Wavelets of Federbush–Lemarié Type

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

201, 274]296 Ž1996.

0255

Wavelets of Federbush]Lemarie ´ TypeU Guy Battle Department of Mathematics, Texas A&M Uni¨ ersity, College Station, Texas 77843 Communicated by Joseph D. Ward Received July 6, 1993

1. INTRODUCTION Three of the most basic properties of a wavelet that reflect its usefulness are its degree of regularity, the estimation law for its decay over distance, and the order up to which its moments vanish. The third property is often an important one in mathematical physics. Perhaps the simplest example of this can be found in electrostatics. If a charge distribution is expanded in wavelets}and arbitrarily large-scale wavelets must appear if they have more vanishing moments than the charge distribution does}then the expansion induces a multi-scale decomposition of the Coulomb potential into potentials of wavelets. If the wavelets themselves have compact support or exponential decay, then their potentials obey an inverse power law with exponent comparable to the number of vanishing moments. In this way, the long-distance analysis of a long-range interaction is reduced to controlling arbitrarily large-scale contributions from interactions that decay more rapidly in the scale-commensurate sense. It is clear from this example alone that one would very much like to increase the number of vanishing moments in the construction of a wavelet, preferably without spreading it out}i.e., without increasing the support in the case of a sharply localized wavelet or reducing the exponential decay rate in the case of an exponentially localized wavelet. This has actually been done in the former case by Strang, Massopust, Geronimo, et al. through an ingenious use of fractal interpolation functions w1]4x. These wavelets are more useful}computationally}than their exponentially localized analogs, but the latter can be constructed from more basic principles, and in this paper we construct this latter type of wavelet. U

Supported in part by the National Science Foundation under Grant DMS 9024867. 274

0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

275

A fundamental theorem on interscale-orthogonal wavelets states that the degree of smoothness guarantees at least a comparable number of vanishing moments w5x. On the other hand, it appears to be well-established by all known constructions that one cannot increase the degree of smoothness without spreading the wavelet w6]11x. However, the theorem does not rule out the existence of wavelets that have far more vanishing moments than derivatives. The simplest example of this theoretical loophole is the Federbush basis w12x, which we briefly review here. We restrict our attention to one dimension for convenience. For a given positive integer N, there are N q 1 ‘‘mother wavelets’’ in the construction associated with N. They have the form

cm Ž x . s PmŽ0. Ž x . x Ž 2 x . q PmŽ1. Ž x . x Ž 2 x y 1 . ,

Ž 1.1.

where PmŽ i . Ž x . are polynomials of degree F N and x is the characteristic function for the unit interval. In a word, the basis generated is a polynomial generalization of the Haar basis, but we should say several words. Consider the Ž2 N q 2.-dimensional space of discontinuous functions supported by w0, 1x and equal to polynomials of degree F N on each of the sub-intervals w0, 12 . and Ž 12 , 1x. Clearly, the space of such functions equal to polynomials on the whole interval is an Ž N q 1.-dimensional subspace. The functions cm are chosen to form an orthonormal basis for the orthogonal complement of this subspace. This is the crux of the Federbush scheme, and the beauty of it is that the interscale orthogonality property is precisely the condition that all moments of order 0, 1, . . . , N vanish. The point is that cmŽ2yr x y n. is a polynomial Žor zero. on w m, m q 1x }the support of cmŽ x y m. }for all positive integers r. Remark. A specific choice of the Federbush basis was later constructed independently by Alpert, Beylkin, Coifman, and Rokhlin w13]15x in the context of developing fast multi-scale algorithms for computing potential interactions. Some Žbut not all. of the N q 1 functions can have even more vanishing moments, and this was a computational advantage. What we wish to emphasize about this example is that the mother wavelets cm have the same support as the single mother wavelet for the Haar basis}and yet the moments of order 0, 1, . . . , N vanish. Since only the 0th-order moment of the Haar wavelet vanishes, the Federbush basis is the kind of example that interests us. Indications are that Daubechies wavelets of the Federbush type also exist}i.e., compactly supported class C My « wavelets with M q N y 1 vanishing moments and support width 0Ž M .. Some of these wavelets have been constructed by Strang, Geronimo, Massopust, and their cohorts.

276

GUY BATTLE

The aim of this paper is to construct Lemarie ´ wavelets w7x of the Federbush type, and we stick to one dimension throughout our description. The real obstacle in the case of class C M wavelets is that the interscale orthogonality is no longer equivalent to the vanishing of moments. Nevertheless, our block spin construction w6x of Lemarie ´ wavelets can be adapted to yield wavelets with M q N y 1 vanishing moments. Let Fm Ž x . s x mx Ž x .

Ž 1.2.

and check that we have the scaling relation N

Fm Ž x . s

N

Ý

ns0

Ž0. Cmn Fn Ž 2 x . q

Ý CmnŽ1. Fn Ž 2 x y 1. ,

Ž0. Cmn s 2ymdmn ,

¡ ¢0,

2 s~

ym

Ž1. Cmn

Ž 1.3.

n s0

Ž 1.4.

ž mn /

,

mGn,

Ž 1.5.

m -n.

Remark. In most wavelet constructions, there is only one scaling function, even when there are several mother wavelets. In this construction there are N q 1 scaling functions, so the coefficients for the scaling relation are matrices. The crux of our scheme is to minimize 5 w 5 22 with respect to the constraints

Hw Ž x . f Ž x y n . dx s 2 m

m

n

, Ž y1. KmŽ1yn. l

Ž 1.6.

M Ž m. Ž0. Kmn s M Cmn q C Ž1. , m m y 1 mn

ž

ž /

/

1 F m F M, Ž 1.7.

Ž0. Ž0. Kmn s Cmn ,

Ž 1.8.

Ž Mq1. Ž1. Kmn s Cmn ,

Ž 1.9.

Ž m. Kmn s 0,

other m g Z, M

fˆm Ž p . s x ˆ Ž p . Fˆm Ž p . ,

Ž 1.10. Ž 1.11.

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

277

where the guiding star is the scaling relation N

fm Ž x . s

Ý Ý KmnŽ n. fn Ž 2 x y n . .

Ž 1.12.

n n s0

The solution wl is a mother wavelet, and the set

 wl Ž 2yr x y 2 n . : n, r g Z and l s 0, 1, . . . , N 4 is a basis. The functions here are interscale orthogonal but not intrascale orthogonal. Moments through order M q N vanish. Remark. The reason for the 2 n-translation is that the mother wavelets we construct are actually associated with the interval 0 F x F 2. This is the unit-scale convention that this author has always used, but it is not the usual one. We immediately give the reason why the functions are interscale orthogonal and discuss the vanishing moments in Section 4. One key observation is that N

Hw

s

Ž 2 x . fm Ž x y m . dx s Ý

Ý KmnŽ n.Hws Ž 2 x . fn Ž 2 x y 2 m y n . dx

n n s0

s

1 2

N

Ý Ý KmnŽ n.Hws Ž x . fn Ž x y 2 m y n . dx n n s0

Ž 1.13. and therefore

Hw

s

Ž 2 x . fn Ž x y m . dx s

N

1

n Ž n. Ž1y2 myn. Kns . Ž 1.14. Ž y1. Ý 2 n Kmn Ý 2

n s0

n

However, the most important observation is that}although C Ž0. does not commute with C Ž1. }we do have the weighted commutation N

N

ns0

n s0

Ž1. Ž1. Ž0. s Ý 2 n Cmn Cns , Ý 2 n CmnŽ0. CnsŽ1. s Cms

Ž 1.15.

278

GUY BATTLE

and this is precisely what we need. Since the K Ž n. are just various linear combinations of C Ž0. and C Ž1., we have the same relation for any pair of the K Ž n. : N

X

Ý

ns0

Ž n. Ž n . 2 n Kmn Kns s

N

X

Ý 2 n KnnŽ n .KnsŽ n. .

Ž 1.16.

n s0

The immediate consequence is that the sum over n on the r.h.s. of Ž1.14. is actually reversed in sign by the mere index change n ¬ 1 y 2 m y n. Thus

Hw

s

Ž 2 x . fm Ž x y m . dx s 0

Ž 1.17.

for all m and m. The third observation is that the minimizer wl is automatically orthogonal to anything annihilated by the bounded linear functionals

w ¬ w Ž x . fm Ž x y m . dx,

H

so in particular, wlŽ x . is orthogonal to ws Ž2 x .. It is easy to extend Ž1.17. to ws Ž2 r x y l . for l g Z and positive integers r. This establishes the interscale orthogonality, and this means the dual basis can be computed scale by scale. Recall that in the case of ordinary Lemarie ´ wavelets Ž N s 0. the exponential decay rate of the dual wavelet is half the decay rate of the one that is initially constructed w8x. Such a simple relationship cannot hold in the Federbush]Lemarie ´ case, as we shall see in Section 6, where we derive the momentum expression for the dual wavelets. We find the momentum space solution for the wl in Section 2 and establish exponential localization of wlŽ x . in Section 3. Naturally this means proving the real analyticity of w ˆlŽ p . and}as so often happens w16]18x }this in turn involves a subtle cancellation of poles in the momentum expression. The key in this case is to control the poles of an inverse matrix expression at the points where the matrix is singular. This is technically different from w16]18x, where the matrix to be inverted was singular everywhere with a p-independent kernel that the matrix algebra in the total p-expression ‘‘missed.’’ Although the construction of these wavelets is motivated by the study of long-range potentials of wavelets, it is very interesting to study how the exponential decay rate of the wavelet itself is affected by this enhancement of the vanishing-moment property. In Section 5 we compute this rate of decay for the M s N s 1 case: it turns out to be lnŽ3 q 2'2 ., while the rate of decay for the M s 1, N s 0 case is known w8x to be lnŽ2 q '3 .. Thus the decay rate is larger for N s 1 than for N s 0. An interesting observation about the dual wavelet in the M s N s 1 case is that the rate of exponential decay is equal to that of the original wavelet in the M s 1, N s 0 case. We show how the dual wavelets are

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

279

constructed in Section 6, but as always, they are obtained from inversion of the overlap matrix, which, in our case, is given by Ž m n. Slk s wl Ž x y 2 m . wk Ž x y 2 n . dx.

Ž 1.18.

H

ŽRemember our unit-scale convention: the ‘‘mother-scale’’ wavelets are generated by 2 n-translates of our choice of mother wavelets, which are splines with integer knots..

2. CONSTRUCTION OF THE MOTHER WAVELETS Our basic construction tool is constrained minimization w6, 8, 16, 17, 19x, which we often refer to as a ‘‘block spin’’ construction. It is related to the computation of renormalization group transformations w20x but also has much in common with the multi-scale resolution analysis of Mallat and Meyer w10, 11, 21, 22x. As we have already indicated in the Introduction we start with scaling functions Žas they do., but we minimize the L2-norm with respect to averaging constraints based on the scaling functions. In this section we fix the index l s 0, 1, . . . , N for the mother wavelet we construct. We also write our constraints in a way slightly different from the way they were written in the Introduction. We did this in w6, 8x as well, because it made the constraints easier to visualize. It turns out to be even more convenient here. U Let the superscript denote complex conjugation. Consider the problem 2 of minimizing 5 w 5 2 with respect to the constraints 1

Hp

U

M

w ˆ Ž p . eyi n p Fˆm Ž p . dp s 2 m Ž CmŽ1.l dn0 y CmŽ0.l dn1 . .

Ž 2.1.

As usual, we first try to minimize 5 w 5 22 q a 2 Ý n

Ý m

žH

1 pM

2 yi n p

w ˆŽ p. e

U Fˆm Ž p . dp y 2 m Ž CmŽ1.l dn0 y CmŽ0.l dn1 .

/

Ž 2.2. for finite a ) 0. The condition on the minimizer is

w ˆŽ p. q s

a2 p

1

M

a2 pM

Ý Ý Fˆm Ž p . e i n pH q M wˆ Ž q . eyi n q Fˆm Ž q . n

dq

m

Ž1. y CmŽ0.l e i p . Fˆm Ž p . , Ý 2 m Ž Cml

m

U

Ž 2.3.

280

GUY BATTLE

which can be solved by the Neumann series `

k

1

k

k Ý Ž ya 2 . p M Ł Ý

w ˆa Ž p . s a 2

ž /ž /ž ŁÝ

js1 m j

ks0 k

1

=Ł

q j2 M

js1

js1 n j

e i n j Ž q jy 1yq j . Fˆm jŽ q jy1 . Fˆm jŽ q j .

k

Ý Hdq j js1

/

U

Ž1. = Ý 2 m Ž Cml y CmŽ0.l e i q k . Fˆm Ž qk .

Ž 2.4.

m

for sufficiently small a ) 0, where q0 s p is understood. Clearly, `

w ˆa Ž p . s a 2

Ý

Ž ya 2 .

Ý

n1 , m 1

ks0 k

=

k

1

k

p

ž /

Fˆ p e i n1 p M m 1Ž .

ŁÝ js2 n j

k

ž /

Ł Ý Ł bm js2 m j

js2

jy 1m j

Ž n jy1 y n j .

Ž1. = Ý 2 m Ž Cml bm km Ž n k . y CmŽ0.l bm km Ž n k y 1 . . ,

Ž 2.5.

m

bmn Ž n . s

Hq

1

U

2M

Fˆm Ž q . Fˆn Ž q . eyi n q dq,

Ž 2.6.

and so we have

w ˆa Ž p . s a 2

`

k Ý Ž ya 2 . Ý

n1 , m 1

ks0

= Ý Ž BŽ t. m

sa2

1

Ý

n1 , m 1

pM

k

1 p

M

Fˆm 1Ž p . e i n1 p

2p yi n t 1

H0

e

. m m 2 m Ž CmlŽ1. y CmŽ0.l e it . dt 1

Fˆm1Ž p . e i n1 p

2p yi n t 1

H0

e

y1

= Ý Ž 1 q a 2 B Ž t . . m1 m 2 m Ž CmŽ1.l y CmŽ0.l e it . dt,

Ž 2.7.

m

B Ž t . mn s

Ý bmn Ž n . e i n t s n

Ý m

Fˆm Ž t q 2p m .

Ž t q 2p m .

U

2M

Fˆn Ž t q 2p m . .

Ž 2.8.

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

281

The last equation follows from the Poisson summation formula.

w ˆa Ž p . s

a2 pM

y1 Ý Fˆm Ž p . Ž 1 q a 2 B Ž p . . m m2 m Ž CmŽ1.l y CmŽ0.l e i p . Ž 2.9.

m1 , m

1

1

obviously holds for all a ) 0, and the existence of the a s ` limit depends on the invertibility of the matrix B Ž p .. LEMMA 2.1.

B Ž t . is in¨ ertible all t f 2p Z.

Proof. Suppose B Ž t 0 . is singular for some t 0 . Then there is an Ž N q 1.© vector © ¨ / 0 with B Ž t 0 . ¨ s 0. Since ©

©

©

¨ ? B Ž t0 . ¨ s

FˆŽ t 0 q 2p m . ?© ¨

Ý

Ž t 0 q 2p m .

m

2

,

2M

Ž 2.10.

© it follows that FˆŽ t 0 q 2p m. ?© ¨ s 0 for all m. On the other hand, we have

Fˆm Ž p . s i m si

d m ei p y 1 dp m

my1

ip m

Ý

ns0

s m!

i m y1 p mq1

n

m

m i m y n e i pn ! Ž y1 . y Ž y1 . i m y1m ! n p nq1 p m q1

ž /

m ip

Ž y1. e

m

Ý

Ž yip . n!

ns0

n

yi

m y1

Ž y1.

m

p m q1

m !, Ž 2.11.

where we have used the index change n ¬ m y n in the last equation. Consider the rational function

Qm Ž w . s m !

Ž y1.

mq1

w m q1

e

it0

m

wn

Ý

n!

ns0

y m!

Ž y1.

m q1

w m q1

Ž 2.12.

and note that ©

FˆŽ t 0 q 2p m . ?© ¨s

N

Ý ¨m Qm Ž yit0 y i2p m . .

ms0

Ž 2.13.

282

GUY BATTLE

But ÝmNs0¨m QmŽ z . can have only N q 1 zeros, so we have the desired contradiction unless this linear combination is identically zero. But this possibility generates the conditions N

e

it 0

Ý

ms l y1

¨m

m ! Ž y1 .

mq1 l

Ž m q 1 y l. !

y Ž l y 1 . ! Ž y1 . ¨ly1 s 0,

Ž 2.14.

which annihilates the components of © ¨ by recursion, provided e it 0 / 1 Žwhich is assumed.. We still have to show that the singularities remaining in the momentum expression are removable, if we expect to show that our solution w` has exponential decay in position space. We have to control the behavior of an inverse matrix close to the points p s 2p n where the invertibility breaks down, and we accomplish this in the next section.

3. REAL ANALYTICITY IN MOMENTUM SPACE We now consider the singularities p s 2p n, and we find it convenient to write our momentum expression for w` as

w ˆ`Ž p . s xˆ Ž p . Ž eyi p y 1 . M

M

Ý Fˆm Ž p . Ž AŽ p . y1 . m

m1 , m

1

1

m

Ž1. = 2 m Ž Cml y CmŽ0.l e i p . ,

Ž 3.1.

A Ž t . mn s < e i t y 1 < 2 M B Ž t . mn s

Ý xˆ Ž t q 2p m .

2M

U Fˆm Ž t q 2p m . Fˆn Ž t q 2p m .

m

s

U Ý fˆm Ž t q 2p m . fˆn Ž t q 2p m . .

Ž 3.2.

m

We now have a matrix-valued function that is continuous everywhere and still invertible for t f 2p Z. The serious issue to be addressed is that det A Ž 2p n . s 0,

n g Z,

Ž 3.3.

adj A Ž t .

Ž 3.4.

and we need to know the behavior of AŽ t .

y1

s Ž det A Ž t . .

y1

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

283

for t close to 2p n. Clearly, U

2M

A Ž t . mn y fm Ž t y 2p n . fn Ž t y 2p n . s 0 Ž Ž t y 2p n .

m , n / 0,

.,

Ž 3.5. for such values of t, because the m / yn terms in Ž3.2. have zeros at t s 2p n that are of multiplicity 2 M. Similarly, U

A Ž t . m 0 y fˆm Ž t y 2p n . fˆ0 Ž t y 2p n . s 0 Ž Ž t y 2p n .

2 Mq1

.,

m / 0,

Ž 3.6. A Ž t . 00 y fˆ0 Ž t y 2p n .

2

s 0 Ž Ž t y 2p n .

2 Mq2

..

Ž 3.7.

On the other hand, N

sgn p Ý Ž y1. Ł AŽ t . mp Ž m .

det A Ž t . s

ms0

p gS Nq1

s

sgn p Ý Ž y1.

Ý

L; 0, 1, . . . , N 4 p gS Nq1

= =

Ł

mfL

ž AŽ t .

mp Ž m .

U y fˆm Ž t y 2p n . fˆp Ž m . Ž t y 2p n .

/

U Ł ž fˆm Ž t y 2p n . fˆp Ž m . Ž t y 2p n . /

mgL

'

Ý

det AL , n Ž t . .

Ž 3.8.

L; 0, 1, . . . , N 4

Since the rows of AL , n Ž t . that lie in L are proportional, the determinant vanishes for < L < G 2, so we have the reduction N

det A Ž t . s

Ý

det An 4, n Ž t .

ns0 N

s

Ý

sgn p Ý Ž y1.

ns0 p gS Nq1

=

Ł

m/ n

ž AŽ t .

mp Ž m . U

U

y fˆm Ž t y 2p n . fˆp Ž m . Ž t y 2p n .

= fˆn Ž t y 2p n . fˆp Ž n . Ž t y 2p n . .

/ Ž 3.9.

284

GUY BATTLE

Combining this with the estimates above, we conclude: For t close to 2p n,

THEOREM 3.1.

det A Ž t . s 0 Ž Ž t y 2p n .

2 MN

..

Ž 3.10.

These are the zeros in the denominator of the momentum expression that must be cancelled. Since the n s 0 case turns out to be distinctive, we assume first that n / 0. Then

xˆ Ž p . s 0 Ž p y 2p n . ,

Ž 3.11.

and so M

x ˆ Ž p . Ž eyi p y 1 .

M

s 0 Ž Ž p y 2p n .

2M

..

Ž 3.12.

On the other hand, Ž . Ž adj AŽ p . . mn s 0 Ž Ž p y 2p n . 2 M Ny1 .

Ž 3.13.

by the same reasoning as above. Hence

Ž AŽ p . y1 . mn s 0 Ž Ž p y 2p n . y2 M . ,

Ž 3.14.

and we have the desired cancellation. We have only the n s 0 case to consider before we have proven: THEOREM 3.2.

w` has exponential decay.

Proof. The remaining obstacle is to control the momentum expression for p close to 0, and the problem is that

xˆ Ž 0 . s 1.

Ž 3.15.

x ˆ Ž p . M plays a more subtle role in the cancellation now. Moreover, for a reason that will be clear in Section 3, we cannot afford to burn up the factors M

Ž eyi p y 1 . s 0 Ž p M .

Ž 3.16.

at p s 0. All of the cancellation must come from

xˆ Ž p .

M

Ý Fˆm Ž p . Ž adj AŽ p . . m m s Ý fˆm Ž p . Ž adj AŽ p . . m m1

1

1

m1

1

s det AŽ m . Ž p . ,

1

m

Ž 3.17.

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

285

where AŽ m . Ž p . is the matrix obtained from AŽ p . by replacing the m th row with the entries fˆm1Ž p ., 0 F m 1 F N Žsince Žadj AŽ p ..m 1m is Žy1. m1qm times the mm 1th minor.. Consider the same differences U X A Ž p . mm y fˆmX Ž p . fˆm 1Ž p . s 0 Ž p 2 Mq dmX 0q dm 10 . 1

Ž 3.18.

X

for m / m that were considered above. In this case, det AŽ m . Ž p . s

sgn p Ý Ž y1. Ł AŽ p . m p Ž m . fˆp Ž m . Ž p . X

p gS Nq1

s

sgn p Ý Ž y1. Ł

p gS Nq1

X

mX/ m

X

m/

ž AŽ p . m

mXp Ž mX .

U

y fˆmX Ž p . fˆm 1Ž p .

= fˆp Ž m . Ž p .

/

Ž 3.19.

because if we expand in subsets L ;  0, 1, . . . , N 4 _  m4 , the L / f terms are determinants of matrices with at least one row proportional to the m th row, so only the L s f term does not vanish. Hence det AŽ m . Ž p . s 0 Ž p 2 M N . ,

Ž 3.20.

and the cancellation at p s 0 is accomplished. Given these arguments, it is now easy to see that the momentum expression for w` is analytic on some strip centered around the real axis. The only possible poles in the complex plane are zeros of det AŽ p ., and we have shown that these zeros contribute no poles on the real axis. Since AŽ p . is periodic, we can therefore obtain a lower bound on the width of the neighborhood of analyticity about the real axis. The degree of regularity for w`Ž x . is obvious from the momentum expression. Since 1 Fˆm1Ž p . s 0 Ž 3.21. p

ž /

for large p, the factor x ˆ Ž p . M yields class C My « smoothness in position space.

4. VANISHING MOMENTS AND INTERSCALE ORTHOGONALITY We now determine the number of vanishing moments, which motivated this construction. Recall that in the last section we removed the singularity at p s 0 without using the factors M

Ž eyi p y 1 . s 0 Ž p M . ,

Ž 4.1.

286

GUY BATTLE

so we know that at least the moments of order 0, 1, . . . , M y 1 vanish. The point is that there are more vanishing moments, and the easiest way to see this is to examine the construction instead of the momentum expression. THEOREM 4.1.

For k s 0, 1, . . . , M q N y 1,

Hw Ž x . x `

k

dx s 0.

Ž 4.2.

Proof. Because of the vanishing moments we clearly do have, we know that at least

w ˆ`Ž p . s 0 Ž p M .

Ž 4.3.

for small p. Let g be defined by

ˆg Ž p . s

1 pM

w ˆ`Ž p . ,

Ž 4.4.

which is analytic on a strip centered on the real axis. Hence g Ž x . has exponential decay as well as class C 2 My « smoothness. We need to show that

Hg Ž x . x

k

dx s 0,

k s 0, 1, . . . , N.

Ž 4.5.

But the averaging constraints satisfied by w` imply

Hg Ž x . x

k

dx s

Ý Hg Ž x . x Ž x y n . x k dx n k

s

k

n ky m

Hg Ž x . F Ž x y n . dx

k

n ky m

Hp

žm/ Ý Ý žm/ Ý Ý žm/ Ý Ý

n m s0 k

s

n m s0 k

s

k

n m s0

s 0,

1 M

U

w ˆ`Ž p . eyi n p Fˆm Ž p . dp

Ž1. n ky m 2 m Ž Cml dn0 y CmŽ0.l dn1 .

k

s 2 k CkŽ1.l y

m

Ý

ms0

k 2 m m CmŽ0.l

ž /

Ž 4.6.

where the last equality follows from Ž1.4. and Ž1.5. in the Introduction.

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

287

We have already discussed the issue of interscale orthogonality in the Introduction. Clearly, the constraints we have used can be written in the form yi n p

Hwˆ Ž p . e

ž

M

xˆ Ž p . Fˆm Ž p .

U

/

M

dp s 2 m

Ý Ž y1. Myl ls0

=

M l

ž /Ž

Ž1. Cml dnql , 0 y CmŽ0.l dnql , 1 . ,

Ž 4.7. which reduces, case by case, to Ž1. , 2 m Cml

2 m Ž y1 .

Mq n

M C ž yn /

Ž1. ml

n s yM,

y 2 m Ž y1 .

Mqny1

ž

M C Ž0. , 1 y n ml

Ž0. y2 m Ž y1 . Cml , M

yM - n F 0,

/

n s 1,

other n.

0, Thus yi n p

Hwˆ Ž p . e

U

fˆm Ž p . dp s Ž y1 .

Mqn

KmŽ1yn. . l

Ž 4.8.

5. EXPONENTIAL DECAY RATE FOR N s M s 1 As we have already mentioned in the Introduction, there is the interesting question of how the exponential localization is affected by our vanishing-moment enhancement of the wavelet construction. We initiate the investigation here by examining the N s M s 1 case, but the answer depends on finding the roots of polynomials given by determinants of matrix arrays of Euler]Frobenius polynomials w23x. The mother wavelet wl has the position space representation N

wl Ž x . s Ž n. Glm sc 1

2 p yi n p

H0

e

N

Ý

m s0

Ý Ý

n m 1s0

Ž n. Glm f Ž x y n. , 1 m1

2 m Ž CmŽ1.l y CmŽ0.l e i p . Ž A Ž p .

y1

Ž 5.1.

. m m Ž eyi p y 1. 1

M

dp

Ž 5.2.

288

GUY BATTLE

for arbitrary M and N because AŽ p . is periodic. The mn th entry of this matrix is given by a variation on the familiar class of trigonometric polynomials. A Ž p . mn s

Ý xˆ Ž p q 2p m .

dm

2 M ny m

i

dp m

m

=

dn dp n

x ˆ Ž p q 2p m .

U

xˆ Ž p q 2p m . .

Ž 5.3.

The entry AŽ p . 00 yields the familiar class indexed by M, but the denominator we get in the integrand when we invert the whole matrix is the determinant of AŽ p ., which is much more complicated. In principle, this determinant can be written as a polynomial in z s e i p , so the computation Ž n. of the coefficients Glm can be done by residues. As in w8x, the exponential 1 decay rate of the mother wavelet is

r s ln

1

ž / < a0 <

,

Ž 5.4.

where a 0 is a root of the polynomial which has the largest modulus among the roots in the unit disk’s interior. We now focus on the N s M s 1 case. Thus AŽ p . is the 2 = 2 matrix Ý m J Ž p q 2p m. with

xˆ Ž p .

4

J Ž p. s 2

yi x ˆ Ž p . xˆ Ž p .

2

i x ˆ Ž p . xˆ Ž p . d

dp

xˆ Ž p .

U

x ˆ Ž p.

2

d dp

U

d dp

x ˆ Ž p. 2

.

x ˆ Ž p.

Ž 5.5. We compute J Ž p . 00 s J Ž p . 01 s

16 sin 4 Ž Ž 1r2 . p . p4

4 Ž e i p y 1 . sin 2 Ž Ž 1r2 . p .

q

p4

,

Ž 5.6.0.0.

16 sin 4 Ž Ž 1r2 . p . ip 5

, Ž 5.6.0.1.

U

J Ž p . 10 s J Ž p . 01 , J Ž p . 11 s

4 sin

2

Ž Ž 1r2. p . p

4

y

2

Ž Ž 1r2. p .

p

5

8 sin p sin

Ž 5.6.1.0. q

16 sin

4

Ž Ž 1r2. p . p6

.

Ž 5.6.1.1.

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

289

Using the familiar trigonometric formulas for Ý mŽ1rŽ p q 2p m. 2 . and its derivatives w24x to calculate the entries of the matrix AŽ p ., we obtain A Ž p . 00 s 23 cos 2 Ž 12 p . q 13 ,

Ž 5.7.0.0.

A Ž p . 01 s y 31 cos 2 Ž 21 p . y 61 y i 61 cos Ž 21 p . sin Ž 21 p . ,

Ž 5.7.0.1.

A Ž p . 10 s A Ž p . A Ž p . 11 s

2 15

cos

2

U 01 ,

Ž p. q 1 2

Ž 5.7.1.0. 7 60

,

Ž 5.7.1.1.

where obvious identities such as e i p y 1 s Ž cos p y 1 . q i sin p,

Ž 5.8.

sin p s 2 cos Ž p . sin Ž p . , 1 2

1 2

Ž 5.9.

1 y cos p s 2 sin 2 Ž 12 p .

Ž 5.10.

have reduced the expressions. Finally, AŽ p . s

Ž 2 q cos p . 1 y 6 Ž 2 q cos p . q i 121 sin p 1 3

y 16 Ž 2 q cos p . y i 121 sin p 1 15

cos p q

11 60

Ž 5.11. and therefore det A Ž p . s

1 720

Ž cos 2

p y 4 cos p q 3 . .

Ž 5.12.

Clearly the factors are cos p y 1 and cos p y 3. The former has zeros 2p n of multiplicity 2, as expected from the analysis in Section 3, and it contributes no pole to the integrand of Ž5.2.. The residues in the contour integration are due to the zeros of the polynomial 2 z Ž 12 Ž z q zy1 . y 3 . s z 2 y 6 z q 1

Ž 5.13.

obtained from the substitution z s e i p. But the root in the interior of the unit disk is 3 y 2'2 , so

r s ln

ž

1 3 y 2'2

This is the exponential decay rate.

/

s ln Ž 3 q 2'2 . .

Ž 5.14.

290

GUY BATTLE

6. THE DUAL BASIS Since our basis of wavelets is only interscale orthogonal, we need the dual basis when we expand a function in these wavelets. On the other hand, the dual basis is computed scale by scale, so we may consider only the unit scale without loss. Following the same derivation w8x that we carried out in the Lemarie ´ case, we first consider the overlap matrix Ž m n. Slk s

2p i2Ž myn.t

e

H0

Wlk Ž t . s

Wlk Ž t . dt,

Ž 6.1.

Ý wˆl Ž t q p l . wˆk Ž t q p l .

U

.

Ž 6.2.

l

This expression is an immediate consequence of Ž1.17. in the Introduction. We decompose Ž6.2. into even and odd p-multiple translates: Wlk Ž t . s WlkŽ0. Ž t . q WlkŽ0. Ž t q p . , WlkŽ0. Ž t . s < eyi t y 1 < 2 M

Ž 6.3.

Ý Ý AŽ t . m m X 1

m 1 , m mX1 , mX

1

U

X

y1

X

. m m Ž AŽ t . y1 . m m 2 mq m Ž CmlŽ1. y CmŽ0.l e i t . = Ž CmŽ1.k y CmŽ0.k eyi t . y1 Ž1. s < eyit y 1 < 2 M Ý Ž A Ž t . . m m 2 m q m Ž Cml y CmŽ0.l e it . = Ž AŽ t .

X 1

1

X

X

1

1

m1 , m

= Ž CmŽ1.1k y CmŽ0.1k eyi t . .

Ž 6.4.

The dual basis  ulŽ?y 2 m.4 is given by ul Ž x y 2 m . s

Ž m n.

Ý Ž Sy1 . lk

n, k

wk Ž x y 2 n . ,

Ž 6.5.

and the task is to derive a momentum expression that makes the correlation lengths of ul computable. We begin by introducing the convenient notation Emn s 2 mdmn , C Ž t . s E Ž C Ž1. y e i t C Ž0. . ,

Ž 6.6. Ž 6.7.

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

291

so that we may write our Eq. Ž6.4. as a matrix equation, W Ž0. Ž t . s < eyit y 1 < 2 M C Ž t .

†

Ž AŽ t . y1 .

†

U

CŽ t. ,

Ž 6.8.

where the superscript † denotes transpose. On the other hand, Ž2.11. implies m ip

Fˆm Ž p . s Ž y1 . e

Ý

ns0 m

s e i p Ž y1 .

n

m

m! m n ! Ž y1 . y n q1 m q1 n Ž ip . Ž ip .

ž /

N

U

Ý C Ž p . mn Ž y1.

n!

n

ns0

Ž ip .

nq1

.

Ž 6.9.

If we set m

n

L Ž t . mn s Ž y1 . C Ž t . mn Ž y1 . ,

Ž 6.10.

then we may write ©

FˆŽ p . s e i p L Ž p . © v Ž p. , U

vn Ž p . s

n!

Ž ip .

n q1

Ž 6.11.

.

Ž 6.12.

Combining this with Ž3.2., we obtain A Ž t . s 4 M sin 2 M G Ž p . mn s

1

†

t L Ž t . G Ž t . L Ž t .U ,

ž / 2

1

Ý l

Ž p q 2p l .

Ž 6.13.

U

2M

vm Ž p q 2p l . vn Ž p q 2p l . , Ž 6.14.

and so W Ž0. Ž t . s C Ž t .

†

†

†

U

Ž L Ž t . y1 . Ž G Ž t . y1 . Ž L Ž t . y1 .

U

CŽ t. .

Ž 6.15.

Now to compare the momentum expression of wl and ul , we may first write Ž3.1. as the matrix equation ©

M

ˆ Ž p . s xˆ Ž p . Ž eyi p y 1 . C Ž p . w M

†

Ž AŽ p . y1 .

†©

FˆŽ p .

Ž 6.16.

292

GUY BATTLE

and reduce it to ©

M

ˆ Ž p . s 4yM e i pxˆ Ž p . Ž eyi p y 1 . csc 2 M Ž 12 p . C Ž p . w M

=Ž GŽ p.

†

Ž L Ž p . y1 .

y1 †©

.

v Ž p.

†

Ž 6.17.

via Ž6.11. and Ž6.13.. On the other hand, we can write Ž6.5. in momentum space, ©

M

ˆŽ p . s xˆ Ž p . Ž eyi p y 1 . W Ž p . u M

y1

C Ž p.

†

Ž AŽ p . y1 .

†©

†

Ž L Ž p . y1 .

†

FˆŽ p . Ž 6.18.

and this becomes ©

M

ˆŽ p . s 4yM e i pxˆ Ž p . Ž eyi p y 1 . csc 2 M Ž 12 p . u M

y1

= W Ž0. Ž p . q W Ž0. Ž p q p . =Ž GŽ p.

C Ž p.

y1 †©

v Ž p.

.

M

s 4yM e i px ˆ Ž p . Ž eyi p y 1 . csc 2 M Ž 12 p . M

=Ž C Ž p q p . =

y1 U

U

Ž L Ž p . y1 .

.

U

LŽ p q p . GŽ p q p .

C Ž p.

†

U

U

Ž C Ž p q p . y1 .

†

qG Ž p . L Ž p . Ž C Ž p . =Ž LŽ p q p .

y1 †

.

† U

LŽ p q p . GŽ p q p .

CŽ p q p .

†

y1 y1 † ©

v Ž p. ,

.

†

Ž 6.19.

where the last step eliminates the occurrence of all GŽ t .y1 from the brackets. Since C Ž t . and C Ž s . commute, we can eliminate the appearance of all inverses in the brackets to obtain ©

M

ˆŽ p . s 4yM e i pxˆ Ž p . Ž eyi p y 1 . csc 2 M Ž 12 p . u M

=Ž C Ž p q p .

y1 U

.

= H Ž p. q H Ž p q p . U

U

U

†

†

LŽ p q p . GŽ p q p . LŽ p q p . C Ž p. y1 †

U

U

C Ž p q p . LŽ p. © v Ž p. , †

†

H Ž p. s C Ž p q p . LŽ p. GŽ p. LŽ p. C Ž p q p . .

†

Ž 6.20. Ž 6.21.

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

293

This completes our derivation of the momentum expression for the dual mother wavelets ul. The correlation length of ul is determined by the set of those zeros of det H Ž p . q H Ž p q p . that are not on the real axis. We can disregard Ž C Ž p q p .y1 .U in Ž6.20. because U

det C Ž p q p . s Ž 1 q eyi p .

Nq 1

,

Ž 6.22.

which has zeros only on the real axis. The point here is that all singularities of Ž6.20. on the real axis can be shown to be removable by the same kind of arguments that were employed in Section 3. Unlike the case of ordinary Lemarie ´ wavelets, this case does not dictate the correlation length of the dual wavelet to be twice the correlation length of the wavelet directly constructed. On one hand, Ž6.14. reduces to G Ž p . mn s i my nm !n ! Ý l

s

1 Ž y1 .

1

Ž p q 2p l .

mq n m y n

i

m !n !

mq n q2 Mq2

d 2 Mq m q n

4 Ž 2 M q m q n q 1 . ! dp

2 Mq m q n

csc 2

1

ž / 2

p

Ž 6.23.

and the correlation length of wl is determined by the set of zeros of det GŽ p .. On the other hand, the relation between this set and the set of zeros of the determinant of H Ž p . q H Ž p q p . is not clear at all. We spend the remainder of this section computing the correlation length of ul for the case M s N s 1. To the extent of obtaining the matrix elements of GŽ p ., we have already analyzed this case in Section 5. By the double angle formulas, we have 1

†

GŽ p. s

1920

=

csc 6

1

ž / 2

p

20 Ž 1 y cos p . Ž cos p q 2 .

i5 sin p Ž cos p q 5 .

yi5 sin p Ž cos p q 5 .

cos 2 p q 13 cos p q 16

.

Ž 6.24. Now clearly, C Ž p. s

1 y ei p 1

0 , 1 y ei p

Ž 6.25.

LŽ p. s

1 y ei p y1

0 , 1 y ei p

Ž 6.26.

294

GUY BATTLE

so we have U

i sin p y1

U

C Ž p q p . L Ž p . s 2 eyi p †

0 , i sin p

yi sin p 0

†

LŽ p. C Ž p q p . s 2 e i p

Ž 6.27.

y1 . yi sin p

Ž 6.28.

We may now write everything in terms of z s e i p , so we have the analytic extensions U

z2 y 1 y2 z

U

C Ž p q p . L Ž p . ¬ zy2 †

1 y z2 0

†

LŽ p. C Ž p q p . ¬

†

GŽ p. ¬ y

=

0 , z2 y 1 y2 z , 1 y z2

z6

1

120 Ž z 2 y 1 . 6 20 Ž 2 z y z 2 y 1 .Ž 4 z q z 2 q 1 .

5 Ž z 2 y 1 .Ž 10 z q z 2 q 1 .

y5 Ž z y 1 .Ž 10 z q z q 1 .

Ž z q 1. q 26 z Ž z 2 q 1. q 64 z 2

2

2

2

2

.

Matrix multiplication yields the analytic extension H Ž p. ¬ y

z4

1

120 Ž z 2 y 1 . 5

Ž H 00 Ž z . q H 01 Ž z . q H 10 Ž z . q H 11 Ž z . . ,

2

H 00 Ž z . s y80 z 2 Ž z y 1 . Ž z 2 q 4 z q 1 .

0 0

0 , 1

Ž 6.29.0.0.

H 01 Ž z . s y Ž z 2 y 1 . =

0

0

40 z Ž z y 1 . Ž z 2 q 4 z q 1 .

y10 z Ž z 2 y 1 .Ž z 2 q 10 z q 1 .

2

H 10 Ž z . s Ž z 2 y 1 .

40 z Ž z y 1 . Ž z 2 q 4 z q 1 .

0

10 z Ž z 2 y 1 .Ž z 2 q 10 z q 1 .

,

Ž 6.29.1.0.

2

2

=

Ž 6.29.0.1.

2

0

H 11 Ž z . s y Ž z 2 y 1 .

,

y20 Ž z y 1 . Ž z 2 q 4 z q 1 .

5 Ž z 2 y 1 .Ž z 2 q 10 z q 1 .

y5 Ž z 2 y 1 .Ž z 2 q 10 z q 1 .

Ž z 2 q 1 . q 26 z Ž z 2 q 1 . q 64 z 2

2

.

Ž 6.29.1.1.

WAVELETS OF FEDERBUSH ] LEMARIE ´ TYPE

295

Now as we have already pointed out, the same analysis that was carried out in Section 3 shows that the singularities of u ˆlŽ p . on the real axis are removable, so the zeros and poles of det H Ž p . q H Ž p q p . on the real axis are irrelevant. Hence, the zeros and poles of det y

z4

1

X

120 Ž z 2 y 1 . 6

X

Ý Ž H ii Ž z . q H ii Ž yz . .

i, i

X

on the unit circle < z < s 1 are irrelevant, and so we may consider det

X

X

Ý Ž H ii Ž z . q H ii Ž yz . .

i, i

X

without loss. ŽThe powers of z may be discarded, since the z s 0 root is a contribution corresponding to compact support anyway.. ExtractingX even powers of z, we can easily calculate the matrix expression for H ii Ž z . q ii X Ž H yz .. The determinant of the sum of these matrix expressions is slightly complicated at first glance, but it reduces to det

X

X

Ý Ž H ii Ž z . q H ii Ž yz . .

i , iX

4

s 10 Ž z 2 y 1 . Ž z 8 q 80 z 6 q 926 z 4 q 80 z 2 q 1 . .

Ž 6.30.

Since the factor Ž z 2 y 1. 4 is irrelevant, we have reduced the problem to finding the zeros of z 8 q 80 z 6 q 926 z 4 q 80 z 2 q 1 s Ž z 4 q 66 z 2 q 1 . Ž z 4 q 14 z 2 q 1 . . Ž 6.31. The solutions of z 4 q 14 z 2 q 1 s 0

Ž 6.32.

inside the unit disk are

'

z s "i 7 y 4'3 ,

Ž 6.33.

z 4 q 66 z 2 q 1 s 0

Ž 6.34.

while those of

296

GUY BATTLE

are complex numbers with a smaller modulus:

'

z s "i 33 y 8'17 .

Ž 6.35.

Hence

r s ln

ž'

1

7 y 4'3

/

s

1 2

ln Ž 7 q 4'3 . s ln Ž 2 q '3 . ,

Ž 6.36.

and so we have the exponential decay rate as claimed in the Introduction.

REFERENCES 1. G. Strang and V. Strela, Short wavelets and matrix dilation equations, MIT preprint. 2. G. Donovan, J. Geronimo, D. Hardin, and P. Massopust, Construction of wavelets using fractal interpolation functions, SIAM J. Math. Anal., in press. 3. J. Geronimo and D. Hardin, J. Math. Anal. Appl. 176 Ž1993., 561. 4. J. Geronimo, D. Hardin, and P. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory, in press. 5. G. Battle, J. Math. Phys. 30 Ž1989., 2195. 6. G. Battle, Comm. Math. Phys. 110 Ž1987., 601. 7. P. Lemarie, ´ J. Math. Pures Appl. 67 Ž1988., 227. 8. G. Battle, Cardinal spline interpolation and the block spin construction of wavelets, in ‘‘Wavelets}A Tutorial in Theory and Applications’’ ŽC. Chui, Ed.., p. 73, Academic Press, Boston, 1992. 9. C. Chui and J. Wang, A general framework of compactly supported splines and wavelets, Trans. Amer. Math. Soc., in press. 10. Y. Meyer, Sem. ´ Bourbaki 38 Ž1985]1986., 662. 11. I. Daubechies, Comm. Pure Appl. Math. 41 Ž1988., 909. 12. P. Federbush, Comm. Math. Phys. 81 Ž1981., 327. 13. G. Beylkin, R. Coifman, and V. Rokhlin, Wavelets in numerical analysis, in ‘‘Wavelets and Their Applications’’ ŽG. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, and B. Ruskai, Eds.., p. 181, Bartlett and Jones, Cambridge, 1992. 14. A. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, SIAM J. Sci. Comput. 14, No. 1 Ž1993., 159. 15. A. Alpert, SIAM J. Math. Anal. 24, No. 1 Ž1993., 246. 16. G. Battle and P. Federbush, Divergence-free vector wavelets, Michigan Math. J. 40 Ž1993., 181. 17. G. Battle and P. Federbush, Space-time wavelet basis for the continuity equation, Appl. Comp. Harmonic Anal. 1 Ž1994., 284. 18. P. Federbush and C. Williamson, J. Math. Phys. 28 Ž1987., 1416. 19. G. Battle, Comm. Math. Phys. 114 Ž1988., 93. 20. G. Battle, Wavelets: A renormalization group point of view, in ‘‘Wavelets and Their Applications’’ ŽG. Beylkin, R. Coifman, I Daubechies, S. Mallat, Y. Meyer, L. Raphael, and B. Ruskai, Eds.., p. 323, Bartlett and Jones, Cambridge, 1992. 21. Y. Meyer, Ondelettes et functions splines, in ‘‘Seminaire Equations aux Derivees Par´ tielles, Ecole Polytechnique, Paris, December 1986.’’ 22. S. Mallat, Trans. Amer. Math. Soc. 315 Ž1989., 69. 23. I. Schoenberg, J. Approx. Theory 2 Ž1969., 167. 24. C. Chui, ‘‘An Introduction to Wavelets,’’ Academic Press, Boston, 1992.