POINT VALUES AND NORMALIZATION OF TWO-DIRECTION MULTIWAVELETS AND THEIR DERIVATIVES

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1 November 1, 1 POINT VALUES AND NORMALIZATION OF TWO-DIRECTION MULTIWAVELETS AND THEIR DERIVATIVES FRITZ KEINERT AND SOON-GEOL KWON,1 Abstract Two-direction multiscaling functions φ and two-direction multiwavelets ψ associated with φ are more general and more flexible setting than one-direction multiscaling functions and multiwavelets In this paper, we investigate how to find and normalize point values and those of derivatives of the two-direction multiscaling functions φ and multiwavelets ψ For finding point values, we investigate the eigenvalue approach For normalization, we investigate the normalizing conditions for them by normalizing the zeroth continuous moment of φ Examples for illustrating the general theory are given 1 Introduction Two-direction multiscaling functions φ and two-direction multiwavelets ψ associated with φ, which are more general setting than one-direction multiscaling functions and multiwavelets, have been investigated in a few papers in the literature, for example, see 4, 6, 13, 14, 15 The two-direction setting is more flexible than one-direction setting In multiwavelet theory, computation of point values and those of derivatives for the one- or two-direction multiscaling functions and multiwavelets associated with multiscaling functions is important Point values of multiscaling functions and multiwavelets are needed to reconstruct a function from its expansion coefficients Point values and those of derivatives for the one- or orthogonal two-direction multiscaling functions and multiwavelets associated with multiscaling functions are necessary in many applications such as solution of differential equations, signal processing, and image processing In one-direction multiwavelet theory, two approaches, called the cascade algorithm approach and the eigenvalue approach, are nown for finding point values and those of derivatives of the multiscaling functions and multiwavelets (see 1,, 3) The cascade algorithm approach computes approximate point values and those of derivatives of the multiscaling functions and multiwavelets The eigenvalue approach computes exact point values and those of derivatives of the multiscaling functions and multiwavelets, but may fail in some cases Point values and those of derivatives of the multiscaling functions and multiwavelets can all be computed from the recursion coefficients Both the cascade algorithm approach and the eigenvalue approach can also be considered for finding point values and those of derivatives of the two-direction multiscaling functions φ and multiwavelets ψ Since the cascade algorithm approach for the two-direction case is not much different from that for the one-direction case, we do not pursue the cascade algorithm approach for φ and ψ in this paper We pursue only the eigenvalue approach for φ and ψ in this paper All calculations of norms, point values, and moments only give the answer up to an arbitrary constant This reflects the fact that the refinement equation only defines φ up to an arbitrary factor When we calculate several quantities, how do we mae consistent choices? This is called 1 Mathematics Subject Classification 4C4 Key words and phrases two-direction multiwavelets, point values, normalization, derivatives Corresponding author 1 This paper was supported by (in part) Sunchon National University Research Fund in 1 1

2 normalization and is another issue for the two-direction multiwavelets which is pursued in this paper Normalization of point values and those of derivatives of the one-direction multiscaling functions and multiwavelets associated with multiscaling functions can be achieved by fixing the value of some quantity related to multiscaling functions We investigate how to normalize point values and those of derivatives of the two-direction multiscaling functions φ and multiwavelets ψ via the normalization of the zeroth continuous moment of φ The main objective of this paper is to investigate the following for orthogonal two-direction multiwavelets: compute point values of orthogonal multiscaling function φ via the eigenvalue approach; determine correct normalization for point values of φ in terms of the zeroth continuous moment m := φ(x) dx of φ; compute point values of orthogonal multiwavelets ψ (s), s = 1,,, d 1, associated with φ; compute point values of the nth derivative D n φ via the eigenvalue approach; determine correct normalization for point values of D n φ in terms of the jth continuous moments m j := xj φ(x) dx for j =, 1,, n; compute point values of D n ψ (s), s = 1,,, d 1 This paper is organized as follows Some basic notions and properties of the two-direction multiscaling functions φ and multiwavelets ψ are introduced in section In section 3, the main results for finding and normalizing point values of φ and ψ are introduced In section 4, the main results for finding and normalizing point values of derivatives of φ and ψ are introduced Finally, examples for illustrating the general theory in sections, 3, and 4 are given in section 5 Two-direction multiscaling functions and multiwavelets In this section we review and investigate orthogonal two-direction multiscaling functions and orthogonal two-direction multiwavelets associated with orthogonal two-direction multiscaling functions (see 13, 14, 15) A two-direction multiscaling function of multiplicity r and dilation factor d is a vector of r real-valued functions φ(x) = φ 1 (x), φ (x),, φ r (x) T, x R, which satisfies a recursion relation φ(x) = d P φ(dx ) P φ( dx) (1) and generates a multiresolution approximation of L (R) The P, P, called the positive- and negative-direction recursion coefficients for φ, respectively, are r r matrices For standard one-direction (multi)wavelets, the scaling function is a linear combination of scaled and shifted versions of itself For two-direction (multi)wavelets, it is a linear combination of scaled and shifted versions of itself and of its reverse Two-direction multiwavelets ψ (s), s = 1,,, d 1, associated with φ satisfy ψ (s) (x) = d Q (s) φ(dx ) Q (s) φ( dx), s = 1,,, d 1 () The Q (s), Q (s) are called the positive- and negative-direction recursion coefficients for ψ (s) By taing Fourier transform on both sides of (1), we have φ(ξ) = P (e iξ/d ) φ(ξ/d) P (e iξ/d ) φ(ξ/d), (3)

3 P (z) = 1 P d z and P (z) = 1 P d z, (4) for z = 1 and z C, are called positive- and negative-direction mas symbols, respectively We rewrite the two-direction multiscaling function (1) as P φ( dx ) P φ( dx) (5) φ( x) = d By taing Fourier transform on both sides of (5), we have φ(ξ) = P (e iξ/d ) φ(ξ/d) P (e iξ/d ) φ(ξ/d) (6) From (3) and (6), we have φ(ξ) P (e iξ/d ) P (e iξ/d ) φ(ξ/d) = (7) φ(ξ) P (e iξ/d ) P (e iξ/d ) φ(ξ/d) We see that (1) has a solution if and only if (7) has a solution Let us combine two-direction multiscaling functions φ(x) and φ( x) to construct a onedirection multiscaling function Φ(x) so that all the (one-direction) multiwavelet theory applies to Φ(x) Let Φ(x) = φ(x) = d φ( x) P P P P Φ(dx ) (8) be the matrix refinable equation of Φ Equation (8) is called the deduced d-scale matrix refinement equation of the two-direction refinement equation (1) Then (7) is the d-scale matrix refinement equation in frequency domain of Φ Its refinement mas is P (z) P (z) P Φ (z) = (9) P (z) P (z) If P and P, Z, are real, then P Φ (z) = 1 P d P P P z (1) In this paper we only consider real recursion coefficients P, P, Q(s), and Q (s) in R r r for s = 1,,, d 1 and Z Orthogonality Two-direction multiscaling function φ and multiwavelet ψ (s), s = 1,,, d 1, associated with φ are orthogonal if for all j, Z, and s, t = 1,,, d 1, φ(x j), φ(x ) = δ j I r, φ(x j), φ( x) = O r, ψ (s) (x j), ψ (t) (x ) = δ st δ j I r, ψ (s) (x j), ψ (t) ( x) = O r, φ(x j), ψ (s) (x ) = O r, φ(x j), ψ (s) ( x) = O r, 3

4 I r is the r r identity matrix and O r is the r r zero matrix Here the inner product is defined by φ, ψ (s) = φ(x)ψ (s) (x) dx, * denotes the complex conjugate transpose This inner product is an r r matrix Condition E A matrix A satisfies Condition E if it has a simple eigenvalue of 1, and all other eigenvalues are smaller than 1 in absolute value Condition E for φ defined in (1) means that the matrix P (1) P (1) P (1) P (1) = 1 d P P P P (11) satisfies Condition E (see 1, Theorem 3 for d = ) For the scalar case r = 1, since the eigenvalues of the matrix in (11) are P (1) P (1) and P (1) P (1), Condition E for φ is equivalent to P (1) P (1) = 1, P (1) P (1) < 1, or P (1) P (1) = 1, P (1) P (1) < 1 (see 13, Theorem ) Moments It is well-nown that continuous moments of the one-direction orthogonal (or biorthogonal) multiscaling function and multiwavelets can be computed if the recurrence coefficients of φ and ψ are given (see 3, 7) An algorithm for computing continuous moments of the two-direction multiscaling function φ and multiwavelets ψ was derived in 6 Continuous moments of φ and ψ can be computed if the recurrence coefficients of φ and ψ are given Here we review some results for moments derived from 6 Let us define the jth discrete moment of two-direction multiscaling function φ by an r r matrix M j = 1 j P P, j =, 1,, (1) d Now we separate the jth discrete moment M j into positive and negative parts We define the jth positive and negative discrete moments, M j and M j respectively, of the discrete moment M j of φ by M j = 1 j P, M j = 1 j P, (13) d d for j =, 1,, so that M j = M j M j The jth continuous moment m j of two-direction multiscaling function φ is a vector of size r defined by m j = x j φ(x) dx, j =, 1,, (14) The following lemma was proved in 6 For the completeness of the paper, we provide it here again Lemma 1 (6) (Normalization for m ) If a two-direction multiscaling function φ L 1 L is orthogonal and Condition E for φ is satisfied, then m m = 1 (15) 4

5 Proof By expanding the constant 1 in the two-direction orthogonal multiscaling function series, we have 1 = 1, φ(x ) φ(x ) 1, φ( x) φ( x) = m φ(x ) φ( x) By integrating (16) on, 1, we have = 1 dx = m φ(x ) φ( x) dx = m Hence, we have equation (15) (16) φ(x) φ( x) dx = m m In the orthogonal two-direction scalar case, the normalization is m = (17) Theorem (6) The continuous and discrete moments of two-direction multiscaling function φ are related by j ( ) j M m j = d j l j l ( 1)l Mj l ml, j =, 1,, (18) In particular, Moreover, l= m j = ( d j I r M ( 1) j M ) 1 j 1 l= m = M m ( ) j M l j l ( 1)j Mj l ml, j = 1,,, (19) I r is the identity matrix of size r r Once m has been chosen, all other continuous moments are uniquely defined and can be computed from these relations Approximation Order The two-direction multiscaling function approximation to a function f at resolution d j is given by the series P j f = φ f, j φ j f, φ j φ j, () φ j (x) = d j/ φ(d j x ) and φ j (x) = d j/ φ( d j x) The two-direction multiscaling function φ provides approximation order p if f P n f = O(d np ), whenever f has p continuous derivatives The two-direction multiscaling function φ has accuracy p if all polynomials of degree up to p 1 can be expressed locally as linear combinations of integer shifts of φ(x) and φ( x) That is, there exist row vectors (c j, ) and (c j, ), j =,, p 1, Z, so that x j = (c j, ) φ(x ) (c j, ) φ( x) (1) 5

6 Let y l = c l, and y l = c l, It is assumed that at least one of y and y is not the zero vector This assumption will be clear after the basic regularity condition in Theorem 5 The vectors y l and y l are called the approximation vectors Theorem 3 Approximation vectors and continuous moments of φ are related by and for j =, 1,, y j = m j, y j = ( 1) j y j = ( 1) j m j, (y j y j ) = 1 ( 1) j m j, (y j y j ) = 1 ( 1) j m j, Proof We multiply equation (1) by φ(x) and integrate to obtain () (3) (y j ) = (c j, ) = x j, φ(x) = m j (4) We multiply equation (1) by φ( x) and integrate to obtain (y j ) = (c j, ) = x j, φ( x) = ( 1) j m j (5) By adding and subtracting equations in (), we obtain (3) According to (), for j =, 1,,, y j can be expressed in terms of y j, that is, y j = ( 1) j y j We use only one notation y j instead of y j from now on and do not use y j any more by removing positive, that is, we let y j := y j Theorem 4 The coefficients c j, and c j, in equation (1) have the form j ( ) j c j, = j l y l l, l= j ( ) j c j, = j l ( 1) l y l l, y j = c j, = ( 1)j c j,, and y is not the zero vector l= Here y is the same vector as in Theorem 5 Proof Replace x j by (x ) j in equation (1) and expand (6) A high approximation order is desirable in applications A minimum approximation order of 1 is a required condition in many theorems The following theorem will be used in proving theorem 31 Theorem 5 (4) Assume that two-direction multiscaling function φ is a compactly supported L -solution of the refinement equations with nonzero integral and linearly independent shifts Then there exists a nonzero vector y R r such that φ(x ) φ( x) = c, (7) y c is a constant Moreover, the nonzero vector y is related to the zeroth continuous moment of φ by y = m 6

7 Proof Sine the two-direction recursion relation (1) has a compactly supported, L (R)-stable solution vector φ of size r, its deduced d-scale matrix refinement equation (8) has a one-direction compactly supported, L (R)-stable solution vector Φ of size r It is well-nown in multiwavelet theory that Φ satisfies the following condition (see 3, 1): There exists a nonzero vector y R r such that y Φ(x ) = c, c is a constant Let y R r be a vector of the upper-half part of y R r in (b) By (4) and (5), we have a structure of y such that y = 1, Φ = m, m = y, y (8) Since y is a nonzero vector, y is also a nonzero vector by (8) By (8) and (7), we have y ) φ( x) = y φ(x, y φ(x ) = y φ( x) Φ(x ) = c, c is a constant This completes the proof Throughout this paper we assume that the two-direction multiscaling function φ is orthogonal, has compact support, is continuous (which implies approximation order at least 1), and satisfies Condition E 3 Point values and normalization of the two-direction multiwavelets In multiwavelet theory, computation of point values and those of derivatives for the one- or two-direction multiscaling functions and multiwavelets is important, since it provides a method for plotting multiscaling functions and multiwavelets It is well-nown that point values and normalization of the one-direction multiscaling function and multiwavelets can be computed if the recurrence coefficients of multiscaling function and multiwavelets are given (see 3) In this section we investigate how to compute point values and their normalization of the orthogonal two-direction multiscaling functions φ and multiwavelets ψ (s), s = 1,,, d 1, in terms of the recurrence coefficients P, P of φ and Q(s), Q (s) of ψ (s) The cascade algorithm approach is a practical way for finding approximate point values of φ by iteration: for n =, 1,,, φ (n1) (x) = d P φ(n) (dx ) P φ(n) ( dx), (31) φ () (x) = max{1 x, } is the hat function and φ (n) stands for φ at the nth iteration There is another approach, called the eigenvalue approach, which produces exact point values of φ and ψ (s), s = 1,,, d 1 It usually wors, but may fail in some cases Since the cascade algorithm approach for the two-direction case is not much different from that for the one-direction case, we do not pursue the cascade algorithm approach but the eigenvalue approach in this section In this section we investigate how to compute point values of orthogonal two-direction multiscaling function φ via the eigenvalue approach; normalize point values of φ; compute point values of ψ (s), s = 1,,, d 1 7

8 Point values of the two-direction multiscaling functions Let N be the largest positive integer such that P N, P N, P N, or P N in (1) are nonzero matrices of size r r Then φ is a two-direction multiscaling function satisfying the refinement equation φ(x) = N d P φ(dx ) P φ( dx) (3) = N Suppose that φ in (3) satisfies Condition E for φ Then it is easy to prove that the support of φ in (3) is also contained in N/(d 1), N/(d 1) (Proof is given for φ(x) = N d P φ(dx ) P φ( dx) = = N in 14, Theorem 4) Let a and b be the smallest and largest integers in this interval For integer l, a l b, the refinement equation (3) leads to φ(l) = d = d N = N dl =dl N P φ(dl ) P N dl P dl φ() = dl φ( dl) P dl φ() Since φ() = for every Z \ a, b, we have φ(l) = b d P dl P dl φ(), a l b (33) This is an eigenvalue problem =a φ(a) φ(a) φ(a 1) = T φ(a 1) φ, (34) φ(b) φ(b) (T φ ) l = d (P dl P dl ), a l, b (35) We note that each column of T φ contains all of the P d l P dl for some fixed l The basic regularity condition (iii) in Theorem 5 implies that (y, y,, y ) is a left eigenvector to eigenvalue 1, so a right eigenvector also exists We assume that this eigenvalue is simple, so that the solution is unique (This is the place the algorithm could fail) Let 1 and be the smallest and largest integers such that P 1 and P are nonzero matrices, respectively Let 3 and 4 be the smallest and largest integers such that P 3 and P 4 are nonzero matrices, respectively In the case of dilation d =, the support of φ is contained in a, b N, N The first row of matrix equation (34) is either φ(a) = d P 1 φ(a) or φ(a) = d P 3 φ(b) The last row of matrix equation (34) is either φ(b) = d P φ(b) or φ(b) = d P 4 φ(a) 8

9 For the case that the first row is φ(a) = d P 1 φ(a), unless P 1 has an eigenvalue of 1/ d, the value of φ at the left endpoint is zero, and we can reduce the size of φ(a) φ(a 1) φ(b) T and T φ For the case that the last row is φ(b) = d P φ(b), unless P has an eigenvalue of 1/ d, the value of φ at the right endpoint is zero, and we can reduce the size of φ(a) φ(a 1) φ(b) T and T φ All calculations of norms, point values, and moments only give the answer up to an arbitrary constant This reflects the fact that the refinement equation only defines φ up to an arbitrary factor When we calculate several quantities, how do we mae consistent choices? This is called normalization and an issue to be pursued in this section Normalization of point values of the two-direction multiscaling function The normalization for point values of φ is given in the following Theorem Theorem 31 (Normalization of point values of φ) The correct normalization for point values of two-direction multiscaling function φ is ( ) φ() = m m (36) m Proof If φ satisfies the basic regularity conditions, then we have, by setting x = in (7), ) ) c = (y ) ( φ() = (m ) ( φ() By integrating equation (7) in theorem 5 on, 1, we have c = 1 c dx = y 1 Hence, we have (36) φ(x ) φ( x) dx = y φ(x) φ( x) dx = m m According to equation (36), unlie the scalar case, the sum of point values at the integers and the integral do not have to be the same They just have to have the same inner product with m For the orthogonal two-direction scalar case, the correct normalization for point values of φ is φ() = (37) Once the values of φ at the integers have been determined, we can use the refinement equation (3) to obtain values of φ at points of the form /d, Z, then /d, and so on to any desired resolution Point values of the two-direction multiwavelets Suppose that ψ (s), s = 1,,, d 1, are the two-direction multiwavelets associated with φ defined by ψ (s) (x) = d for some positive integer N N = N Q (s) φ(dx ) Q (s) φ( dx) 9 (38)

10 For s = 1,,, d 1 and integer l, a l b, the refinement equation (38) leads to ψ (s) (l) = N d Q (s) φ(dl ) Q (s) φ( dl) = d = N dl =dl N N dl Q (s) dl ψ() = dl Q (s) dl ψ() Since φ() = for every Z \ a, b, we have ψ (s) (l) = b d Q (s) dl Q(s) dl φ(), a l b (39) This is a multiplication =a ψ (s) (a) φ(a) ψ (s) (a 1) = T φ(a 1) ψ (s), (31) ψ (s) (b) φ(b) (T ψ (s)) l = d (Q (s) dl Q(s) dl ), a l, b (311) Note that each column of T ψ (s) contains all of the Q (s) d l Q(s) dl for some fixed l Once the values of φ at the integers have been determined, we can use the refinement equation (38) to obtain values of ψ (s) at points of the form /d, Z, then /d, and so on to any desired resolution 4 Point values and normalization of derivatives of the two-direction multiwavelets In this section we investigate how to compute point values of derivatives of two-direction multiscaling function φ and two-direction multiwavelets ψ (s), s = 1,,, d 1 associated with φ The main idea of the method is similar to the eigenvalue problem in Section 3 We can use the eigenvalue approach to compute point values of derivatives of φ (assuming they exist) In this section we investigate how to compute point values of the nth derivative D n φ via the eigenvalue approach; determine correct normalization for point values of D n φ in terms of the jth continuous moments m j := xj φ(x) dx for j =, 1,, n; compute point values of D n ψ (s), s = 1,,, d 1 Point values of derivatives of the two-direction multiscaling functions Let us recall (3) that φ satisfies the refinement equation φ(x) = N d P φ(dx ) P φ( dx) for some positive integer N Then (D n φ)(x) = d n N d = N = N P (Dn φ)(dx ) ( 1) n P (Dn φ)( dx) (41) 1

11 For integer l, a l b, the refinement equation (41) leads to (D n φ)(l) = d n d = d n d N = N dl =dl N Since D n φ() = for every Z \ a, b, we have (D n φ)(l) = d n d This is an eigenvalue problem P (Dn φ)(dl ) ( 1) n P (Dn φ)( dl) N dl P dl Dn φ() ( 1) n b =a = dl P dl Dn φ() P dl ( 1)n P dl D n φ() (4) D n φ = d n T D n φ D n φ, (43) (T D n φ) l = d (P dl ( 1)n P dl ), a l, b (44) So D n φ is an eigenvector of T D n φ to the eigenvalue 1/d n This eigenvalue must exist if φ is n times differentiable Normalization of point values of the nth derivatives of the two-direction multiscaling function The normalization for point values of the nth derivative D n φ is given in the following Theorem Theorem 41 (Normalization for point values of the nth derivative D n φ) The correct normalization for the nth derivative D n φ of the two-direction multiscaling function φ is given by n! n ( ( ) n l) = ( 1) n l m l n l D n φ() (45) l= Proof It is well-nown in multiwavelet theory that if φ has n continuous derivatives, it has approximation order at least n 1 By (1) and Theorem 4, we have x n = (c n, ) φ(x ) (c n, ) φ( x) = n ( n l) n l (y l ) φ(x ) ( 1) l φ( x) l= By differentiating n times and setting x = on both sides of the above equation, we have n ( n n! = l) n l (y l ) D n φ( ) ( 1) nl D n φ() = = l= n l= l= ( n l) ( 1) n l (y l ) 1 ( 1) l ( ) n l D n φ() n ( ( ) n ( 1) l) n l m l n l D n φ() 11

12 For future reference, we list some in detail Normalization for the nth derivative D n φ of φ for n = 1,, 3 are ( ) ( ) 1 = m Dφ() m 1 Dφ(), ( ) ( ) ( ) 1 = m D φ() m 1 D φ() m D φ(), ( ) ( ) ( ) 3 = m 3 D 3 φ() 3m 1 D 3 φ() 3m D 3 φ() m 3 ( ) D 3 φ() For the orthogonal two-direction scalar case, the correct normalization for point values of Dφ is ( ) ( ) 1 = Dφ() m 1 Dφ() (46) Once the values of D n φ at the integers have been determined, we can use the refinement equation (41) to obtain values of D n φ at points of the form /d, Z, then /d, and so on to any desired resolution Point values of derivatives of the two-direction multiwavelets Now we wor on point values of derivatives for two-direction multiwavelets Let us recall (38) that ψ (s), s = 1,,, d 1, satisfy the refinement equation ψ (s) (x) = d for some positive integer N Then (D n ψ (s) )(x) = d n N d = N N = N Q (s) Q (s) φ(dx ) Q (s) φ( dx) (D n φ)(dx ) ( 1) n Q (s) (D n φ)( dx) (47) For s = 1,,, d 1 and integer l, a l b, the refinement equation (47) leads to (D n ψ (s) )(l) = d n N d Q (s) (D n φ)(dx ) ( 1) n Q (s) (D n φ)( dx) = d n d = N dl =dl N N dl Q (s) dl (Dn φ)() ( 1) n = dl Q (s) dl (Dn φ)() Since D n ψ (s) () = for every Z \ a, b, we have (D n ψ (s) )(l) = d n b d Q (s) dl ( 1)n Q (s) dl (D n φ)() (48) This is a multiplication =a D n ψ (s) = d n T D n ψ (s) Dn φ, (49) (T D n ψ (s)) l = d (Q (s) dl ( 1)n Q (s) dl ), a l, b (41) 1

13 This D n ψ (s) must exist if D n φ exists Once the values of D n φ at the integers have been determined, we can use the refinement equation (47) to obtain values of D n ψ (s) at points of the form /d, Z, then /d, and so on to any desired resolution 5 Examples In this section, we give two examples for illustrating the general theory in sections, 3, and 4 Example 51 In this example we tae orthogonal two-direction scaling function φ of approximation order derived from BAT O (orthogonal balanced one-direction multiscaling function of order 8, 9) given in 5 We note that multiplicity r = 1 and dilation factor d = The nonzero recursion coefficients for φ are P 1 = , P = , P 3 = , P 4 = , P4 = , P5 = , P6 = , P 7 = The nonzero recursion coefficients for ψ are Q 1 = , Q = , Q 3 = , Q 4 = , Q 4 = Q 4, Q 5 = Q 3, Q 6 = Q, Q 7 = Q 1, Condition E for φ is satisfied φ and ψ are supported on 1/, 7/, 4 φ is orthogonal and ψ is also orthogonal Sobolev exponent of φ is 1631, same as that for BAT O (orthogonal balanced one-direction multiscaling function of order ) given in 8, 9 By (17) and (19) the jth continuous moments m j of φ for j =, 1, are m = 771, m 1 = 7 8 Point values of φ The eigenvalue problem in (34) is φ() φ() φ(1) φ() φ(3) = T φ(1) φ φ() φ(3), φ(4) φ(4) 1374 m = P4 P P 1 P4 P5 P 6 T φ = T φ T φ = P 4 P4 P 3 P5 P P6 P 1 P7 P6 P7 P 4 P 3 P P 4 The matrix T φ has the eigenvalues 1, 1783, 1536, 116, and The matrix T φ has the eigenvector T T φ() φ(1) φ() φ(3) φ(4)

14 to the eigenvalue 1 Since the correct normalization for φ is φ() = the normalizing constant is 1 Hence, φ is normalized already Point values of normalized φ at integers are φ() φ(1) φ() φ(3) φ(4) T T, Once the values of φ at the integers have been determined, we can use the refinement equation (3) to obtain values at points of the form /d, Z, then /d, and so on to any desired resolution Point values of ψ The multiplication in (31) is ψ() φ() ψ(1) ψ() ψ(3) = T φ(1) ψ φ() φ(3), ψ(4) φ(4) Q 4 Q Q 1 Q 4 Q 5 Q 6 T ψ = Q 4 Q 4 Q 3 Q 5 Q Q 6 Q 1 Q 7 Q 6 Q 7 Q 4 Q 3 Q Q 4 Hence, we have T T ψ() ψ(1) ψ() ψ(3) ψ(4) Once the values of φ and ψ at the integers have been determined, we can use the equation (39) to obtain values of ψ at points of the form /d, Z, then /d, and so on to any desired resolution See Figure 51 for the graphs of φ and ψ Point values of the first derivative Dφ Since not only T φ has the eigenvalue 1/, but also Sobolev exponent of φ, 1631, is greater than 15, the existence of the first derivative Dφ of φ is guaranteed The eigenvalue problem in (43) is Dφ() Dφ() Dφ(1) Dφ() Dφ(3) = d T Dφ(1) Dφ Dφ() Dφ(3), Dφ(4) Dφ(4) P4 P P 1 P4 P5 P 6 T Dφ = T φ T φ = P 4 P4 P 3 P5 P P6 P 1 P7 P6 P7 P 4 P 3 P P 4 14

15 (a) (b) (c) (d) Figure 51 (a) Graph of orthogonal two-direction scaling function φ of order derived from BAT O (b) Graph of orthogonal two-direction wavelet function ψ derived from BAT O (c) Graph of the first derivative Dφ of φ (d) Graph of the first derivative Dψ of ψ The matrix T Dφ has the eigenvalues 1/, 116, 359, 148, and The matrix T Dφ has the eigenvector Dφ() Dφ(1) Dφ() Dφ(3) Dφ(4) T T to the eigenvalue 1/ Since the correct normalization for point values of Dφ is Dφ() =, The normalizing constant is 1 /( ) 646 Hence, point values of normalized Dφ at integers are Dφ() Dφ(1) Dφ() Dφ(3) Dφ(4) T T 15

16 Point values of the first derivative Dψ The multiplication in (49) is Dψ() Dφ() Dψ(1) Dψ() Dψ(3) = d T Dφ(1) Dψ Dφ() Dφ(3), Dψ(4) Dφ(4) Q 4 Q Q 1 Q 4 Q 5 Q 6 T Dψ = T ψ T ψ = Q 4 Q 4 Q 3 Q 5 Q Q 6 Q 1 Q 7 Q 6 Q 7 Q 4 Q 3 Q Q 4 Hence, we have Dψ() Dψ(1) Dψ() Dψ(3) T Dψ(4) T See Figure 51 for the graphs of Dψ and Dψ Point values of the second derivative D φ and D ψ Since T φ does not have eigenvalue 1/4, we can not find point values of the second derivatives D φ and D ψ Example 5 In this example we tae two-direction multiscaling function φ and two-direction multiwavelet ψ associated with φ given in 11 We note that multiplicity r = and dilation factor d = We shift P, P, Q and Q so that the supports of φ and ψ are the same, while orthogonality of φ and ψ is ept The nonzero recursion coefficients for φ are P 1 = , P = , 7 P = , P3 = The nonzero recursion coefficients for ψ are Q 1 = , Q = , 1 Q = , Q 3 = These are shifted and differ from Wang, Zhou, and Wang 11 by a factor of 1/, due to differences in notation Condition E for φ is satisfied 16

17 By (17) and (19) (detailed computation can be found in 6, Example 4), the jth continuous moments m j of φ for j =, 1, are 1 m =, m 1 = , m = 1 98(4 14) (4 6 4) Point values of φ The eigenvalue problem in (34) is φ() φ(1) = T φ φ() φ(1), φ() φ() T φ = 1 P P P P 1 P3, P is the zero matrix of size The eigenvalues of T φ are 1, 1/, 87, 1614, and (twice) Hence, we can find point values of φ, ψ, Dφ and Dψ, but not D φ and D ψ The matrix T φ has the eigenvector φ() T φ(1) T φ() T T = T to the eigenvalue 1 Since the correct normalization for φ is ( ) φ() = m, m the normalizing constant is 6/ 147 Hence, point values of normalized φ at integers are 771 φ() =, φ(1) =, φ() = Once the values of φ at the integers have been determined, we can use the refinement equation (3) to obtain values at points of the form /d, Z, then /d, and so on to any desired resolution Point values of ψ The multiplication in (31) is ψ() ψ(1) = T ψ φ() φ(1), ψ() φ() T ψ = 1 Q Q Q Q 1 Q 3 Q Hence, point values of ψ at integers are ψ() =, ψ(1) =, ψ() =

18 Once the values of φ and ψ at the integers have been determined, we can use the equation (39) to obtain values of ψ at points of the form /d, Z, then /d, and so on to any desired resolution See Figure 5 for the graphs of φ and ψ (a) (b) (c) (d) Figure 5 Orthogonal two-direction multiscaling function of order and multiwavelet function: (a) Graph of φ 1 (b) Graph of φ (c) Graph of ψ 1 (d) Graph of ψ However, T φ has the eigenvalue 1/, we do not pursue Dφ The existence of the first derivative Dφ of φ is guaranteed if Sobolev exponent of φ is greater than or equal to 15 Sobolev exponent of φ in this example is 5 and the existence of Dφ is not guaranteed References 1 I Daubechies Orthonormal bases of compactly supported wavelets Comm Pure Appl Math, 41(7):99 996, 1988 I Daubechies Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, F Keinert Wavelets and multiwavelets Studies in Advanced Mathematics Chapman & Hall/CRC, Boca Raton, FL, 4 4 S-G Kwon Approximation order of two-direction multiscaling functions Unpublished results 5 S-G Kwon High order orthogonal two-direction scaling functions from orthogonal balanced multiscaling functions Unpublished results 6 S-G Kwon Two-direction multiwavelet moments Appl Math Comput doi:1116/jamc

19 7 S-G Kwon Characterization of orthonormal high-order balanced multiwavelets in terms of moments Bull Korean Math Soc, 46(1): , 9 8 J Lebrun and M Vetterli High order balanced multiwavelets In Proc IEEE ICASSP, Seatle, WA, pages , May J Lebrun and M Vetterli High-order balanced multiwavelets: theory, factorization, and design IEEE Trans Signal Process, 49(9): , 1 1 G Plona and V Strela From wavelets to multiwavelets In Mathematical methods for curves and surfaces, II (Lillehammer, 1997), Innov Appl Math, pages Vanderbilt Univ Press, Nashville, TN, G Wang, X Zhou, and B Wang The construction of orthogonal two-direction multiwavelet from orthogonal two-direction wavelet Unpublished results 1 C Xie and S Yang Orthogonal two-direction multiscaling functions Front Math China, 1(4):64 611, 6 13 S Yang and Y Li Two-direction refinable functions and two-direction wavelets with dilation factor m Appl Math Comput, 188():198 19, 7 14 S Yang and Y Li Two-direction refinable functions and two-direction wavelets with high approximation order and regularity Sci China Ser A, 5(1): , 7 15 S Yang and C Xie A class of orthogonal two-direction refinable functions and two-direction wavelets Int J Wavelets Multiresolut Inf Process, 6(6): , 8 Fritz Keinert, Department of Mathematics, Iowa State University, Ames, IA address: einert@iastateedu Soon-Geol Kwon, Department of Mathematics Education, Sunchon National University, Sunchon, 54-74, Korea address: sgwon@sunchonacr 19