (I ), where (M3) reads as follows: The systems Bj; ((Nj =2) = 'j ) (j 2 N ) are Lw (I ){stable with optimal constants = 1=2 and = 1, i.e., for all j 2 N and for any ( j;k )Nk 2 RN we have the sharp estimate N N N
X X 1X
= 2 k "j;k j;k k "j;k j;k (Nj =2) j;k 'j k "j;k j;k : 0
0
2
1 2
2
0
j
=0
j
2
1 2
=0
j+1
j
0
j
2
2
=0
=0
Using (7.1), we nd the Chebyshev transformed two{scale relation of 'j an['j ] = Aj (n) an ['j ] (n 2 N ) with the corresponding two{scale symbol 8 < 2 n = 0; : : :; Nj ? 1 ; Aj (n) := : 1 n = Nj ; 0 n = Nj + 1; : : : ; Nj : Let Wj := Vj Vj be the wavelet space of level j . Thus, dim Wj = Nj . Consider the polynomials j 2 Vj (j 2 N ) given by their Chebyshev coecients 8 < 2 n = Nj + 1; : : :; Nj ? 1 ; Nj an[ j ] := : 1 n = Nj ; (7.4) 0 otherwise : +1
+1
0
+1
+1
+1
+1
0
+1
+1
22 2
1
0.8
1.5
0.6 1 0.4 0.5
-1
0.2
00
-0.5
-1
1
0.5 x
00
-0.5
1
0.5 x
-0.2 -0.5 -0.4 -1
-0.6
-0.8
-1.5
Fig. 3. Wavelet .
Fig. 4. Shifted wavelet :
5
6 33
5
Then the corresponding bracket product reads as follows 8 < 0 k = 0; : : : ; Nj ; Nj [a[ j ]; a[ j ]]j ;k = : 4 k = Nj + 1; : : :; Nj ? 1 ; 2 k = Nj : The shifted polynomials j ; r j satisfy the interpolation properties 2
+1
.
(7.5)
+1
+1
+1 2 +1
j
;r
+1 2 +1
j (hj +1;2s+1 )
(r; s = 0; : : : ; Nj ? 1) :
= r;s
Figure 3 shows the wavelet , and Figure 4 presents the shifted wavelet ; 5
6 33
5
.
The wavelet space Wj is a shift{invariant subspace of Lw (I ) of type 1 generated by j . The systems Bj; ((Nj =2) = j ) (j 2 N ) are Lw (I ){stable with optimal constants = 1=2 and = 1, i.e., for all j 2 N and for any ( j;r)rN ? 2 RN , we have the sharp estimate 2
1 2
1
0
2
j
0
1
? ?
NX 1 NX
= 2 r j;r r j;r (Nj =2) j j
j
=0
1
j
1
2
1 2
=0
;r
+1 2 +1
2
j
=0
NX ?1 j
r=0
j;r : 2
Using (7.1) and (7.4), we obtain the Chebyshev transformed two{scale relation of
an[ j ] = Bj (n) an ['j ] +1
+1
j
(n 2 N ) 0
with the corresponding two{scale symbol 0; : : : ; N j ; Bj (n) := 02 nn = = Nj + 1; : : :; Nj : +1
+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 2j + 1 function values of fj 2 Vj are given. The decomposition algorithm for linear spline wavelets in [0; 1] needs 6 2j real multiplications in order to compute all wavelet coecients of gj 2 Wj . For the same problem, the decomposition algorithm for cubic spline wavelets in [13] can be implemented using 14 2j real multiplications. Compared +1
+1
+1 +1
+1
23 30
25
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
20
15
10
5 2
4
6
8
10
12
14
16
18
20
j
Fig. 5. Comparison of needed real multiplications (divided by 2j ) for a complete +1
decomposition.
to that, our algorithm 6.2 requires fewer real multiplications up to the level j = 14. { Now we consider the complete decomposition of fj 2 Vj . Here we have to determine all coecients of the related functions in Wj ; Wj? ; : : :; W and V . Figure 5 shows the numbers of needed real multiplications (divided by 2j ) 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 2 f7; : : :; 11g is often used in praxis, our algorithm is an interesting alternative to the method in [13]. +1
1
+1
+1
3
3
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 denote the cardinal linear B{spline. Interpolating the function f (x) := B (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 (4x + 4) (x 2 I ) shows that f has singularities at ?0:75 and ?0:5, but not at ?1 (see Figure 7). 2
2
2
We can generalize this example of polynomial wavelets in a similar manner as done for periodic functions in [14]. Set Nj = d 2j (j 2 N ) with xed d 2 N. Further let for xed 0
24 1
0.002
0
0 -0.003 -1
0
1
-1
0
1
0
1
Fig. 6. Sample part in V and wavelet part W of f (x) = N (4x + 3:96). 6
6
1
2
0.0015
0
0 -0.003 -1
0
1
-1
Fig. 7. Sample part in V and wavelet part W of f (x) = N (4x + 4). 6
2N
6
2
1 j ; j ? 2 j > ; where 3 2d is ful lled. Then, Nj + Mj Nj ? Mj . Let the scaling function 'j of level j be given by its Chebyshev coecients 0
Mj :=
+1
Nj an['j ] :=
8 >
:
0
j
j
+1
0 n N j ? Mj ; Nj ? Mj < n < Nj + Mj ; n N j + Mj :
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 Vj = Sj; ('j ) can be described by n o Vj = N ?M span MM +1k TN ?k + MM ?+1k TN k : k = 0; : : : ; Mj ? 1 ; i.e., N ?M Vj N M ? . The corresponding wavelet space Wj (j 2 N ) is of type 1 generated by the polynomial j := 2 'j ? 'j 2 Vj such that 0
j+
j
j
j
j
j+
j
j
j
j
j
1
0
+1
8 n?N +M > > M > > > > M +1 > : j
0
j+
j
j
+1
N j ? Mj < n < N j + M j ; N j + Mj n N j ? Mj ; N j ? Mj < n < N j + M j ; otherwise: +1
+1
+1
+1
+1
+1
25 The shifted polynomials j
j
;r
+1 2 +1
;r
also satisfy the interpolation properties
j
+1 2 +1
j (hj +1;2s+1 )
(r; s = 0; : : : ; Nj ? 1) :
= r;s
For the Chebyshev transformed two{scale relations of 'j and j , we obtain the two{scale symbols 8 0 n N j ? Mj ;
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 re ected by large wavelet coecients for high levels near the endpoints 0 and 1, even if f is smooth on [0; 1]. Thus the regularity of f is not characterized by the decay of wavelet 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 [0; 1], then the derivative of f~ has discontinuities at the integers. The smoothness 1
2 of f is again not characterized by the decay of wavelet 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 Lw (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 to multiresolution of Lw (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 (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 lead to a uni ed 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 Sa F := 21 (F ( + a) + F ( ? a)) (a 2 R) for F 2 L ; . The isomorphism between L ; and Lw (I ) can be exploited in order to construct new sample and wavelet spaces in Lw (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 brie y introduce the Chebyshev transform, related shifts and the DCT. In Section 3 we analyze shift{invariant subspaces of Lw (I ). The scalar product of functions from shift{invariant subspaces can be simpli ed to a nite sum by means of the so{called bracket product. In Section 4 we consider a nonstationary multiresolution of Lw (I ) consisting of shift{invariant subspaces Vj (j 2 N ) generated by shifts of scaling functions 'j . The required conditions for the multiresolution of Lw (I ) and their consequences for the scaling functions 'j are analyzed in detail. In Section 5 we introduce the wavelet space Wj (j 2 N ) as the orthogonal complement of Vj in Vj . Then Wj 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 , the wavelet j is characterized in Theorem 5.3. Section 6 provides fast, numerically stable decomposition 2
2
2
2 2
2 2
2 2
2 2
0
0
0
2
2
2
2
0
2
0
+1
+1
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. Note that the transformed spline wavelets are supported on small subintervals of I . The examples show that periodic multiresolutions of L with even scaling functions 'j can be transformed into a multiresolution of Lw (I ). 2 2
2
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 )? = for x 2 (?1; 1). Let Lw (I ) be the weighted Hilbert space of all measurable functions f : I ! R with the property 2
Z
1 2
2
w(y) f (y) dy < 1 : 2
I
For f; g 2 Lw (I ), the corresponding inner product and norm are de ned by Z 2 hf; gi := w(y) f (y) g(y) dy ; kf k := hf; f i = : 2
1 2
I
Let l denote the Hilbert space of all real, square summable sequences a = (an)1n , b = (bn)1n with the weighted inner product and norm given by 2
=0
=0
1 X (a; b)l2 := 21 a b + anbn ; n
kakl2 := (a; a)l2= : 1 2
0 0
=1
Let C (I ) be the set of all continuous functions f : I ! R. By n (n 2 N ) we denote the set of all real polynomials of degree at most n restricted on I . As known, the Chebyshev polynomials Tn := cos (n arccos) 2 n (n 2 N ) form a complete orthogonal system in Lw (I ). Note that arccos : I ! [0; ] is the inverse function of cos restricted on [0; ]. For m; n 2 N we have 8 < 2 m = n = 0; hTm; Tni = : 1 m = n > 0; 0 m 6= n: Further, we use the Chebyshev transform of Lw (I ) into l mapping f 2 Lw (I ) into a[f ] := (an[f ])1 n 2 l with the Chebyshev coecients an[f ] := hf; Tni (n 2 N ) : Then for f; g 2 Lw (I ), we have the Parseval identities (2.1) hf; gi = (a[f ]; a[g])l2 ; kf k = ka[f ]kl2 : Note that the Chebyshev transform is a linear bijective mapping of Lw (I ) onto l . For more details on the Chebyshev transform see [2, 10]. 0
0
2
0
2
=0
2
2
2
0
2
2
2
4 The Chebyshev transform is strongly related with the Fourier cosine transform. Let L be the Hilbert space of all 2{periodic, square integrable functions F; G : R ! R with the inner product Z 1 (F; G) := 2 F (s) G(s) ds : 2 2
2
?
Let L ; be the subspace of all even functions of L . For a given function f 2 Lw (I ), the cos{transformed function F := f (cos) 2 L ; has the Fourier expansion 2 2
0
2 2
2 2
2
0
1 X F = 21 a (F ) + an (F ) cos(n ) n
(2.2)
0
=1
with the Fourier cosine coecients
Z 2 an(F ) := F (s) cos(ns) ds (n 2 N ) :
(2.3)
0
0
In order to adapt the concept of shifts to the interval I , we consider the even shift SaF of F 2 L ; by a 2 R, which is de ned as the even part of the translated function F ( ? a), i.e. SaF := 12 (F ( + a) + F ( ? a)) 2 L ; : (2.4) Observe that for n 2 N Sa cos(n ) = cos(na) cos(n ) ; an(SaF ) = cos(na) an(F ) : Restricting F = f (cos) on [0; ], the arccos{transformed function F (arccos) coincides with f 2 Lw (I ). From (2.2) { (2.3) it follows directly the Chebyshev expansion 2 2
0
2 2
0
0
2
1 X 1 f = 2 a [f ] + an[f ] Tn ; n 0
an[f ] = an(f (cos))
(n 2 N ) : 0
=1
Further, the even shift Sa of F = f (cos) (a 2 R) goes into the Chebyshev shift shf of f with h := cos a 2 I , i.e. (2.5) (sh f )(x) := 21 f (xh ? v(x)v(h)) + 12 f (xh + v(x)v(h)) (x 2 I ) with v(x) := (1 ? x ) = (x 2 I ). For the realization of the Chebyshev transform in nite dimensional subspaces of Lw (I ), we will use fast algorithms of the discrete cosine transform (DCT). In the following, we brie y introduce the di erent types of DCT. Let Nj := d 2j , where j 2 N stands for the level and d 2 N is a constant depending on the application. Further, let k;l be the Kronecker symbol and "j; = "j;N := 2? , "j;k := 1 (k = 1; : : : ; Nj ? 1). We introduce the matrices 2 1 2
2
0
1
0
j
N Dj := diag ("j;k )Nk ; I j := (k;l)Nk;l ; C j := ( cos kl N )k;l ; C~ j := ( cos Ns +1r )Nr;s ? ; D~ j := diag ("j;s)Ns ? ; I~j := (r;s)Nr;s ? ; j
=0
=0
=0
(2 +1)
j
j
j
j
1 =0
j
j
=0
1
1 =0
j
5 which ful l the relations
This follows from N X
C j Dj C j Dj = N2j I j ; C~ Tj D~ j C~ j = C~ j C~ Tj D~ j = N2j I~j :
(2.6) (2.7)
Nj u 0 mod Nj ; (2.8) 0 otherwise ; k 8 NX ? u 0 mod Nj ; < Nj (2 k + 1) u cos N = : ?Nj u Nj mod Nj ; (2.9) j k 0 otherwise (cf. [16]). For further development, we de ne some variants of the DCT. The type I{DCT of length Nj + 1 (DCT{I (Nj + 1)) is a mapping of RN into itself de ned by x^I := C Ij x (x 2 RN ) (2.10) with C Ij := C j Dj . By Nj (C Ij )? = 2 C Ij , this mapping is bijective. Note that (2.6) and (2.10) imply (x^I)TDj x^I = xT(C Ij )TDj C Ij x = xTDj C j Dj C j Dj x = N2j xTDj x : (2.11) The type II{DCT of length Nj (DCT{II (Nj )) is a mapping of RN into itself de ned by y~II := C IIj y (y 2 RN ) (2.12) j
=0
j
"j;k cos ku Nj =
+1
1
=0
+2
+1
+1
+2
j+1
j +1
1
j
j
with C IIj := C~ j . Then by (2.7) and (2.12), we obtain
(2.13) (y~II)T D~ j y~II = y T C~ Tj D~ j C~ j y = N2j y T y : The type III{DCT of length Nj (DCT{III (Nj )) is a mapping of RN into itself de ned by j
y~III := C~ IIIj y
(y 2 RN ) j
with C IIIj := C~ Tj D~ j . By (2.7), the inverse of the DCT{II (Nj ) is the mapping (2=Nj ) DCT{III (Nj ). Fast and numerically stable algorithms for the DCT{I (Nj + 1), DCT{II (Nj ) and DCT{III (Nj ), which work in real arithmetic, are described in [1, 15].
3 Shift{Invariant Subspaces
Using the Gauss{Chebyshev nodes hj;u := cos (u=Nj ) (u 2 Z) of level j (j 2 N ) and the Chebyshev shift (2.5), we obtain the shifts of level j j;u := sh (u 2 Z) ; which possess the following properties (see [2, 16]): 0
j;u
6
Lemma 3.1 For j 2 N , u; v 2 Z and f; g 2 Lw (I ) we have 2
0
(i) j;u N +1 = j;u = j ; u ; (ii) 2 j;u j;v = 2 j;v j;u = j;u v + j;u?v ; (iii) hj;uf; gi = hf; j;ugi ; (iv) j;u Tn = cos (nu=Nj ) Tn ; an[j;uf ] = cos (nu=Nj ) an[f ] (n 2 N ) ; (v) j;uf 2 n for f 2 n (n 2 N ) . Note that j; f = f and j;N f = f (? ) for f 2 Lw (I ). Further, for f 2 C (I ) we have (j;u f )(1) = f (hj;u ) (u 2 Z) : (3.1) A linear subspace S of Lw (I ) is called shift{invariant of level j (j 2 N ), if for each f 2 S all shifted functions j;l f (l = 0; : : : ; Nj ) are contained in S . The shift{invariant subspace of level j Sj; (') := span fj;l' : l = 0; : : :; Nj g is said to be of type 0 generated by ' 2 Lw (I ). The shift{invariant subspace of level j Sj; (') := span fj ; l ' : l = 0; : : : ; Nj ? 1g is said to be of type 1 generated by ' 2 Lw (I ). It is obvious by Lemma 3.1, (i) { (ii) that Sj; (') Sj ; (') and Sj; (') = Sj; (j ; ') Sj ; ('). By de nition, f 2 Sj ; (') can be represented in the form +
+1 2
j
+
0
0
2
0
j
2
0
0
2
1
+1 2 +1 2
0
+1 0
1
f =
NX +1 j
k=0
0
"j
+1 1
+1 0
;k j +1;k (f ) j +1;k '
+1
( j
+1 0
;k (f ) 2 R) :
(3.2)
+1
Using Lemma 3.1, (iv) and Chebyshev transform, we obtain the Chebyshev coecients an[f ] = ^j ;n (f ) an ['] (n 2 N ) (3.3) with NX +1 "j ;k j ;k (f ) cos Nkn : ^j ;n(f ) := (3.4) +1
0
j
+1
k=0
+1
+1
j +1
Observe that (^ j ;n (f ))nN +1 is the DCT{I (Nj + 1) of ( j ;k (f ))kN +1 and that the following properties of periodicity and symmetry hold for n 2 N and k = 0; : : : ; Nj ? 1 ^j ;n(f ) = ^j ;N +2 n(f ) ; ^j ;k (f ) = ^j;N +2 ?k (f ) : In particular, for f 2 Sj; (') we get the representation (3.2) with j ; l (f ) := 0 (l = 0; : : :; Nj ? 1) : (3.5) j
+1
+1
=0
j
+1
=0
0
+1
+1
j
+
+1
+1
j
0
+1 2 +1
Then it follows that the vector (^ j ;n(f ))Nn with components (3.4) is the DCT{I (Nj +1) of ( j ; l(f ))Nl . For f 2 Sj; (') we obtain (3.2) with j ; l(f ) := 0 (l = 0; : : :; Nj ) ; (3.6) +1
+1 2
j
=0
1
+1 2
j
=0
7 and the corresponding vector (^ j ;n (f ))Nn ? is the DCT{II (Nj ) of ( j i.e. NX ? j ; l (f ) cos (2lN+ 1)n : ^j ;n(f ) = =0
j
+1
1
j
+1
;l
+1 2 +1
(f ))Nl ? , j
1
=0
1
+1 2 +1
l=0
j +1
In the following, we derive some important properties of the subspaces Sj; ('). We characterize Sj; (') ( 2 f0; 1g) by the Chebyshev transform:
Lemma 3.2 Let j 2 N , 2 f0; 1g and '; f 2 Lw (I ) be given. (i) Then f 2 Sj; (') if and only if there exist ^j ;n (f ) 2 R (n 2 N ) with ^j ;n(f ) = ^ j ;N +2 n (f ) (n 2 N ) ; ^ j ;N +1n (f ) = (?1) ^j ;n (f ) (n = 0; : : : ; Nj ) ; 2
0
+1
+1
+1
+1
0
+
j
(3.7)
0
+1
j
such that (3.3) is satis ed. (ii) Let j+1; f 2 Sj; ('). Then Sj; (f ) = Sj; (') if and only if
supp a[j ; f ] = supp a[j ; '] ; where supp a[f ] := fn 2 N : an [f ] 6= 0g is the support of a[f ]. +1
(3.8)
+1
0
Proof: As mentioned before, if f 2 Sj; ('), then (3.7) is satis ed. Since the Chebyshev transform is a linear bijective mapping, the proof of the reversed direction is straightforward. Hence (i) is valid. Now we show (ii) for = 0. 1. If Sj; (f ) = Sj; ('), then ' 2 Sj; (f ). From (i) it follows that supp a['] supp a[f ]. Analogously, by f 2 Sj; (') we nd supp a[f ] supp a[']. Hence we obtain (3.8). 2. Assume that (3.8) is satis ed. We only need to show that ' 2 Sj; (f ). Since f 2 Sj; ('), we have (3.3) with (3.7). By supp a[f ] = supp a['], we conclude that an['] = ^j ;n an[f ] (n 2 N ) with ? if ^ j ;n (f ) 6= 0 ; ^j ;n := 0^j ;n (f ) otherwise , 0
0
0
0
0
0
+1
0
1
+1
+1
+1
for which (3.7) is also satis ed. For = 1, the assertion follows immediately from Sj; (') = Sj; (j ; ') and Sj; (f ) = Sj; (j ; f ). For a further analysis of the shift{invariant subspaces of Lw (I ), we introduce the bracket 1 product of a := (an)1 n and b := (bn )n 2 l . Let for k = 0; : : : ; Nj 1
0
0
+1 1
1
+1 1
2
=0
[a; b]j;k :=
2
=0
1 X m=0
(amN +1 k bmN +1 j
+
j
+1
+ am
k
+
(
N +1 ?k b(m+1)N +1 ?k ) :
+1)
j
j
Observe that [a; b]j;k satis es the symmetry property [a; b]j;N +1 ?l = [a; b]j;l (l = 0; : : : ; Nj ? 1) : We extend the values [a; b]j;k (k = 0; : : : ; Nj ) to an Nj {periodic sequence, i.e. [a; b]j;k = [a; b]j;k N +1 (k 2 N ) : +1
j
+1 +
+1
j
0
(3.9)
8 Then the type I{bracket product of length Nj + 1 is de ned by [a; b]Ij := ([a; b]j;k )Nk 2 j
=0
RN +1 ; j
and the type II{bracket product of length Nj by [a; b]IIj := ([a; b]j;k )Nk ? 2 j
1
=0
RN
j
:
Lemma 3.3 Let j 2 N , ; 2 f0; 1g and '; 2 Lw (I ) be given. Further, let f 2 Sj; ('), g 2 Sj;( ) with an[f ] = ^j ;n (f ) an['] ; an[g] = ^j ;n(g) an [ ] (n 2 N ) be given, where ^ j ;n (f ), ^j ;n (g) 2 R possess the properties (3.7). Then we have 2
0
+1
+1
+1
0
+1
hf; gi =
NX +1 j
k=0
"j
^ j+1;k (f ) ^j+1;k (g) [a[']; a[ ;k
] ]j
+1
;k :
+1
In particular, for = ,
hf; gi =
NX ? j
k=0
"j;k ^j
;k (f ) ^j +1;k (g ) [a[']; a[
+1
] ]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 (I ), we obtain for l; m = 0; : : : ; Nj the relations 2
N X j
hj;l'; j;m i =
k=0
i.e.,
km [a[']; a[ ] ] ; "j;k cos kl cos j;k Nj Nj
(3.10) ( hj;l'; j;m i )Nl;m = C j Dj diag [a[']; a[ ] ]Ij C j : Analogously, for f := j ; l ', g := j ; m we have for l; m = 0; : : : ; Nj ? 1 j
=0
+1 2 +1
hj
;l
+1 2 +1
'; j
; m+1
+1 2
i=
+1 2
NX ?1 j
k=0
+1
+ 1) [a[']; a[ ] ] ; "j;k cos k(2Nl + 1) cos k(2m j;k N j +1
j +1
and thus
i )Nl;m? = C~ Tj D~ j diag [a[']; a[ ] ]IIj C~ j : Corollary 3.4 For j 2 N and '; 2 Lw (I ), we have (i) Sj; (') ? Sj; ( ) ( 2 f0; 1g) if and only if [a[']; a[ ] ]j;k = 0 (k = 0; : : :; Nj ? ) ; (hj
;l
+1 2 +1
'; j 0
; m+1
+1 2
j
1 =0
(3.11)
2
(3.12)
9 (ii) Sj; (') ? Sj; ( ) if and only if [a[']; a[ ] ]j 0
1
;k
+1
(k = 0; : : :; Nj ) :
= 0
+1
For ' 2 Lw (I ), we consider the systems Bj; (') := fj;l' : l = 0; : : : ; Nj g and Bj; (') := fj ; l ' : l = 0; : : :; Nj ? 1g. For Bj; ('), we de ne a special orthonormality criterion. We say that Bj; (') is orthonormal, if the modi ed Gramian matrix ful ls (3.13) ("j;m hj;l'; j;m'i )Nl;m = I j : The system Bj; (') is called orthonormal, if the Gramian matrix satis es N? = I~ j : (hj ; l '; j ; m 'i)l;m Then we obtain the following characterizations for the bases Bj; (') ( 2 f0; 1g) in terms of the bracket products. Lemma 3.5 Let 2 f0; 1g and j 2 N be given. (i) The system Bj; (') is a basis of Sj; (') if and only if for all k = 0; : : : ; Nj ? [a[']; a['] ]j;k > 0 : (3.14) 2
0
+1 2 +1
1
0
0
j
=0
1
+1 2 +1
+1 2
j
+1
1 =0
0
(ii) The system Bj; (') is an orthonormal basis of Sj; (') if and only if [a[']; a[']]j;k = 2=Nj (k = 0; : : : ; Nj ? ):
(3.15)
(iii) If ' satis es (3.14), and if '? 2 Lw (I ) is de ned by an['?] := c^j ;n ('?) an ['] (n 2 N ) with coecients c^j ;n ('?) determined by (3.7) and (n = 0; : : : ; Nj ? ); c^j ;n ('?) := (2=Nj ) = [a[']; a[']]?j;n= then Bj; ('?) is an orthonormal basis of Sj; ('). Proof: Let = 0. The system Bj; (') forms a basis of Sj; (') if and only if the Gramian matrix ( hj;l'; j;m'i )Nl;m is regular. Since C j and Dj are regular, by (3.10) this is the case if and only if diag [a[']; a['] ]Ij is regular, i.e., if and only if (3.14) is satis ed. By de nition, Bj; (') is an orthonormal basis of Sj; (') if and only if ("j;m hj;l'; j;m'i )Nl;m = ( hj;l'; j;m'i )Nl;m Dj = I j : By (3.10) and (2.6), this is true if and only if (3.15) holds. Finally, by verifying [a['?]; a['?]]j;k = 2=Nj (k = 0; : : : ; Nj ) ; we see that by construction Bj; ('?) is an orthonormal basis of Sj; ('?). By Lemma 3.2, the de nition of '? implies that Sj; ('?) = Sj; ('). 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 Pj; ( = 0; 1) of Lw (I ) onto Sj; ('). 2
+1
0
+1
1 2
1 2
+1
0
0
j
=0
0
0
j
j
=0
=0
0
0
0
0
2
10
Lemma 3.6 Let j 2 N , 2 f0; 1g and let ' 2 Lw (I ) with (3.14) be given. Then for f 2 Lw (I ) we have an[Pj; f ] = c^j ;n(Pj; f ) an ['] (n 2 N ) ; 2
0
2
+1
0
where the coecients c^j+1;n (Pj; f ) satisfy the relations (3.7) and
c^j ;l(Pj; f ) := [[aa[['f ]];; aa[['']]]]j;l (l = 0; : : : ; Nj ? ): (3.16) j;l The projector Pj; is shift{invariant of level j , i.e., for all f 2 Lw (I ) and k = 0; : : : ; Nj ? +1
2
j;k (Pj; f ) = Pj; (j;k f ) :
Proof: We show the assertion only for = 0. For f 2 Lw (I ), the orthogonal projection Pj; f 2 Sj; (') is determined by f ? Pj; f ? Sj; ('). Then there are coecients c^j ;n (Pj; f ) (n 2 N ) satisfying the properties (3.7) of symmetry and periodicity with an[Pj; f ] = c^j ;n(Pj; f ) an ['] (n 2 N ): 2
0
+1
0
0
0
0
0
0
+1
0
0
Using Lemma 3.3, we obtain for all l = 0 : : : ; Nj 0 = hf ? Pj; f; j;l'i N X "j;k cos kl [a[f ? Pj; f ]; a[']]j;k = N j k 0
j
0
=0
=
N X j
k=0
"j;k cos lk N ([a[f ]; a[']]j;k ? c^j
Hence the coecients c^j
j
;k (Pj;0 f ) [a['];
+1
;k (Pj;0 f ) satisfy (3.16).
a[']]j;k) :
The shift{invariance of Pj; follows from an[j;l(Pj; f )] = c^j ;n(Pj; f ) an ['] cos ln Nj = c^j ;n(Pj; (j;lf )) an ['] = an[Pj; (j;lf )] (n 2 N ): +1
0
+1
0
+1
0
0
0
0
4 Multiresolution of L2w (I )
We form shift{invariant subspaces Vj := Sj; ('j ) with 'j 2 Lw (I ) for each level j 2 N . The sequence of subspaces Vj (j 2 N ) is called a nonstationary multiresolution of Lw (I ), if the following three conditions are satis ed: (M1) Vj Vj (j 2 N ): 0
0
+1
(M2) clos
1 S j =0
!
0
Vj = Lw (I ): 2
2
2
0
11 (M3) The systems Bj; ((Nj =2) = 'j ) (j 2 N ) are Lw (I ){stable, i.e., there exist positive constants ; independent of j such that for all j 2 N and for any ( j;n)Nn 2 RN , 1 2
0
2
0
j
0
j+1
N X j
n=0
"j;n j;n 2
N
2
X
"j;n j;n (Nj =2)1=2 j;n'j
j
n=0
=0
N X j
n=0
"j;n j;n:
(4.1)
2
By (M3), Bj; ((Nj =2) = 'j ) is a basis of Vj . Note that dim Vj = Nj + 1. The shift{ invariant subspace Vj is called sample space of level j . The function 'j of Vj is said to be the scaling function of Vj . If all systems Bj; ((Nj =2) = 'j ) are orthonormal bases of Vj (j 2 N ) in the sense of (3.13), then we say that (Nj =2) = 'j (j 2 N ) are orthonormal scaling functions. In this case the constants in condition (M3) are = = 1. Concerning (M2), we observe the following Theorem 4.1 Let fVj g1j be a nested sequence of shift{invariant subspaces Vj := Sj; ('j ) with 'j 2 Lw (I ), i.e., (M1) is valid. Then the condition (M2) is satis ed if and only if 0
1 2
1 2
0
1 2
0
2
0
0
=0
1 [
j =0
supp a['j ] =
N0 :
(4.2)
Proof: 1. Suppose that (4.2) is not satis ed. Then there is a number n 2 0
N0
n
1 [ j =0
supp a['j ]
such that for the Chebyshev polynomial Tn0 it holds that
Tn0 ? clos
1 [ j =0
!
Vj :
Thus, (M2) is not satis ed. 2. Assume that (4.2) holds. By (M1) and Lemma 3.2, we have supp a['j ] supp a['j ] (j 2 N ): Suppose that there exists f 2 Lw (I ) (f 6= 0) with +1
(4.3)
0
2
f ? clos
1 [ j =0
!
Vj :
(4.4)
By k 2 N , we denote an index for which jak0 [f ]j = max fjak [f ]j : k 2 N g > 0 : By (4.2) { (4.3) we conclude that there is an index j 2 N such that k 2 supp a['j0 ] and Nj0 k . Since 'j0 2 Vj for all j j , we nd that f ? Sj; ('j0 ) (j j ). Hence, for j j , we have by (3.12) [a[f ]; a['j0 ]]j;k0 = 0 ; 0
0
0
0
0
0
0
0
0
0
0
12 i.e.,
ak0 [f ] ak0 ['j0 ] + aN +1 ?k0 [f ] aN +1?k0 ['j0 ] j
+
1 P n=1
(ak0
j
nN +1 [f ] ak0
+
j
nN +1 ['j0 ] + a n
+
and choose j j such that 1
( +1)
j
Put
N +1 ?k0 [f ] a n
N +1 ?k0 ['j0 ]) = 0 :
( +1)
j
(4.5)
j
:= jak0 [f ] ak0 ['j0 ]j > 0 ; 0
0
X nN 1
jan[f ] an['j0 ]j =2:
(4.6)
0
j
This choice of j is possible, since by Cauchy{Schwarz inequality 1
1 X n=1
jan[f ] an['j0 ]j ka[f ]kl2 ka['j0 ]kl2 < 1:
But (4.6) contradicts equation (4.5) for j = j . This implies that f = 0, i.e., (M2) is satis ed. 1
Theorem 4.2 The system fBj; ((Nj =2) = 'j ) : j 2 N g is Lw (I ){stable with positive constants ; independent of j if and only if for all k = 0; : : : ; Nj and for all j 2 N 1 2
0
2
0
0
N 4j [a['j ]; a['j ]]j;k : 2
(4.7)
Proof: 1. From Lemma 3.3, it follows that for j 2 N and ( j;k )Nk 2 RN j
0
N
2
X
1=2 "j;k j;k (Nj =2) j;k 'j
j
k=0
with
N X j
^j;n :=
k=0
By (2.10) { (2.11) we have
N X j
k=0
N X
= N2j
j
n=0
"j;k j;k cos kn N j
"j;k j;k 2
=0
j +1
"j;n ^ j;n [a['j ]; a['j ]]j;n 2
(n 2 N ) : 0
N X 2 "j;n ^ j;n : = N jn j
2
=0
With the considerations above, (4.1) reads as follows
N X j
n=0
"j;n ^j;n 2
N 4j
2
N X j
n=0
"j;n ^j;n [a['j ]; a['j ]]j;n 2
N X j
n=0
"j;n ^j;n ; 2
13 with arbitrary (^ j;n )Nn 2 RN
and j 2 N , which is equivalent to (4.7).
j +1
j
=0
0
In the following, we assume that (M1) { (M3) are satis ed. From (M1) it follows 'j 2 Vj , i.e., there exist unique coecients j ;k ('j ) 2 R (k = 0; : : : ; Nj ) such that +1
+1
NX +1 j
'j =
k=0
"j
+1
;k j +1;k ('j ) j +1;k 'j +1 :
+1
This is the so{called two{scale relation or re nement equation of 'j . The Chebyshev transformed two{scale relation of 'j reads an['j ] = Aj (n) an ['j ] (n 2 N ) (4.8) with the two{scale symbol or re nement mask of 'j +1
Aj (n) := +1
NX +1 j
k=0
"j
+1
0
cos kn Nj
;k j +1;k ('j )
+1
(n 2 N ): 0
+1
By de nition we obtain the relations of periodicity and symmetry for all n 2 N and l = 0; : : : ; Nj ? 1, Aj (n) = Aj (Nj + n) ; Aj (Nj ? l) = Aj (l) : (4.9) If a scaling function 'j (j 2 N ) satisfying (4.7) is given, then an orthonormal basis Bj; ((Nj =2) = '?j) (j 2 N ) can be easily obtained by Lemma 3.5, (iii). Let '?j (j 2 N ) be de ned by its Chebyshev coecients Nj a ['?] := ([a[' ]; a[' ]] )? = a [' ] (n 2 N ): j j j;n n j 2 n j Then Bj; ((Nj =2) = '?j) is an orthonormal basis of Vj = Sj; ('j ). The two{scale symbol A?j satisfying an['?j] = A?j (n) an['?j ] (n 2 N ) is connected with Aj by 0
+2
+1
+1
+1
+2
+1
0
1 2
0
+2
0
0
1 2
0
1 2
0
0
+1
+1
0
+1
+1
A?j+1(n)
:= 2 [a['[aj ['];]a; a['['j ]]]]j j j j;n +1
+1
;n
+1
1=2
(n 2 N ):
Aj (n) +1
0
The following connection between the bracket product [a['j ]; a['j ]]Ij and the two{scale symbol Aj can be observed: Lemma 4.3 For j 2 N and k = 0; : : : ; Nj , we have [a['j ]; a['j ]]j;k = Aj (k) [a['j ]; a['j ]]j ;k + Aj (Nj ? k) [a['j ]; a['j ]]j ;N +1?k : In particular, if (Nj =2) = '?j is an orthonormal scaling function and if A?j is the two{ scale symbol of '?j, then (4.10) A?j (k) + A?j (Nj ? k) = 4 (k = 0; : : : ; Nj ): +1
0
2
+1
+1
+1
+1
1 2
+1
2
+1
2
+1
+1
+1
+1
j
+1
+1
+1
2
14
Proof: By the de nition of the bracket product and by (4.8) { (4.9), we obtain for k = 0; : : :; Nj
[a['j ]; a['j ]]j;k 1 X
(anN +2 k ['j ] + a n
=
j
2
+
n=0 Aj+1(k)2 [a['j+1];
=
N +2 ?k ['j ]
( +1)
j
a['j ]]j +1
2
+ anN +2 j
N +1 +k ['j ]
+
2
j
;k + Aj +1 (Nj +1 ? k )
+1
2
+ anN +2 j
N +1 ?k ['j ]
[a['j ]; a['j ]]j +1
2
+
+1
j
)
;N +1 ?k :
+1
j
For orthonormal scaling functions, the assertion follows by Lemma 3.5, (ii).
5 Wavelet Spaces
Let the wavelet space Wj of level j (j 2 N ) be de ned as the orthogonal complement of Vj in Vj , i.e. Wj := Vj Vj (j 2 N ): Then it follows dim Wj = (Nj +1) ? (Nj +1) = Nj . By de nition, the wavelet spaces Wj (j 2 N ) are orthogonal. By (M1) { (M2), we obtain the orthogonal sum decomposition 0
+1
+1
0
+1
0
Lw (I ) = V 2
0
1 M j =0
Wj :
Further, Wj can be characterized by the orthogonal projector Pj; of Lw (I ) onto Vj , namely by Wj = ff ? Pj; f : f 2 Vj g: The subspace Wj is shift{invariant of level j , since by Lemma 3.6 we have for g := f ?Pj; f (f 2 Vj ), j;lg = j;lf ? j;l(Pj; f ) = j;lf ? Pj; (j;lf ) 2 Wj : Assume that the shift{invariant subspace Wj can be of type 1 generated by a function j 2 Vj such that Wj = Sj; ( j ). Further, we suppose that the set Bj; ((Nj =2) = j ) = f(Nj =2) = j ; l j : l = 0; : : : ; Nj ? 1g is Lw (I ){stable, i.e., there are constants 0
Nj : Then it holds that N Nj ' = X "j;k Tk 2 N : 2 j 0
j
j
k=0
By (7.1), the corresponding bracket product reads as follows 0; : : :; Nj ? 1 ; Nj [a['j ]; a['j ]]j;k = 42 kk = = Nj : Using (2.8), we obtain the following interpolation property of 'j
(7.2)
2
N X 2 "j;k cos kl 'j (hj;l ) = j;l'j (1) = N Nj = 2 ;l (l = 0; : : : ; Nj ) : j k j
(7.3)
0
=0
By (7.1), the shifted scaling functions j;k 'j (k = 0; : : : ; Nj ) are contained in N . Further, these functions j;k 'j (k = 0; : : : ; Nj ) are modi ed Lagrange fundamental polynomials with respect to the Gauss{Chebyshev nodes hj;l (l = 0; : : : ; Nj ), since for k; l = 0; : : :; Nj from Lemma 3.1, (ii) and (3.1) it follows j
j;k 'j (hj;l) = (j;l j;k 'j )(1) = (j;l k 'j (1) + j;jl?kj 'j (1)) = "?j;k k;l : Figure 1 shows the scaling function ' , and Figure 2 presents the shifted function ; ' . The function j;k 'j (k = 0; : : : ; Nj ) is supported on the whole interval I , and has signi cant values in a small neighbourhood of hj;k , if j is large enough. 1 2
1
5
+
5 16
5
21 2
1
0.8 1.5
0.6 1 0.4
0.5 0.2
-1
00
-0.5
1
0.5 x
-1
00
-0.5
1
0.5 x
-0.2
Fig. 1. Scaling function ' .
Fig. 2. Shifted scaling function ; ' .
5
5 16
5
Let Vj := Sj; ('j ) be the sample space of level j . Consequently by Lemma 3.5, (i), the polynomials j;k 'j (k = 0; : : : ; Nj ) form a basis of Vj , i.e., Vj = N ; dim Vj = Nj + 1 : Note that the operator Lj : C (I ) ! Vj de ned by 0
j
N X j
Lj f :=
k=0
(f 2 C (I ))
"j;k f (hj;k ) j;k 'j
is an interpolation operator, which maps C (I ) onto Vj with the property Lj f (hj;l ) = f (hj;l) (l = 0; : : : ; Nj ) : All sample spaces Vj (j 2 N ) form a multiresolution of Lw (I ), where (M3) reads as follows: The systems Bj; ((Nj =2) = 'j ) (j 2 N ) are Lw (I ){stable with optimal constants = 1=2 and = 1, i.e., for all j 2 N and for any ( j;k )Nk 2 RN we have the sharp estimate N N N
X X 1X
= 2 k "j;k j;k k "j;k j;k (Nj =2) j;k 'j k "j;k j;k : 0
0
2
1 2
2
0
j
=0
j
2
1 2
=0
j+1
j
0
j
2
2
=0
=0
Using (7.1), we nd the Chebyshev transformed two{scale relation of 'j an['j ] = Aj (n) an ['j ] (n 2 N ) with the corresponding two{scale symbol 8 < 2 n = 0; : : :; Nj ? 1 ; Aj (n) := : 1 n = Nj ; 0 n = Nj + 1; : : : ; Nj : Let Wj := Vj Vj be the wavelet space of level j . Thus, dim Wj = Nj . Consider the polynomials j 2 Vj (j 2 N ) given by their Chebyshev coecients 8 < 2 n = Nj + 1; : : :; Nj ? 1 ; Nj an[ j ] := : 1 n = Nj ; (7.4) 0 otherwise : +1
+1
0
+1
+1
+1
+1
0
+1
+1
22 2
1
0.8
1.5
0.6 1 0.4 0.5
-1
0.2
00
-0.5
-1
1
0.5 x
00
-0.5
1
0.5 x
-0.2 -0.5 -0.4 -1
-0.6
-0.8
-1.5
Fig. 3. Wavelet .
Fig. 4. Shifted wavelet :
5
6 33
5
Then the corresponding bracket product reads as follows 8 < 0 k = 0; : : : ; Nj ; Nj [a[ j ]; a[ j ]]j ;k = : 4 k = Nj + 1; : : :; Nj ? 1 ; 2 k = Nj : The shifted polynomials j ; r j satisfy the interpolation properties 2
+1
.
(7.5)
+1
+1
+1 2 +1
j
;r
+1 2 +1
j (hj +1;2s+1 )
(r; s = 0; : : : ; Nj ? 1) :
= r;s
Figure 3 shows the wavelet , and Figure 4 presents the shifted wavelet ; 5
6 33
5
.
The wavelet space Wj is a shift{invariant subspace of Lw (I ) of type 1 generated by j . The systems Bj; ((Nj =2) = j ) (j 2 N ) are Lw (I ){stable with optimal constants = 1=2 and = 1, i.e., for all j 2 N and for any ( j;r)rN ? 2 RN , we have the sharp estimate 2
1 2
1
0
2
j
0
1
? ?
NX 1 NX
= 2 r j;r r j;r (Nj =2) j j
j
=0
1
j
1
2
1 2
=0
;r
+1 2 +1
2
j
=0
NX ?1 j
r=0
j;r : 2
Using (7.1) and (7.4), we obtain the Chebyshev transformed two{scale relation of
an[ j ] = Bj (n) an ['j ] +1
+1
j
(n 2 N ) 0
with the corresponding two{scale symbol 0; : : : ; N j ; Bj (n) := 02 nn = = Nj + 1; : : :; Nj : +1
+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 2j + 1 function values of fj 2 Vj are given. The decomposition algorithm for linear spline wavelets in [0; 1] needs 6 2j real multiplications in order to compute all wavelet coecients of gj 2 Wj . For the same problem, the decomposition algorithm for cubic spline wavelets in [13] can be implemented using 14 2j real multiplications. Compared +1
+1
+1 +1
+1
23 30
25
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
20
15
10
5 2
4
6
8
10
12
14
16
18
20
j
Fig. 5. Comparison of needed real multiplications (divided by 2j ) for a complete +1
decomposition.
to that, our algorithm 6.2 requires fewer real multiplications up to the level j = 14. { Now we consider the complete decomposition of fj 2 Vj . Here we have to determine all coecients of the related functions in Wj ; Wj? ; : : :; W and V . Figure 5 shows the numbers of needed real multiplications (divided by 2j ) 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 2 f7; : : :; 11g is often used in praxis, our algorithm is an interesting alternative to the method in [13]. +1
1
+1
+1
3
3
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 denote the cardinal linear B{spline. Interpolating the function f (x) := B (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 (4x + 4) (x 2 I ) shows that f has singularities at ?0:75 and ?0:5, but not at ?1 (see Figure 7). 2
2
2
We can generalize this example of polynomial wavelets in a similar manner as done for periodic functions in [14]. Set Nj = d 2j (j 2 N ) with xed d 2 N. Further let for xed 0
24 1
0.002
0
0 -0.003 -1
0
1
-1
0
1
0
1
Fig. 6. Sample part in V and wavelet part W of f (x) = N (4x + 3:96). 6
6
1
2
0.0015
0
0 -0.003 -1
0
1
-1
Fig. 7. Sample part in V and wavelet part W of f (x) = N (4x + 4). 6
2N
6
2
1 j ; j ? 2 j > ; where 3 2d is ful lled. Then, Nj + Mj Nj ? Mj . Let the scaling function 'j of level j be given by its Chebyshev coecients 0
Mj :=
+1
Nj an['j ] :=
8 >
:
0
j
j
+1
0 n N j ? Mj ; Nj ? Mj < n < Nj + Mj ; n N j + Mj :
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 Vj = Sj; ('j ) can be described by n o Vj = N ?M span MM +1k TN ?k + MM ?+1k TN k : k = 0; : : : ; Mj ? 1 ; i.e., N ?M Vj N M ? . The corresponding wavelet space Wj (j 2 N ) is of type 1 generated by the polynomial j := 2 'j ? 'j 2 Vj such that 0
j+
j
j
j
j
j+
j
j
j
j
j
1
0
+1
8 n?N +M > > M > > > > M +1 > : j
0
j+
j
j
+1
N j ? Mj < n < N j + M j ; N j + Mj n N j ? Mj ; N j ? Mj < n < N j + M j ; otherwise: +1
+1
+1
+1
+1
+1
25 The shifted polynomials j
j
;r
+1 2 +1
;r
also satisfy the interpolation properties
j
+1 2 +1
j (hj +1;2s+1 )
(r; s = 0; : : : ; Nj ? 1) :
= r;s
For the Chebyshev transformed two{scale relations of 'j and j , we obtain the two{scale symbols 8 0 n N j ? Mj ;
