QUADRATURES INVOLVING POLYNOMIALS AND DAUBECHIES' WAVELETS *

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1 QUADRATURES INVOLVING POLYNOMIALS AND DAUBECHIES' WAVELETS * WEI-CHANG SHANN AND JANN-CHANG YAN Abstract. Scaling equations are used to derive formulae of quadratures involving polynomials and scaling/wavelet functions with compact supports; in particular, those discovered by Daubechies. It turns out that with a few parameters, which are theoretically exact, these quadratures can be evaluated with algebraic formulae instead of numerical approximations. Those parameters can be obtained with high precision by solving well-conditioned linear systems of equations which involve matrices already seen in the literature of wavelets for other purposes. Key words. scaling equation, wavelet, quadrature, polynomial, Neumann series Subject classications: AMS(MOS): 65A5, 65D3, 65F35, 4C5. Introduction. Because of their similarities with Fourier transforms and nite elements, wavelets are studied for the applicability of being a tool in scientic computation and numerical analysis, such as the numerical solution of dierential equations. Among the recent articles on the later subject, we refer to the works by Bacry, Mallat and Papanicolaou [], Beylkin, Coifman and Rokhlin [4], Glowinski, Lawton, Ravachol and Tenenbaum [8], Jaard [9], Maday, Perrier and Ravel [4], and u and Shann [8]. One of the fundamental computing steps of these applications is the numerical quadratures f(x)(x) dx. As pointed out by Dahmen and Micchelli [5], since the derivatives of also satisfy (other) scaling equations, the techniques derived for the foregoing integrals can be carried to those with derivatives of. Beylkin, Coifman and Rokhlin [] studied numerical methods for the rst two kinds of integrals. There the idea is that (x) can be related to ( j x R? k) by the fast pyramid algorithm [3]; and the integral f(x)( j x? k) dx can be approximated by numerical quadrature rules if j is large enough (that is, if the support of ( j x) is narrow enough). The fundation for the derivation of such quadrature points and parameters is the integrals for which f(x) are polynomials. That is, the moments of the forms R f(x)(x) dx, R f(x)(x)(x? k) dx, or R b a (:) x m (x) dx; and the integrals (:) x m (x)(x? k) dx; (:3) b a x m (x) dx: In [], (:) were approximated by a recursively generated sequence of vectors, based on the -periodic characteristic function of ^(). Sweldens and Piessens [7] went on to show a better * Technical Report 93, Department of Mathematics, National Central University, Taiwan. Department of Mathematics, National Central University, Chung-Li, Taiwan, R.O.C. This work was supported by the National Science Coucil in Taiwan; grant # 8-8-M address: shann@math.ncu.edu.tw. address: yan@math.ncu.edu.tw

2 W. Shann and C. Yen approximation result for the one-point rule in [], and to extend the multi-point rules. They also started with the relation, which states that (:) can be done theoretically exact. Beylkin [3] found the integrals (:) are associated with matrices. In particular, he found the values of R (x) (x? k) dx consist a normalized eigenvector of a certain matrix. This technique turns out to be a special case of the theorems in [5], which guarateed this sort of integrals always associate with eigenproblems and there are always unique solutions with appropriate assumptions. In this article we extend the idea to treat (:3). We also show that the matrices associated with these integrals have uniform bounds in vector `-norm. That is, the upper bound of the norms do not change with the order of wavelets. In particular, the assocated linear systems can be solved with high precision eciently by Neumann series. Wavelets in this article are those discovered by Daubechies [6], which have compact supports and form an orthonormal basis of L (R). In this case, the wavelet (x) is derived from the scaling function (x) which is normalized to have R = and satises the scaling equation (:4) (x) = k c k (x? k): The coecients c k were computed by constructing a certain trigonometric polynomial m () = P ck e?ik, see Daubechies' \Ten Lectures" [7]. However, Strang pointed out [6] that they can also be solved by the following relations: (:5) c k = for k 6 f; ; : : :; p? g; (:6) (:7) and (:8) k k c k? = ; (?) k k m c k = ; for m p? ; ( := ) k c k c k?m? m = : Here p is a given positive integer. We shall call respectively such dened c k, and the scaling coecients, scaling functions, and wavelets of order p. This parameter p plays several roles in the theory of wavelets. What we need now is that supp = [; p? ]. Once (x) is determined, the multiresolution approach asserts that (:9) (x) = k (?) k c?k (x? k) R will be a wavelet. One of the properties of is that x m dx = for m p?. Reducing the computations involving wavelets to the manipulation of c k is a general strategy. It is sought for every kind of application for wavelets. Examples have been seen in numerical algorithms and signal decomposition [, ], among others. Thus the numerical values of c k heavily determine the accuracy of these algorithms. In the next section we discuss the computation of scaling coecients. Then we derive the formulae for R x m. As the immediate applications, polynomials P k m (x? k), for m p?, R are constructed and formulae for x m are derived. Next we evaluate the integrals R on the half-lines [; ) or (?; ]. Finally we present that the similar idea can treat convolutions x m (x)(x?k).

3 Wavelets quadratures 3. Computation of scaling coecients. Note that (:8) contributes exactly p? equations in the system, namely m p? : The situations for m p and m?p are consequences from (:5); those for? p m? are dual to the cases of jmj; and the case of m = can be derived from (:6), the m = case of (:7), and the m 6= part of (:8). Therefore, we can write conditions (:6){(:8) into p equations in x = (x ; : : :; x p? ) T. Let it be a vector-valued function F p (x). Algebraic solutions for p =, and 3 are known; but it is not likely to nd those with higher p. Numerical solutions for F p, p, are listed in a table [7, p. 95] with 6 digits. 3 Let d be the solution of F listed in the table (with (:6) changed to p ). We have kf (d)k 8?6. This is due to the diculty introduced in (:7). For p = and m = 9, the smallest and the largest number in that equation is dierent by a scale of 8. Observe that the set of equations in (:7) can be replaced by (:) k (?) k (k? `) m c k = ; for m p? : for any xed integer `. In particular, we take ` = p to have the most advantage. With (:7) replaced by (:), the foregoing diculty is slightly improved. Let F ~ p (x) be the alternate system of equations. One can write down the Jacobian of F ~ p and perform the Newton's iteration to solve ~F p (x) = (we use the roots of Fp? ~ appended with two zeros as the initial guess). Our FORTRAN code is able to squeeze out more information about the scaling coecients. First, let c be the numerical solution of F ~, we have kf (c)k?8. Secondly, we can calculate the scaling coecients up to p = 4. Some results are listed in Table. All values quoted hereafter are computed in quartic precision on a VA 65. Numerical values of c k of order p 4, together with those values and programs mentioned later in this article, are available from the rst author by request. It makes more sense to query and obtain the data electronically. 3. Integrals on half lines. While solving dierential equations, the domain of interest is usually R a bounded interval, on which boundary conditions are imposed. In this situation, integrals b like a q(x)(j x? k) dx will come in order. In practice one can rescale the problem and only use those wavelets and scaling functions in such ne scales that in the support of each ( j x? k) (or ( j x? k)), there lies at most one boundary point. Since the situations are analogous at either boundary, we now suppose x = is the left boundary point and consider the integrals on [; ). As before, the core formulae for this kind of integrals are N p m;k = x m (x? k) dx: For the translation parameters k? p, the integrals are zero. For k, integration over [; ) is the same as over R. Thus N p m;k = M p m;k. We now concentrate on the cases that? p k?. Again, we start from (:4) with a changing of variable: (3:) x m (x? k) dx = m+ ` c` x m (x? (k + `)) dx: 3 Over there, equation (:6) is replaced by P c k = p. Here we follow the notation in [6].

4 4 W. Shann and C. Yen If? p k?; that is, over half of the support is still inside of [; ). Then the supports of (x? (k + `)), for?k ` p?, actually lies entirely in [; ). That means, for these `, the integrals on the right hand side of (3:) can be evaluated as in Section 3. Over all, we have i c`n p m;k+` + : N p m;k = m+ h `<?k `?k c`m p m;k+` Suppose the part of `?k is known, we have a linear system in N p m;k. Given a p (there is nothing to do with p = ) and x an m, we have unknowns N p m;k for? p k?. Now let 8 >< >: x k = N p m;?k ; b k = p? m+ `=k c`m p m;`?k ; k = ; ; : : :; p? : Let b = (b k ) T and A = (a k`) be the matrix of order p? such that a k` = c k?`. Then x = (x k ) T is the solution of the system (I? A) x = b: m+ Note that A is independent of m and it is a nite submatrix of the low pass lter matrix L dened in [6], in which it is also shown that A has an eigenvalue = with an associated eigenvector ((); (); : : :; (p? )) T. Here we claim another property for A. Proposition 3.. Let A be the matrix dened above. k k be the matrix norm induced by the Euclidean norm. We have kak p : Proof. Since A = S = S where c c... c p?3 c p?4 c A and T = c p? c p? c... c p? c p? C A : By (:8), ST T = T S T =. Thus AA T = SS T T T T : Also by (:8), we have SS T + T T T = I. Thus if is an eigenvalue of SS T with an associated eigenvector v, then and? are eigenvalues of AA T with associated eigenvectors v, respectively. Since SS T and T T T are positive semi-denite, we have. Therefore kaa T k. v and

5 Wavelets quadratures 5 (3:) Thus, for any m and p, the Neumann series n= ( A)n m+ is an ecient method for computing the inverse of I? m+ A. We have prepared values for p 4 and m 4. Those lower order values are listed in Table. (The right most integer in each column is the exponent.) The iterations are computed in quartic precision with the stop criterion being that the `-norm of error <?4. As for the right boundary point, one can translate the problem to the integral on (?; ]. Clearly x m (x? k) dx = M p m;k? N p m;k :? 4. Convolutions. R Now we consider the integrals of the form q(x)( j x? `)( k x? m) dx. If j < k, apply the scaling equation (:4) recursively on ( j x? `) will lift the scale to the same level k. Then a change of variable can go back to level j = k =. Thus in practice we do not need the formula for those 's on dierent scales. Again, after changing of variables, we can concentrate on the core: L p m;k = q(x)(x) (?x) (k) = x m (x)(x? k) dx Since L p ;k = k, we start from m =. The support of (x) of order p is [; p? ], hence for each p and m, we have unknowns L p m;k for? p k p?. Then L p m;k = (4:) = = x m n P n;` cnc` m+ h n;` c n (x? n) ` c`(x? k? `) dx R ( x+n )m (x)(x + n? k? `) dx m c n c`l p m m;k+`?n + c n c` r n;` r= The problem is recursively equivalent to a system of linear equations (4:) (I? A)x = b: m+ n r L m?r;k+`?n i : Let N = p?, then x = (L p m;?n ; : : :; Lp m;n )T, b = (b k ) and A = (a kj ) with b k = p? ` m m m+ c n c` r n= `=` r= n r L p m?r;k+`?n ; where ` = maxfn? k? N; g; ` = minfn? k + N; p? g and a kj = n c n c n+j?k ;?N j; k N: This matrix A was considered by Lawton [] for a sucient and necessary condition on c k (and hence m ()) for constructing orthonormal wavelets basis. It is noted that A has an eigenvalue

6 6 W. Shann and C. Yen with an associated eigenvector v = ( k ). The condition is that is a nondefective eigenvalue of A. Daubechies [7] summerized it as one of the three equivalent conditions. Since (x) is innitely many times dierentiable as p!, dierentiate (:4) to have (s) (x) = s k c k (s) (x? k): R Apply this equation into (x) (s) (x? k) dx and repeat the steps in (4:), we nd that the matrix A = A(p) will asymptotically have eigenvalues,,, 4, : : :, as p!. Actually A has a simple pattern. On the even-numbered columns, by (:8), all entries are except a k;k =. On the odd-numbered columns all entries are except for d j?p e k b j+p c, which is the p-element vector P R k c kc k+(p?) (x)(x? p? ) dx h = P P. P k c kc k+3 P k c kc k+ P k c kc k+ k c kc k+3. k c kc k+(p?) C A = R. (x)(x? 3 ) dx R (x)(x? ) dx R (x)(x? ) dx R (x)(x? 3 ) dx R. (x)(x? p? ) dx Proposition 4.. Let A be the matrix dened in this section. k k be the matrix norm induced by the Euclidean norm. We have kak p : C A : Proof. Dene matrices B = c c c p? c c c p? c p?... c c cp? C A (N+)(N+P ) and D = c p? c p?... c c c3 C A : c c That is, B ij = c j?i and D ij = c (p?)+j?i. Observe that (N+)(N+P ) A = D B T : By (:8), DD T is a diagonal matrix and clearly its spectrum radius is. Hence kdk = p.

7 Wavelets quadratures 7 Then Let x be a (N + P ) dimensional vector. Extend x to (~x k ) ` such that kbxk = ~x k = n xk if k (N + P ) otherwise. = = N+ i= N+ i= N+ i= = ( j;k ( k j j;k j;k 4 kxk : c j?i ~x j c j?i c k?i ~x j ~x k c j c k ~x j+i ~x k+i N+ c j c k ) ( i= N+ c k ) ( i= ~x j+i ~x k+i ) N+ ~x j+i ) = ( i= ~x k+i ) = Therefore kbk. Fortunately the linear system (4:) is not needed for m = (actually b = and I? A is singular). Hence for all cases with p (p = is an empty case) and m, equation (4:) can be solved by the Neumann series (3:). As in the previous section, we list values of L p m;k with smaller p and m on Table 3. For each m, values for k =? p; : : :; p? are listed from top down and from left to right. Except those for m =, only those for k =? p; : : :; are listed. Because L p ;k = (x + k)(x + k) (x) dx = x(x)(x + k) dx = L p ;?k : The integrals x m (x)(x? k) dx and x m (x)(x? k) dx? can be evaluated by the combination of techniques described in Sections 4 and 5. It is slightly tedious but otherwise straight forward. REFERENCES E. Bacry, S. Mallat and G. Papanicolaou, A wavelet based space-time adaptive numerical method for partial dierential equations, RAIRO Math Modelling Numer. Anal., 6: 793{834 (99) G. Beylkin R. Coifman and V. Rokhlin, Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math., 44: 4{83 (99) 3 G. Beylkin, On the presentation of operators in bases of compactly supported wavelets, SIAM J. Numer. Anal., 6: 76{74 (99)

8 8 W. Shann and C. Yen 4 G. Beylkin, R. Coifman and V. Rokhlin, Wavelets in numerical analysis, in Wavelets and their applications, Jones and Barlett Publishers, Boston, W. Dahmen and C. A. Micchelli, Using the renement equation for evaluating integrals of wavelets, SIAM J. Num. Anal., 3: 57{537 (993) 6 I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 4: 99{996 (988) 7 I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, R. Glowinski, W. M. Lawton, M. Ravachol and E. Tenenbaum, Wavelets solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension, in Computing methods in applied sciences and engineering, SIAM, Philadelphia, S. Jaard, Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal., 9: 965{986 (99) B. Jawerth and W. Sweldens, An overview of wavelet based multiresolution analysis, Math. Rev. to appear. W. Lawton, Necessary and sucient conditions for constructing orthonormal wavelet bases, J. Math. Phys., 3: 57{6 (99) S. Mallat, Multifrequency channel decompositions of images and wavelet models, IEEE Trans. Acoust. Speech Signal Process, 37: 9{ (989) 3 S. Mallat, A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. on Patt. Anal. Mach. Intell., : 674{693 (989) 4 Y. Maday, V. Perrier and J.-C. Ravel, Adaptativite dynamique sur bases d'ondelettes pour l'approximation d'equations aux derivees partielles, C. R. Acad. Sci. Paris, Serie I, 3: 45{ 4 (99) 5 V. Perrier, Towards a method for solving partial dierential equations using wavelet bases, in Wavelets, Time-Frequency Methods and Phase Space, Springer-Verlag, Berlin, G. Strang, Wavelets and dilation equations: A brief introduction, SIAM Review, 3: 64{67 (989) 7 W. Sweldens and R. Piessens, Quadrature formulae for the calculation of the wavelet decomposition, SIAM J. Numer. Anal. to appear. 8 J. u and W.-C. Shann, Galerkin-wavelets methods for two-point boundary value problems, Numer. Math., 63: 3{44 (99)

9 Wavelets quadratures 9 p c k mantissa E ? ? ? ?? ? ? ?? ?? ? ? ?3? ? ? ? ? ? ? ? ? ? ?? ?? ? ?? ?? ? ? ?? ?3? ? ?4? ?4? ? ?5? ?6 p c k mantissa E ? ? ? ? ? ? ? ?? ? ? ?? ?3? ? ?3? ? ? ? ? ? ? ? ? ?? ? ? ?? ?? ? ? ?4? ? ? ?4? ? ?6 p c k mantissa E ? ? ? ? ? ?? ? ? ?? ?? ? ?? ?3? ? ? ?4? ? ? ? ? ?? ? ? ?? ? ?? ?? ? ? ?? ? ? ?3? ?3? ? ?4? ?5 Table. Values of c ; : : :; c p? computed by solving ~ F p, for p = 9; : : :; 4.

10 W. Shann and C. Yen m p = ? ?? ? ?? ? ? 3? ? ?3 4? ? ?3 5? ? ?4 m p = ? ? ? ? ?? ? ? ?5? ? ?? ?3? ?5 3? ? ?? ?3? ? ? ? ? ? ? ? ?6 m p = ? ?? ?? ? ? ? ? ?? ?3? ? ?6? ? ?? ? ? ?5? ?8 3? ? ?? ? ? ?5? ?8 4? ? ?? ? ? ?5? ? ? ?? ?? ? ?5? ?9 m p = ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ?? ? ? ? ? ? 3? ?? ? ?? ?4? ?5? ?8? ? 4? ? ?? ? ?? ?4? ?5? ?8? ? Table. Values of N p m;k, index k is assumed from top down for k =?;?; : : :;? p.

11 Wavelets quadratures m p = ?? ? ?? ? ? ? ?3? ? ? ? ?3? ? ? m p = 3? ?5? ? ?? ? ?6? ? ?? ? ? ?? ? ? ? ?? ? ? ?? ? ? ? ?? ? ? ?? ? m p = 4? ? ? ?3? ? ?? ? ? ?5? ?5? ? ?? ? ? ? ? ?4? ?7 3? ? ?5? ?3? ? ? ? ? ? ?3? ?6 4? ? ?5? ?3? ? ? ? ? ? ?? ?5 m p = 5? ?? ?8? ?5? ? ?? ? ?? ? ?9? ? ? ?? ? ? ? ? ?? ?3? ?4? ?7? ? Table 3. Values of L p m;k, index k is assumed from top down for k = :p; : : :; p:, except for m =.

2 W. LAWTON, S. L. LEE AND ZUOWEI SHEN is called the fundamental condition, and a sequence which satises the fundamental condition will be called a fu

2 W. LAWTON, S. L. LEE AND ZUOWEI SHEN is called the fundamental condition, and a sequence which satises the fundamental condition will be called a fu CONVERGENCE OF MULTIDIMENSIONAL CASCADE ALGORITHM W. LAWTON, S. L. LEE AND ZUOWEI SHEN Abstract. Necessary and sucient conditions on the spectrum of the restricted transition operators are given for the