Technische Universität Chemnitz

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1 Technische Universität Chemnitz Sonderforschungsbereich 393 Numerische Simuation auf massiv paraeen Rechnern Zhanav T. Some choices of moments of renabe function and appications Preprint SFB393/03-11 Preprint-Reihe des Chemnitzer SFB 393 ISSN (Print ISSN (Internet SFB393/03-11 Juni 2003

2 Contents 1 Introduction 1 2 Choice of moments. Approximation theorem 2 3 Quadrature formua based on the waveet expansion 6 4 Evauation of the vaues of scaing function and its derivatives 13 Author's addresses: Zhanav T. Nationa University of Mongoia P.O Box 46/145 Uaanbaatar e-mai address: zhanav@chinggis.com

3 Some choices of moments of renabe function and appications T.Zhanav Abstract We propose a recursive formua for moments of scaing function and sum rue. It is shown that some quadrature formuae has a higher degree of accuracy under proposed moment condition. On this basis we obtain higher accuracy formua for waveet expansion coefcients which are needed to start the fast waveet transform and estimate convergence rate of waveet approximation and samping of smooth functions. We aso present a direct agorithm for soving renement equation. AMS subject cassication: 42C40, 42C15, 41A25, 65D32, 65D15. Key words: waveet approximation, quadrature rue, renabe function evauation. 1 Introduction This paper concerns the construction of compacty supported orthonorma waveets.it is we nown that the moment condition for erne is essentiay equivaent to good approximation properties.at present there are a numerous necessary and sufcient conditions for erne moment condition [1,2,3]. In this paper we propose another necessary and sufcient condition for erne moment condition,which is constructive in sense that using this we nd recursivey moments of the scaing function.these conditions aso aow us to construct efcient quadrature formuas for evauation of coefcients of the waveet expansion,which are needed to start the fast waveet transform and to estimate convergence rate of waveet and samping approximations for smooth functions.on the basis of the proposed moment conditions we suggested an unied agorithm for the exact evauation of the renabe function and its derivatives.we organize our paper as foows.in section 2 is considered the construction of compacty supported scaing function that generates mutiresoution anaysis of L 2 (R. Section 3 dea with the construction of higher accuracy formuas for the waveet expansion coefcients and samping approximation.in section 4 is presented an unied and exact agorithm for evauation of renabe function and its derivatives.some numerica resuts are aso presented. This wor has been supported in part by the Deutsche Forschungsgemeinschaft. 1

4 2 Choice of moments. Approximation theorem Let ϕ(x be a scaing function that generates a mutiresoution anaysis of L 2 (R and K(x, y be a periodic projection erne of the form K(x, y = ϕ(x ϕ(y. (2.1 For a measurabe function f dene an operator associated with the erne K j f(x = K j (x, yf(ydy, (2.2 where K j (x, y = 1 h K( x h, y h, h = 2 j. Let us introduce some conditions on ernes used in the seque [1,2,3]. Let N 0 be an integer. Condition H(N. y, x, y R and There exists an integrabe function F (x such that K(x, y F (x x N F (xdx <. Condition M(N (moment condition. Condition H(N is satised and K(x, y(y x s dy = δ 0s, s = 0, 1,..., N, x R, (2.3 where δ j is the Kronecer deta. Condition S(N. There exists a bounded non increasing function φ such that φ( u du <, ϕ(u φ( u and φ( u u N du <. In seque we wi assumed that condition S(N is satised. It is we nown that, condition S(N being satised, the condition H(Nhods as we, and the foowing quantities are we dened M n = ϕ(xx n dx, (2.4 µ n (x = K(x, y(y x n dy, c n (x = ϕ(x (x n, n = 0, 1,..., N. Moreover [3] the foowing reations are equivaent 2

5 Each of these reations impies that We assume that c n (x = c n (a.e, n = 0, 1,..., N, µ n (x = µ n (a.e, n = 0, 1,..., N. c n = M n, n = 0, 1,..., N. ϕ(x (x n = M n, n = 0, 1,..., N. (2.5 The moment condition for the erne is essentiay equivaent to good approximation properties. Next theorem gives a necessary and sufcient condition for the condition M(N. Theorem 2.1 and The erne (2.1 satises moment condition M(N if and ony if (2.5 hods s ( s m ( 1 m M s m M m = δ 0s, s = 0, 1,..., N. (2.6 Proof. First, prove that (2.5 and (2.6 are a necessary condition. Let K(x, y be a erne satisfying moment condition (2.3. Then by Proposition 8.5 [3] we have ϕ(x (x s = const = M s, s = 0, 1,..., N If we tae into account (1.1 and (1.5 the eft hand side of (1.3 may be rewritten as s ( s K(x, y(y x s dy = ( 1 m M m s m M m = δ 0s, s = 0, 1,..., N. The converse is obvious. Obviousy the condition (2.6 gives us reationship between moments M i and using this we can determine a moments. It shoud be mentioned that the equaity (2.6 turns out to identity for odd s. This means that moments with odd indices can be chosen as a free parameters, whie the moments with even indices are given by M 0 M 2e = 2 m=1 ( 2 m We denote by m i the discrete moments of {h } i.e., ( 1 m M 2 m M m, = 1, 2,... (2.7 m i = h i, i = 0, 1,..., N, (2.8 where h is the coefcients of renement equation ϕ(x = 2 h ϕ(2x. (2.9 3

6 To nd this discrete moments we use we-nown equaity M 0 m = (2 1 1 (2M i=1 ( i m i M i, = 1, 2,..., N, m 0 = (2. (2.10 Using (2.7 and (2.10 we can recursivey determine a m i. Since the moments M 2+1, are free parameters in (2.7 we have possibiity to obtain a set of renabe function. For exampe, if we choose the odd moments by formua M 2i+1 = ( M 1 M 0 2i+1 M0, i = 0, 1,... Then from (2.7 and (2.10 immediatey we obtain and In generay we see for m i and M i in the form and M i = ( M 1 M 0 i M0, i = 0, 1,..., N (2.11 m i = (2 ( M 1 M 0 i M0, i = 0, 1,..., N. (2.12 m i = (2a i ( M1 M 0 i M0, i = 0, 1,..., N, a 0 = 1 (2.13 M i = b i ( M1 M 0 i M0, i = 0, 1,..., N, b 0 = 1. (2.14 as (2.11 and (2.12 respectivey. Substituting (2.13 and (2.14 into (2.10 and (2.7 we get and a = 2 b j=1 ( j 2 ( 2 b = ( 1 m m=1 a j b j, = 1, 2,..., N (2.15 b m b 2 m, = 1,... (2.16 Thus we obtain two recursive reations for determining the coefcients a i and b i. Now we consider the construction of waveet system.as is nown, the moment condition M(N for erne (2.1 is equivaent to x s ψ(xdx = 0, s = 0, 1,..., N (2.17 or λ s = 0, s = 0, 1,..., N, (2.18 4

7 where λ are the coefcients of equation ψ(x = (2 λ ϕ(2x, λ = ( 1 +1 h 1. The equaity (2.18 gives (2 s h 2 = We coect a necessary equations for determining coefcients h : (2 i h 2 = (2 + 1 s h 2+1, s = 0, 1,..., N. (2.19 h h +2 = δ 0, N, ( h = 1, (2.21 (2 (2 + 1 i h 2+1 = (2 2 a i ( M 1 M 0 i, i = 0, 1,..., N. (2.22 Soving noninear system of equations (2.20, (2.21 and (2.22 we obtain coefcients of renement equation (2.9. It shoud be mentioned that if we choose M 1 = 0 in (2.22 then the system of equations (2.20 (2.22 turns out to ones for a coiet [6]. Now we consider approximation properties of the waveet expansion. We denote by P j the orthogona projection operator onto V j i.e. P j f(x = α j ϕ j (x. (2.23 where α j are given by inner product α j = (f, ϕ j. (2.24 Theorem 2.2 Assume that the conditions (2.5 and (2.6 hod and ϕ satises the condition S(N + 1. Then for f C N+1 is vaid: P j f(x f(x = O(h N+1, h = 2 j. (2.25 Proof. If we use of (2.24 then P j f can be rewritten as P j f(x = 2 j f(y ϕ(2 j x ϕ(2 j dy = K j ( x h, y f(ydy. (2.26 h Substituting the Tayor expansion for f(x f(y = N f (m (x m! (y x m + f (N+1 (ξ (y xn+1 (N + 1! 5

8 into (2.26 we obtain P j f(x = N f (m (x h m ( K(x, y(y x m dy+ m! f (N+1 (s h N+1 K(x, y(y x N+1 dy = N f (m (x h m δ (N+1! m! 0m + O(h N+1, in which we have used the condition (2.5, (2.6 and condition S(N + 1. This competes the proof of the theorem. 3 Quadrature formua based on the waveet expansion We consider a quadrature formua I = f(xϕ(xdx Q[f(x] = r w f(x, (3.1 =0 where the weights w and abscissae x are to be determined. Here ϕ(x is the father waveet that generates a mutiresoution anaysis of L 2. That is compacty supported scaing function ϕ(x satises the equation [1,2] ϕ(x = 2 c φ(2x, c 2 < (3.2 with nite number nonzero coefcients c and the system of function ϕ j (x = 2 j 2 ϕ(2 j x, Z (3.33 forms a orthonorma system in space V j. If the orthogona projection operator onto V j denote by P j then it can be written as P J f(x = α j ϕ j (x, α j = (f, ϕ j. (3.4 Reca that the degree of accuracy of the quadrature formua (3.1 is q if it yieds the exact resut for every poynomia of degree ess than or equa to q. As mentioned in [4], the quadrature formua is usuay constructed by demanding that Q[x i ] = M i for O i q (3.5 which eads to an agebraic system with respect to unnowns w and x, = 0,..., r. Here M i are the moments of the scaing function, i.e., M i = x i ϕ(xdx. (3.6 6

9 The scaing function is usuay normaized with M o = 1. Another way to construct a quadrature formua is to use of expansion (3.4, in which we dwe more detai. Let f(x C n+1. Then using Tayor expansion of f(x it is easy to show that α j = (f, ϕ j = h Here and in sequence we assume { N } f (m (h h m M m + O(h N+1 m!, h = 2 j. (3.7 ϕ(x (x S = M S, S = 0, 1,..., N. (3.8 We consider next quadrature formua I Q[f(x] = 1 ( + M c f 2 2 (3.9 with nown weights w = c 2 and abscissae x = (+M. We have 2 Theorem 3.1 Let f(x C N+1 and the condition (3.8 is fued. Then the error of quadrature formua (3.9 is of order O(h N+ 3 2 i.e., f(xϕ(xdx 1 ( + M c f = O(h N+ 3 2, h = 0.5. ( Proof. Taing into account the condition (3.8 we can write M i in the form. M i = +1 ϕ(xx i dx = 1 0 ( ϕ(x + (x dx i = M i, i = 0, 1,..., N. (3.11 Thereby, from (3.7 we get α j = h { N } f (m (h (hm m + O(h N+1 = m! = h{f( + Mh + O(h N+1 }, h = 1 2 j. (3.12 On the other hand, using scaing equation (3.2 and (3.4 in integra I, we have I = 2 c f(xϕ(2x dx = c α 1. (3.13 7

10 Substituting α 1 from (3.12 into (3.13 we obtain I = 1 ( ( + M c f + O(h N which competes the proof of theorem. Theorem 3.2 The degree of accuracy of the quadrature formua (3.9 equa to N under condition (3.8, i.e., the equaity (3.5 hods for i = 0, 1,..., N. Proof. Set f(x x i, i = 0, 1,..., N in (3.9. Then by denition of moments and assumption (3.8 the eft hand side of (3.9 gives (3.11, i.e., The right hand side of (3.9 gives M i = M i, i = 0, 1,..., N. Q[x i ] 1 c (+M i h i = hi c [ i +Ci 1 i 1 M+Ci 2 i 2 M 2 + +C i 1 i M i 1 +M i ]. 2 2 By virtue of emma [5] we have Then c i = 2M i, i = 0, 1,..., N. Q[x i ] = h i M i (1 + C 1 i + C 2 i + + C i 1 i + 1 = h i M i 2 i = M i = M i, which competes the proof of theorem 3.2. Note that the proposed quadrature formua (3.9 is stabe. Indeed from normaization condition M o = 1 it foows that w = 1 c = 1. 2 Athough among these coefcients c may be occurred negative ones, the foowing uniform bound hods: 1 c < C, ( where C is constant not depending on, due to (3.2. The inequaity (3.14 proves the stabiity of formua (3.9. By virtue of (3.4 we have α 0,0 = (f, ϕ 0,0 = f(xϕ(xdx = I. (3.15 8

11 Therefore in order to cacuate the integra (3.1 or (3.15 with higher accuracy than formua (3.9 we need to use of we-nown Maat agorithm: α mo = e c e α m+1,e, m = j, j 1,..., 0. (3.16 At the high eve j we can use the approximate formua (3.12, i.e., α j+1,e 2 j+1 2 f(( + Mh, h = 2 (j+1 (3.17 with accuracy O(h N By this agorithm the integra (3.15 wi be evauated with higher accuracy. It shoud be mentioned that in [4] were proposed some quadrature formua with r degree of accuracy. But these agorithms ead to sove noninear system of (r + 1 unnowns, which is i-conditioned. Unie this the formua (3.9 is a more simpe one with nown weights and abscissae. Moreover we can construct the extension of formua (3.9. Indeed, it is easy to show that ϕ jo (x = c ϕ j+1, (x. (3.18 As above, we can consider By virtue of (3.12 we obtain I j = f(xϕ jo (xdx = c α j+1, (3.19 or I j = f(xhϕ(xdx 1 ( + M c f 2 2 f(xϕ(2 j xdx h ( + M c f 2 2 h, h = 2 j (3.20a h. (3.20b Quadrature formua (3.20 with j = 0 eads to (3.9. Obviousy, the theorem 3.1 and theorem 3.2 remain true aso for (3.20 with h = 2 j. Obviousy,the smaer the number of abscissae r,the more efcient the quadrature formua (3.1 since the number of function evauations and agebraic operations for one coefcient is proportiona to r. Therefore the idea of a one-point quadrature is attractive because of its simpicity. However, its degree of accuracy is imited. More precisey the degree of accuracy of the onepoint formua [ ] I = f(xϕ(xdx Q f(x, x 1 = M 1 (3.21 9

12 is two, when ϕ is an orthogona scaing function [4]. It is aso nown, that for the coiets with N vanishing moments M p = 0, 1 p N (3.22 the one-point formua (3.21 with x 1 = 0 has a degree of accuracy of N. This concusion is aso true for (3.21 with a scaing function ϕ(x, satisfying the condition (3.8. Indeed, setting f(x = x s in (3.21 and using (3.11 we see that the eft and right hand sides of (3.21 are equa to each other for s = 0, 1,..., N. Therefore we can use the one-point quadrature formua (3.21 for an evauation of the waveet coefcients α j+1, = (f, ϕ j+1, = 2 j+1 2 ( y + ( f ϕ(ydy 2 j+1 M + 2 f. 2 2 j+1 That is we arrive at formua (3.17. This means that the approximate formua (3.17 may be considered as a resut of using one-point quadrature formua (3.21 for evauation α j+1,. Let M i = c, i = 1,..., N. Then from (3.7 it foows α j = (h[(1 cf(h + cf(( + 1h] + O(h N (3.23 Thus we have a two-point quadrature formua: α j = f(xϕ j (xdx (h where ω 1 = 1 c; ω 2 = c, x 1 = h, x 2 = ( + 1h. Using (3.18, (3.19 and (3.23 we easy to seen that I j = 2 ω f(x ; h = 2 j, (3.24 =1 f(xhϕ(xdx 1 ((1 cc + cc 1 f(h, h = 2 j (3.25 (2 or f(xϕ(2 j xdx h ( ((1 cc + cc 1 f (2 2 h (3.26 with error O(2 (j+1(n When j = 0 the formua (3.26 eads to I = f(xϕ(xdx 1 ( ((1 cc + cc 1 f (2 2 (

13 with error O(2 (N Now we consider the trapezoida rue I = f(xϕ(xdx Q[f(x] = f(ϕ( (3.28 It is easy to show that the degree of accuracy of the rue (3.28 is equa to N, if the scaing function satises condition (2.5. Then we use this rue for evauating the coefcients of the waveet expansion α j = 2 j 2 f(xϕ(2 j x dy = 2 j 2 f((y + hϕ(ydy 2 j 2 f(( + hϕ(. Thus we have approximate expression for coefcients: α j = 2 j 2 f(( + hϕ(, h = 2 j. (3.29 This means that α j represents a discrete convoution of f and ϕ. Now we are interested in error estimate of this approximate expression (3.29. Assume that f C N+1. Then using Tayor expansion f(( + h = f (m (h (h m + O(h N+1 m! m and (2.10 we obtain α j = 2 j 2 N Comparing this with (3.7 we concude that f (m (h (h m M m + O(h N (3.30 m! α j = α j + O(h N+ 3 2, h = 2 j. (3.31 In the case of M i = ( M 1 M 0 i M 0 for i = 0, 1,..., N the formua (3.30 yieds α j = (hm 0 f(( + M 1 h + O(h N (3.32 M 0 Thus in this case we can use a simpe formua (3.32 instead of (3.29. It is easy to seen that the direct appication of formua (3.29 to the evauation of I gives I = f(xϕ(xdx = c α ( + c f ϕ(. (3.33 (2 2

14 Of course, the degree of accuracy of this formua (3.33 is N and I 1 ( + c f ϕ( = O(h N+ 3 2 = O(2 (N (2 2 As before, if the accuracy of this formua is not satisfactory then we can use the Maat agorithm: α j = c m 2 α j+1,m, j = j 1,..., 0 and using formua (3.29 at higher eve j + 1. As a resut we obtain integra I with higher accuracy O(2 j(n Let R(x be an error of waveet expansion (2.23,i.e., R(x = f(x α j ϕ j (x. (3.34 We consider two method for determining coefcients of the decomposition. Interpoation. (i Setting x = ( + nh, h = 2 j in (3.34 we require that R(( + nh = 0, Z, (3.35 which is a system of equations 2 j 2 α j ϕ( + n = f(( + nh, Z. (3.36 Since ϕ(x is a scaing function with compacted support then the ast system turns out to a nite system of inear equations with respect to coefcients α j. The system (3.36 can be soved by eimination method. (ii Discrete Gaerin method We require that This is a discrete version of Gaerin method. The ast system can be rewritten as α j ( ϕ( + n ϕ( = 2 j 2 R(( + nhϕ( = 0. (3.37 ϕ(f(( + nh. (3.38 Again we obtain a system of equations w.r.t α j. The soution of this system can be found expicity by formua α j = 2 j 2 ϕ(f(( + nh (

15 if ϕ( + ϕ( = δ 0. (3.40 The ast condition can be considered as a discrete version of the orthogonaity condition for ϕ(x. Thus we again arrive at formua (3.29. But in this case we have an exact formua (3.39. Now we are interested in approximation properties of samping approximation Sf(x = α j ϕ j (x, (3.41 where the coefcients of this expression are given by (3.29. From (3.41 it cear the samping operator S is oca when ϕ(xis compacty supported and it is easy to impement. As before, we assume that f C N+1 and the conditions (2.5, (2.6 hod and ϕ(x satises the condition S(N + 1.Using Tayor expansion for f(x and condition (2.5, (2.6 in (3.41 it is easy to show that Sf(x f(x = O(h N+1, h = 2 j. (3.42 Obviousy the estimate (3.42 is aso vaid for (3.41 with coefcients given by (3.17 and (3.23. From the estimate (3.42 it is cear that the samping operator (3.41 is a quasi-interpoating one [4]. That is it produces every poynomias of degree ess or equa to N. Note that estimate of type (3.42 for a coiet was proven by J.Tian and R.O. Wes,Jr [6,10,11] and for generaized coiet by E-B Lin and X Zhou [10,11]. If f(x is continuous then from (3.29 it foows that 2 j 2 αj f(h as j. Then it is easy to show that f(hϕ(2 j x f(x = O(h α (3.43 provided that ϕ satises condition S(1 and f is Höder continuous with exponent α. 4 Evauation of the vaues of scaing function and its derivatives As is nown the waveet basis in the mutiresoution anaysis is dened by transation and diations of the scaing function φ(x and waveet function ψ(x. So it is often needed the vaues of the function φ(x at any points. However no expicit expression of the scaing function is avaiabe. There are many agorithms for numerica evauating of vaues of the scaing function. (see, for exampe, [1], [2]. Another method for computing the vaues φ(x at integer points was 13

16 given by A. Garba. In [5,8] the required vaues of φ(x are obtained as a soution of eigensystem. The disadvantage of this method is that it has a time consuming for the higher dimension case of the eigensystem. In this section we present an exact method for computing vaues of scaing function at integer as we as at dyadic points. N 1], where N is posi- Assume that the scaing function has a support in the interva [0, tive integer, and satises the reation φ(x = N 1 =0 c φ(2x, c = 2h. (4.1 Equation (4.1 is caed the renement equation. Since supp(φ [0, N 1], we ony need to compute the vaues φ at the integer points i = 1, 2,..., N 2. This is aso true for the derivatives because they are aso compacty supported, as can be seen by deriving the scaing equation (4.1. If we use the denotation Φ (n = (φ (n 1, φ (n 2,..., φ (n N 2 T then Φ (n satises an eigensystem [5]. HΦ (n = λφ n, λ = 2 n, (4.2 where the (N 2 by (N 2 matrix H has a form: c 1 c c 3 c 2 c 1 c 0.. c 3 c 2 H = c N 1 c N c N 1 c N 2... c 1 c c N 1 c N 2 (4.3 The vector Φ (n is an eigenvector corresponding to the nown eigenvaue λ = 2 n. It is understood that the derivative of order 0 corresponds to the scaing function φ(x. The coefcients h are given by Daubechies in [1]. In order to nd vector Φ (n we use, in addition Eq.(4.2, a partition of unity satised by φ, i.e., φ(x 1 for x R. (4.4 Differentiating both side of (4.4 and setting x = 2i, for i = 1, 2,..., N 2 in the obtained equation, we have N 2 i=1 φ (n i = δ on, n = 0, 1..., (4.5 14

17 where δ on is a Kronecer symbo. The eigensystem (4.2 together with (4.5, as in [5], may be soved by numerica methods. Since the eigenvaue of this system is nown, it is usefu to consider this as an system of inear agebraic equations. Dividing both side of Eq.(4.2 by 2 n φ (n 1 0 and ignoring one of these equations invoving a the unnowns we obtain the nonhomogeneous inear system of (N 3 equations with (N 3 unnowns φ (n = φ(n i, i = 2, 3,..., N 2. This system can be soved, for instance, by Gaussian eimination method. After this, substituting = φ (n 1 φ n i, i = 2,..., N 2 (4.6 in equations (4.5, we have From this we nd φ 1 = φ (n i φ (n 1 φ (n (n (n (n 1 (1 + φ 2 + φ φ N 2 δ on. ( φ 2 + φ φ N 2, when n = 0 (4.8 The remainder vaues of φ i are dened by formua (4.6 for n = 0. Thus for n = 0 the vaues of φ(x at integer points determined competey. Substituting x = h, h = 2 j in equation (4.1, it is easy to show that φ( N 1 2 = j c m φ( m, j = 0, 1,... (4.9 2j 1 φ( N 1 j 2 = c j+1 m φ( m, j = 0, 1,.... (4.10 j 1 2j Thus the vaues of φ at the dyadic points are founded recursivey by reations (4.9, (4.10. For exampes, we have φ( + N 1 2 = c j+1 m φ(2 + 2 m, = 1, 2,..., j 2j+1 1, j = 0, 1, 2,... (4.11 For n 1 from (4.7 it is evident that the eigensystem (4.2 has a nonzero soution if and ony if 1 + (n (n (n φ 2 + φ φ N 2 = 0. (4.12 In this case the vaue φ (n 1 is remains unnown. To nd φ (n 1 we need to use of next emmas. Lemma 4.1 Let φ be C n scaing function satisfying conditions φ(x (x S = φ(xx S dx = M S, s = 0, 1,..., N. (

18 Then φ (m (x (x S = ( 1 m s(s 1(s 2... (s m + 1M S m, (4.14 s = 0, 1,..., N, m = 0, 1, 2,..., n Proof. We wi prove (4.14 by induction. For m = 0 the equaity (4.14 becomes (4.13. Differentiating (4.13 we get φ (x (x S = s φ(x (x S 1, s = 0, 1,..., N. By virtue of (4.13, we have φ (x (x S = sm S 1, s = 0, 1,..., N. This proved the reation (4.14 for m = 1. Suppose the reation (4.14 is vaid for m = 1, 2,..., j. We prove (4.14 for m = j + 1. To this end differentiating reation (4.13 (j + 1- times, we get { j+1 ( j + 1 } φ j+1 (x ((x S ( = 0. =0 From this we nd φ (j+1 (x (x S = { j+1 =1 ( j + 1 φ (j+1 (x s(s 1... (s + 1(x S } j+1 ( j + 1 = =1 { s(s 1... (s + 1 φ (j+1 (x (x S }. (4.15 By induction for = 1,..., j hods next reation φ (j+1 (x (x S = ( 1 j+1 (s (s 1... (s jm S j 1. (4.16 Substituting (4.16 into (4.15 we get j+1 φ (j+1 (x (x S = ( 1 j+1 s(s 1... (s jm S j 1 =1 ( j + 1 (

19 Since n =1 =0 ( n j+1 ( j + 1 ( 1 = 0 we have j+1 ( 1 +1 = =1 ( j + 1 j+1 ( j + 1 ( 1 ± 1 = 1 =0 ( 1 = 1 This competes the proof of Lemma 4.1. Lemma 4.2 Assume the same conditions as in Lemma 4.1. Then φ (n n φ (n n φ (n (N 2 n φ (n N 2 = ( 1n n!. (4.17 Proof. Tae m = s = n in (4.14. Then we have φ (n (x (x n = ( 1 n n!. (4.18 Since suppφ (n [0, N 1] as φ(x the summation on in (4.18 is run from = i N + 2 to = i 1. Setting x = i in (4.18 we get (4.17. From (4.17 we nd φ (n 1 = ( 1 n n! n φ n n φ n (N 2 n φ. (4.19 n N 2 For n = 0 the formua (4.19 coincides with (4.8. Remar 4.1 It is easy to show that the formua (4.19 is aso vaid for Daubechies scaing function for n = 0, 1 and for n = 2 since M 2 = M 2 1. Remar 4.2 If suppφ [N 0, N 1 ] (φ(n 0 = φ(n 1 = 0 then it is easy to show that formuae (4.17 and (4.19 ead to φ (n N 0 +1 (N n + φ (n N 0 +2 (N n + + φ (n N 1 1 (N 1 1 n = ( 1 n n! and φ (n N 0 +1 = ( 1 n n! (N n + (N n (n φ N (N 1 1 n (n φ N 1 1, φ (n i where = φ(n i, i = N φ (n 0 + 2,..., N 1 1, respectivey. N 0 +1 The vaues of φ (n at dyadic points are determined by reation φ (n ( + = 2n 2j+1 n 1 c m φ (n (2K + 2 j m, = 1, 2,... 2j+1 1, j = 0, 1, 2,... (4.20 The proposed agorithm for evauation of the renabe function is immediatey extended to the mutidimensiona cases.for brevity we wi restrict ourseves ony by two dimensiona case (d = 17

20 2 an n = 0. Let ϕ(x = ϕ 1 (x 1 ϕ 2 (x 2 (4.21 be a tensor product of scaing functions ϕ 1, ϕ 2, whose factors fu one-dimensiona renement equations with coefcients h 1 and h 2, 1, 2 Z. Then ϕ satises the renement equation [7,8] ϕ(x = det M 1 2 h ϕ(mx, = ( 1, 2. (4.22 Z 2 Let ϕ be normaized R 2 ϕ(xdx = 1. (4.23 For simpicity we sha assume that the diation matrix M = 2I. Then the renement equation becomes as ϕ(x 1, x 2 = 1, 2 h 1, 2 ϕ(2x 1 1, 2x 2 2. (4.24 Assume supp(ϕ 1, ϕ 2 [0, N 1]. Then supp(ϕ [0, N 1] 2.Setting x 1 =, x 2 = j in (4.24 and denoting φ = (φ,j then resuting system of equations can be written as (A Iφ = 0, (4.25 where A is a matrix with eements h 1, 2 of system (4.24. The ast system can be written in the extended form N 2 =1 N 2 =1 a 1 φ j = φ 1j a 2 φ j = φ 2j... N 2 a N 2, φ j = φ N 2,j =1 (4.26 for j = 1, 2,..., N 2. If we denote φ j φ 1j = φ j, = 2,..., N 2 then (4.26 form a non homogeneous system w.r.t φ j, = 2,..., N 2 in which is aced one of equations that contain a unnowns. Since each system of (4.26 can be soved independenty of one another, a computations are inherenty parae. Thus soving the system (4.26 for a j = 1,..., N 2 we nd φ j, = 2,..., N 1, j = 1,..., N 2.It remains to nd ony φ 1j, j = 1,..., N 2. To this end we use the partition of unity [9]. φ ij 1. (4.27 i,j 18

21 Obviousy,the equaity (4.27 hods,if (for exampe N 2 i=1 φ ij = ϕ 2j, j = 1,..., N 2 (4.28 because of The system (4.28 can be rewritten as N 2 j=1 ϕ 2j 1. φ 1j (1 + φ 2j φ N 2,j = ϕ 2j, j = 1,.., N 2. From this we nd Using (4.29 we obtain φ 1j = ϕ 2j 1 + φ 2j φ N 2,j. (4.29 φ j = φ 1j φ j, = 2,..., N 2, j = 1,..., N 2. (4.30 Thus the soution of the system (4.25 is given by (4.29, (4.30. In concusion we present numerica resuts. We were soved the system of equations (4.2, (4.5 for Daubechies scaing function and N = 4(220 and for n = 0, 1, 2,.... The resuts coincide competey with those given by Garba in [5]. In [5] was present the same vaues for the rst and second derivatives of the Daubechies seaing function for N = 6. Perhaps, this is a technica mistae. For correctness here we present ony these vaues for N = 6.. Tabe 1. i Φ (0 Φ (1 Φ ( E E E E E E E E E E E E-0002 References [1] Daubechies, I. Orthonorma bases of compacty supported waveets. Commun. Pure App. Math. 41, No.7, (1988. [2] Daubechies, I. Ten ectures on waveets. CBMS-NSF Regiona Conference Series in Appied Mathematics. 61. Phiadephia, PA: SIAM, Society for Industria and Appied Mathematics., 357 p. (

22 [3] Härde, W. and ets. Waveets, approximation, and statistica appications. Lecture Notes in Statistics (Springer, 265 p. (1998. [4] Swedens, W.; Piessens, R. Quadrature formuae and asymptotic error expansions for waveet approximations of smooth functions. SIAM J. Numer. Ana. 31, No.4, (1994. [5] Garba, A. Waveet coocation approximation of differentia operators. IC/96/212. Miramare-Triesta. [6] Resnioff, H.L.; Wes, R.O. Waveet anaysis. The scaabe structure of information. New Yor, NY: Springer. 435 p. (1998. [7] Lin, E-B.; Zhou, X. Coiet interpoation and approximate soutions of eiptic partia differentia equations. Numer. Methods Partia Differ. Equations 13, No.4, (1997. [8] Lin, E-B.; Xiao, Z. Muti-scaing function interpoation and approximation. Adroubi, Aram (ed. et a., Waveets, mutiwaveets, and their appications. AMS specia session, January 1997, San Diego, CA, USA. Providence, RI: American Mathematica Society. Contemp. Math. 216, (1998. [9] Dahmen, W. Waveet and mutiscae methods for operator equations. Iseres, A. (ed., Acta Numerica Vo. 6, Cambridge: Cambridge University Press (1997. [10] Louis, A.K.; Maa, P.; Rieder, A. Waveets. Theory and appications. Pure and Appied Mathematics. A Wiey-Interscience Series of Texts, Monographs, 324 p. (1997. [11] Lawton, W.; Lee, S.L.; Shen, Z. Convergence of mutidimensiona cascade agorithm. Numer. Math. 78, No.3, (