Construction of non separable
dyadic compactly supported
orthonormal wavelet bases
for
L 2(
R 2) of arbitrarily
high regularity
Antoine Ayache
Abstract.
By means of simple computations, we construct new classes of non separable QMF's. Some of these QMF's will lead to non separa-ble dyadic compactly supported orthonormal wavelet bases for L2(R 2)
of arbitrarily high regularity.
1. Introduction.
In the most general sense, wavelet bases consist of discrete families of functions obtained by dilations and translations of well chosen fun-damental functions 8], 9]. In this paper we will focus on compactly supported dyadic orthonormal wavelet bases forL2(
R
2), they are of the
form f2ji(2jx 1 ;ki2jx 2 ;li) : jkili 2Z i= 123g:
I. Daubechies has constructed compactly supported wavelet bases for
L2(
R) of arbitrarily high regularity, generalising the classic Haar basis 37
6]. The most commonly used method to construct compactly sup-ported wavelet bases for L2(
R
2) of arbitrarily high regularity, is the
tensor product method 9]. It leads to the scaling function '(x1x2) =
'1(x1)'2(x2) and to the fundamental wavelets
a(x1x2) ='1(x1)2(x2)
b(x1x2) =1(x1)'2(x2)
and
c(x1x2) =1(x1)2(x2)
(where '1 (respectively '2) is a scaling function for L 2(
R) and 1
(re-spectively 2) is the corresponding fundamental wavelet). The scaling
functons and the wavelets that result from the tensor product method are called separable. In this paper, we will also call separable the scaling functions and the wavelets that are the images of separable scaling func-tions and wavelets by an isometry ofL2(
R
2) of the typef(x)
7;!f(Bx)
(B2SL(2Z)).
Let us now give an outline of the present article.
In the second section, by means of simple computations we con-struct new classes of bidimensional non separable QMF's (Theorems 2.2 and 2.3).
In the third section, we show that some of these QMF's gener-ate non separable, compactly supported, orthonormal wavelet bases for
L2( R
2) of arbitrarily high regularity. These wavelets will be constructed
by two methods:
The rst method consists in perturbing the separable I. Daubechies QMF's (Theorem 3.4). Thus it leads to wavelets that are close to the I. Daubechies separable wavelets with the same number of vanishing moments (for theL1 norm).
The second method permits to construct wavelets that are not near to the I. Daubechies separable wavelets (Theorem 3.5).
All the results of the second section and some of the results of the third section may be adapted to multidimensional compactly supported orthonormal wavelets bases for L2(
Rd 1 Rd 2) of dilation matrix A1 0 0 A2
where fori= 12, Ai is a matrixdidi such that all the eigenvalues
2. New classes of non separable QMF's.
Ad-dimensional QMF is a trigonometric polynomial onRd, M 0() such that (2.1) 8 > < > : M0(0) = 1 X s2f01gd jM 0(+ s) j 2= 1:
The conjugate lters are 2d;1 trigonometric polynomials on Rd,
M1()::: M
2d;1()
such that the matrix
U() = (Mk(+ s))s2f01gdk2f0:::2d;1g
is unitary for all 2Rd.
When d= 1, we will take M1() = ;e
;iM
0(+ ).
If '(x) is a compactly supported scaling function with d vari-ables, there exists a unique d-dimensional QMF M0() such that the
Fourier transform of '(x) satises ^'() = M0(2
;1) ^'(2;1). Thus,
we have a one-to-one correspondance between the multiresolution anal-yses for L2(
Rd) with a compactly supported scaling function and the
d-dimensional QMF's that satisfy the A. Cohen's criterion 3], 4]. A. Cohen's criterion is satised when jM
0()
j > 0, for all 2
; =2 =2]d.
One can notice that, in general, it is not clear that one may always associate compactly supported wavelets to a multidimensional multires-olution analysis, even if its scaling function is compactly supported 7]. In this paper this diculty will be solved by ad hoc constructions (The-orems 2.2 and 2.3).
The bidimensional QMF that corresponds to a separable wavelet basis is also said separable. It can be written
(2.2) M(12) =m1(a111+a122)m2(a211+a222)
where (aij) belongs toSL(2Z) and m
1(x)m2(x) are two
monodimen-sional QMF's.
Let us now study a class of bidimensional QMF's , which is rather easy to construct. This class seems to be a natural extension of the class of the separable QMF's for (aij) =I2.
2.1. The class of the semi separable QMF's.
Theorem 2.1.
Let a(x) b(x) m(x) be three monodimensional QMF's and ~a(x)~b(x)m~(x) their conjugate lters. If c(x) is a trigonometric polynomial, we set ce(x) = 12 (c(x) +c(x+ )) and co(x) = 12 (c(x);c(x+ )): Then (2.3) M0(12) =a(1)me(2) +b(1)mo(2)is a bidimensionalQMFcalled a semi separable QMFand its conjugate lters are (2.4) 8 > < > : M1(12) =a(1) fm~ge( 2) +b(1) fm~go( 2) M2(12) = ~a(1)me(2) + ~b(1)mo(2) M3(12) = ~a(1) fm~ge( 2) + ~b(1) fm~go( 2):
If a6=b and if m(x) has at least three non vanishing coecients, then
M0(12) is non separable.
Proof. An obvious calculus shows that M0(12) is a QMF and that
M1(12), M2(12), M3(12) are its conjugate lters.
It is a bit technical to prove thatM0(12) is non separable.
Sup-pose that M0(12) = m1(a111 + a122)m2(a211 + a222) where
(aij) belongs to SL(2Z) and m
1(x), m2(x) are two monodimensional
QMF's. Taking1 = 0, we get ( ) m( 2) =m1(a122)m2(a222): If a12 a
22 + 1 (mod 2), we will suppose that a12 is even and a22 is
odd, the other case being similiar, then
jm 1(a122) j 2 = jm( 2) j 2+ jm( 2+ ) j 2= 1
thus we have a12 = 0. If a12 a 22 (mod 2), since (aij) 2 SL(2Z) then a 12 and a22 are odd and jm 1(a122) j 2 jm 2(a222) j 2+ jm 1(a122+ ) j 2 jm 2(a222+ ) j 2 =jm( 2) j 2+ jm( 2+ ) j 2 = 1
but this cannot be true since (jm 1(a122) j 2+ jm 1(a122+ ) j 2)(m 2(a222) j 2+ jm 2(a222+ ) j 2) = 1:
Therefore ( ) implies that a
12a22 = 0.
We will only study the case
"1 0 a21 "2 "i = 1 the case a 11 "1 "2 0
is similar. We have m(x) =m2("2x), thus
M0(12) =m1("11)m("2a211+2):
Taking 2= 0 and then 2 = we get
1 2 (a(1) +b(1)) =m1("11)m("2a211) 1 2 (a(1) ;b( 1)) =m1("11)m("2a211+ ) hence a(1) = 2m1("11)me("2a211) b(1) = 2m1("11)mo("2a211):
Since we must have
ja( 1) j 2+ ja( 1+ ) j 2 = 1 jb( 1) j 2+ jb( 1+ ) j 2 = 1
it follows that 4jme(a 211) j 2 = 1 4jmo(a 211) j 2 = 1: Ifa21 6
= 0 the trigonometric polynomialsme(a211) and mo(a211) are
inversible and thus of the type
me(a211) = 12e ik1 and mo(a211) = 12e il1: If a21 = 0 we have a=b.
When the QMF's a(x) and m(x) satisfy A. Cohen's criterion and when the norm ka;bk
1 is small enough, the corresponding semi
sep-arable QMF satises obviously this criterion. However, the constraint on ka;bk
1 does not seem necessary. Indeed, if for example
a(x) =m(x) = 12 (1+e;ix)
and
b(x) = 12 (1+e;i3x):
We have ka; bk
1 = 1 but the corresponding semi separable QMF
satises A. Cohen's criterion 2]. This example also shows that it is even not necessary that both a(x) and b(x) satisfy A. Cohen's for the associated semi separable QMF satises it. In 2] we have however established that m(x) must satisfy this criterion.
2.2. Other classes of non separable QMF's.
It is clear that many QMF's are not semi separable, even in the weak sense. This means that they are not of the form
where a(x), b(x), m(x) are three monodimensional QMF's and (cij) is a matrix of SL(2Z).
Let 2]01 and (x), (x) the two trigonometric polynomials
in one variable dened by
(2.5) (x) = 1; q(x)
whereq(x) is a trigonometric polynomial vanishing in zero, with values in 01] and such that q6= 0, and by
(2.6) j (x)j
2+
j(x)j 2 = 1:
The existence of(x) is given by the Fejer-Riesz Lemma.
Theorem 2.2.
Let S0(12) =a(1)b(2)be a separable QMF and letS1(12), S2(12), S3(12) be its conjugate lters (~a(x) = ;e
;ixa(x+ ) and ~b(x) = ;e
;ixb(x+ ) will be the conjugate lters
of a(x) and b(x)). If (12) and (12) are two trigonometric
poly-nomials -periodic in 1 and in 2 and such that ( (00) = 1 j( 12) j 2+ j( 12) j 2 = 1: Then (2.7) R0(12) =(12)S0(12) +(12)S1(12)
is a QMF and its conjugate lters are (2.8) 8 > < > : R1(12) =(12)S0(12) ;( 12)S1(12) R2(12) =(12)S2(12) +(12)S3(12) R3(12) =(12)S2(12) ;( 12)S3(12): Moreover i) If (12) = (21) and (12) = (21) (as dened by
(2:5) and (2:6)), the QMF R0(12) is non separable when S1(12)
is not the lter ~a(1)b(2) and R0(12) has zeros of order greater or
equal than 2 in ( 0), (0 ) and ( ).
ii) If (12) = (2(1+2)) and (12) = (2(1+2)) or if (12) = (2(1 ; 2)) and (12) = (2(1 ; 2)), then the QMF R0(12) is non separable.
iii) If S1(12) = ~a(1)b(2), (12) = (21) and (12) =
(21)e ;i2
2, then the QMF R
0(12) is non separable.
Proof. One sees immediately that R0(12) is a QMF and that
R1(12), R2(12), R3(12) are its conjugate lters.
Let us show i).
We will begin by the case where S1(12) = a(1)~b(2). Suppose
that
R0(12) =m1(c111+c122)m2(c211+c222)
where (cij) belongs toSL(2Z) and m
1(x),m2(x) are two
monodimen-sional QMF's. Taking successively (12) = (x0), (0x) and (x )
where x is an arbitrary real one obtains
(2x)a(x) =m1(c11x)m2(c21x) (a) b(x) =m1(c12x)m2(c22x) (b) (2x)a(x) =m1(c11x+c12 )m2(c21x+c22 ): (c)
Since the product of two QMF's in the same variables is never a QMF (see the proof of the Theorem 2.1), it results from (b) that c12c22 = 0.
This implies that
(cij) = "1 0 c21 "2 or (cij) = c11 "1 "2 0
with"i = 1. We will suppose that we are in the rst case, the second
case being similar.
We notice that whatever the value of the integer c21 may be, one
cannot have for all x, j (2x)j 2 = jm 2(c21x) j 2. Indeed, if c 21 = 0
then = 0 and else (2 =c21) = 0. In both cases the hypotheses are
contradicted.
It follows from (a) and (c) that
ja(x)j 2= j (2x)j 2 ja(x)j 2+ j(2x)j 2 ja(x)j 2 =jm 1("1x) j 2 jm 2(c21x) j 2+ jm 1("1x) j 2 jm 2(c21x+ ) j 2 =jm 1("1x) j 2:
Thus by using (a) we obtain the contradiction j (2x)j =jm
2(c21x) j
for all x2R.
Let us study now the case where S1(12) = ~a(1)~b(2). As
pre-viously, we will suppose that
R0(12) =m1(c111+c122)m2(c211+c222):
Taking successively (12) = (x0), (0x) and (x ) where x is an
arbitrary real, one obtains
(2x)a(x) =m1(c11x)m2(c21x) (a) b(x) =m1(c12x)m2(c22x) (b) (2x)~a(x) =m1(c11x+c12 )m2(c21x+c22 ): (c)
It follows from (b), as previously, thatc12c22 = 0 and we can suppose
that
(cij) = "1 0
c21 "2
with "i = 1. (a) and (c) imply then that
(2x)a(x) +(2x)~a(x) = 2m1("1x)m2e(c21x) (2x)a(x);(2x)~a(x) = 2m 1("1x)m2o(c21x) where m2e(c21x) = 12 (m2(c21x) +m2(c21x+ )) and m2o(c21x) = 12 (m2(c21x) ;m 2(c21x+ )): As j (2x)a(x) +(2x)~a(x)j 2+ j (2x)a(x+ ) +(2x)~a(x+ )j 2 = 1 j (2x)a(x);(2x)~a(x)j 2+ j (2x)a(x+ );(2x)~a(x+ )j 2 = 1 we have 4jm 2e(c21x) j 2 = 1 4jm 2o(c21x) j 2= 1:
If c21
6
= 0, it follows from the two last equalities that the QMF m2(x)
has only two non vanishing coecients. This is impossible since
R0(12) has zeros of order greater or equal than 2 in ( 0), (0 )
and ( ). If c21 = 0, it follows from (c) that for all x, (2x)~a(x) = 0.
This is impossible. Let us show ii).
As the variables1 and2 play the same role we will only study the
case where S1(12) = e(1)~b(2) with e = ~aor e =a. As previously,
we will suppose that
R0(12) =m1(c111+c122)m2(c211+c222):
Taking successively (12) = (x0), (0x) and (x ) one obtains
(2x)a(x) =m1(c11x)m2(c21x) (a) (2x)b(x) = m1(c12x)m2(c22x) when e= ~a (b) (2x)b(x) +(2x)~b(x) =m1(c12x)m2(c22x) when e=a (b') (2x)e(x) =m1(c11x+c12 )m2(c21x+c22 ): (c)
Whene = ~a, it follows from (a) and (b) thatc11,c12,c21 andc22 are all
odd. Indeed, suppose for example thatc11 is even,c21 would necessarily
be odd and then (a) would imply that
j (2x)j 2 = j (2x)j 2 ja(x)j 2+ j (2x)j 2 ja(x+ )j 2 =jm 1(c11x) j 2 jm 2(c21x) j 2+ jm 1(c11x) j 2 jm 2(c21x+ ) j 2 =jm 1(c11x) j 2:
But we never have j (2x)j 2 =
jm
1(c11x) j
2 for all x. Thus it results
from (c) that j(2x)j 2 = j(2x)j 2 je(x)j 2+ j(2x)j 2 je(x+ )j 2 =jm 1(c11x+ ) j 2 jm 2(c21x+ ) j 2+ jm 1(c11x) j 2 jm 2(c21x) j 2
and it results from (a) that
j (2x)j 2 = jm 1(c11x) j 2 jm 2(c21x) j 2+ jm 1(c11x+ ) j 2 jm 2(c21x+ ) j 2:
This leads to the contradiction j j 2 =
jj 2.
When e = a, since (2x)b(x) + (2x)~b(x) is a QMF, it follows from (b') that c12c22 = 0. So, as previously we can suppose that
(cij) = "1 0
c21 "2
with "i = 1. Moreover (a) implies thatc
21 is odd.
It results then from (a) and (c) that
ja(x)j 2= j (2x)j 2 ja(x)j 2+ j(2x)j 2 ja(x)j 2 =jm 1("1x) j 2 jm 2(c21x) j 2+ jm 1("1x) j 2 jm 2(c21x+ ) j 2 =jm 1("1x) j 2:
Thus (a) implies thatj (2x)j 2 =
jm
2(c21x) j
2 for all x, which is
impos-sible.
We can prove by the same method that
R0(12) = (2(1 ;
2))S0(12) +(2(1 ;
2))S1(12)
where S1(12) is any conjugate lter of S0(12), is non separable.
Let us show iii).
As previously, we will suppose that
( ) R
0(12) =m1(c111+c122)m2(c211+c222):
Taking (12) = (x ) one obtains
R0(x ) =m1(c11x+c12 )m2(c21x+c22 ) = 0
and it follows thatc11c21 = 0. Thus we may suppose that
(cij) = 0 "1
"2 c22
where "i = 1, the other case being similar. Then taking successively
in ( ), (
12) = (x0) and (0x) one obtains
(2x)a(x) +(2x)~a(x) =m2("2x)
The last equality implies that c22 = 0 and b(x) = m1("1x). Thus we
have
R0(12) =m1("12)m2("22) =b(2)( (21)a(1) +(21)~a(1))
and it follows that
b(2)( (21)a(1) +(21)~a(1)e ;i2
2)
=b(2)( (21)a(1) +(21)~a(1))
which leads to the contradiction: for all2, e ;i2
2 = 1.
3. Some of the previous QMF's lead to wavelet bases for
L2( R
2)
of arbitrarily high regularity.
In this section we give two methods for constructing non separable orthonormal compactly supported wavelet bases forL2(
R
2) of
arbitrar-ily high regularity.
In all this section the norm onR
2will be j( 12) j= supfj 1 jj 2 jg.
3.1. The method by perturbing the I. Daubechies QMF's.
Proposition 3.1.
For all L 1, let DL(12) =dL(1)dL(2) be the
separable I. Daubechies QMF such that
jdL(x)j 2 =c L Z x sin 2L;1 tdt:
For all " >0, one can construct a non separable QMFDL"(12) that
satises
i) kDL";DLk 1
",
ii) DL"(12) has zeros of order L on ( 0),(0 ) and ( ),
iii) the size of DL"(12) is independent on ".
Moreover DL"(12) may be chosen of the type (2:3) or of the
type (2:7). 'L and 'L" will be the scaling functions that correspond to
Proof. See 2, Chapter 3] and see also 1].
It is clear that for " >0 small enough DL"(12) will satisfy the
A. Cohen's criterion.
Proposition 3.1 remains valid, if we replace the QMF DL(12)
by any other separable QMF a(1)b(2) that has zeros of order L on
( 0), (0 ) and ( ).
We can now state the main result of this subsection.
Theorem 3.2.
The QMF's DL"(12) generate for " > 0 smallenough non separable orthonormal compactly supported wavelet bases forL2(
R
2) of arbitrarily high regularity. We will say that these wavelets
are obtained by perturbing the I. Daubechies QMF's.
Proof. The critical Sobolev exponent of f 2L 2( R 2) is by denition (f) = supn : Z R 2 jf^()j 2(1 + jj) 2d < 1 o :
R. Q. Jia has shown in 8] that if M() is a QMF that satises A. Cohen's criterion and that has zeros of order L in ( 0), (0 ) and ( ), then the critical Sobolev exponent of'the corresponding scaling function is ( ) (') =;log 4 TM 2L
where(TM=2L) is the spectral radius of the restriction of the transfer
operator TMf() = X 2f01g 2 M 2 + 2 f 2 +
to the vector space 2L of the trigonometric polynomials that have a
zero of order greater or equal than 2L in (00).
Let ('L) and ('L") the critical Sobolev exponent of the scaling functions 'L and 'L" (as dened in the Proposition 3.1). Since the regularity of the I. Daubechies scaling function 'L can arbitrarily high when L is big enough, we have limL!1 ('L) = +
1. At last, it
follows from ( ) and from the continuity of the spectral radius, that
We have solved the open theorical problem of establishing the existence of non separable orthonormal compactly supported wavelet bases forL2(
R
2) of arbitarily high regularity. However the wavelets we
have obtained are probabely very similar to the I. Daubechies separable wavelets since lim"!0'L"='L(for theL
1 norm) (see 2, Chapter 4]).
3.2. Another method of construction.
The aim of this subsection is to construct non separable orthonor-mal compactly supported wavelets of arbitrarily high regularity that are not near to the I. Daubechies bidimensional wavelets with the same number of vanishing moments (for the L1 norm).
Let us rst give a condition ensuring the decrease at innite of the Fourier transform of a scaling function.
Theorem 3.3.
Given two reals 2]01 and C (2 );1, there exists
an exponent = (C)>0having the following property. If M(12)
is a QMF that satises for some integer N 1
a) jM( 12) j N when 1 22 =34 =3] or 2 22 =34 =3], b) jM( 12) j CNj( 1 ;s 1 2 ;s 2 ) jN for all 1, 2 for all (s1s2) 2f01g 2 and (s 1s2) 6 = (00), c) jM( 12) j=jM(; 1 ; 2) j for all 1,2.
Then 'the scaling function that corresponds to M(12), satises
^ '(12) =O( j( 12) j ;N).
To prove the Theorem 3.3 we need the following lemma.
Lemma 3.4.
Given two reals 2]01 and C 1, there exists anexponent = (C) > 0 having the following property. If f(st) is a continuous function from R
2 to 01], 1-periodic in s and t, which
satises i) 0 f(st) < 1, when s21=32=3] or t 21=32=3], ii) f(st) Cj(s; 1=2t ; 2=2)
j, for all st, for all ( 12) 2 f01g 2 and ( 12) 6 = (00),
Then if j 1 and if (s,t) satises 1=4 j(st)j 1=2 we have the
inequality
f(st)f(2s2t)f(2js2jt) 2 ;j:
Proof. First we set h(st) = f(st)f(2s2t). The function f(st)
satises the property ii) and the inequality i) when s 2 1=65=6] or
when t 2 1=65=6]. Moreover this function being with values in 01]
we have j ;1 Y k=0 h(2ks2kt) j Y k=0 f(2ks2kt) 2
thus it is sucient to show that for some =(C)>0 for allj 1
j;1 Y
k=0
h(2ks2kt) 2 ;j:
Consider now (st) satisfying j(st)j =jsj 2 1=41=2]. Because of the
periodicity of f(st) and because of iii), one can suppose that (st) 2
1=41=2]01]. It follows that s= 14 + 3 8 + t= 1 2 + 2 4 + where j, j 2f01g.
We then deneq1:::qrthe transition indices of the nite vectorial
sequence ( 00)::: ( j+1j+1) where ( 00) = (00) as follows: r
will be the number of the indices q that satisfy, 0 q j ;1 and
( q+1q+1)
6
= ( q+2q+2), q1 = 0 and for all l, 2
l r, ql= minfn: ql ;1 < n j ;1 and ( n +1n+1) 6 = ( n+2n+2) g: Form= 1::: rwe setlm =qm+1 ;qm (qr +1 =j by convention), thus we have Pr m=1lm=j.
We have introduced the transition indices in order to get the in-equalities h(2qms2qmt) (M1) h(2qms2qmt) C2 ;lm: (M2)
We have 2qms 2 1=65=6] or 2qmt 2 1=65=6] therefore i) implies
(M1). To prove (M2) we will suppose that qm+1
6 = qm+2, we then have j2qms ;1=2j 2 ;(lm+1). So, if qm+1 = qm+2 it follows that j2qmt;qm +1 j 2
;(lm+1) and else that
j2qmt;1=2j 2
;(lm+1), in the
both cases ii) implies (M2).
At last, for all j 1 we have
h(st)h(2s2t)h(2j
;1s2j;1t)
h(2q
1s2q1t)
h(2qrs2qrt):
So, if A and are two reals such that 2A C, 2 ;A
and 2A (1;)
C (for example A= log2C+ log2(1=) and = log2(1=)=A) then we
will have
( ) h(2qms2qmt) 2 ;lm
indeed, when lm < A (M1) implies that h(2
qms2qmt) 2 ;A <2;lm and when lm A (M 2) implies that h(2 qms2qmt) C2 ;lm 2 ;lm. SincePr 1lm=j it results from ( ) that h(2q1s2q1t) h(2qrs2qrt) 2 ;j:
Proof (Of the Theorem 3.3). If M(12) is a QMF satisfying
(a), (b) and (c) the function f(st) = jM(2 s2 t)j
1=N satises the
conditions i), ii) and iii) of the Lemma 3.4. Let (12) 2R
2 such that j(
12)
j 2 and let j 1 the integer such that 2j j( 12)
j
2j+1 . Thus, if
(st) = 12j+2 ( 12)
we have 1=4 j(st)j 1=2 and it results from the Lemma 3.4 that j'^M( 12) j (f(st)f(2s2t)f(2js2jt))N 2 ;Nj C( N)j( 12) j ;N:
From now on, our aim will be to construct a sequence of non sep-arable QMF's fAL(
12) gL
1 such that for allL big enough,
i)AL(12) satises the conditions (a), (b) and (c) of the Theorem
ii) AL(12) satises A. Cohen's criterion,
iii) AL(12) is not near to the I. Daubechies QMF DL(12) as
dened in the Proposition 3.1. More precisely we will have liminfL !1 kAL;DLk 1 1 4 : Let AL(12) be a QMF of the form
(3.1) AL(12) =dL(2)( (21)dL(1) +(21) ~dL(1)e ;i2
2)
where,
2]01,
dL(x) is the monodimensional I. Daubechies QMF such that jdL(x)j 2 =c L Z x sin 2L;1 tdt and ~dL(x) =;e ;ixd
L(x+ ) is its conjugate lter,
(x) = 1; q(x) and (x) are the trigonometric polynomials
as dened by (2.5) and (2.6).
Let us rst give some useful properties of the QMF dL(x).
Proposition 3.5.
The monodimensional I. Daubechies QMF dL(x) satises:i) for all real 2]0 =4, one can nd a real 2]01 such that
for all L big enough, for all x2 =2 + 3 =2; ],jdL(x)j L,
ii)there exists a realC (2 )
;1 such that for allx
2R, jdL(x)j
CLjx; jL,
iii) for all x 2]; =2 =2,
lim L!1
jdL(x)j= 1 and lim
L!1
jd~L(x)j= 0:
Proof. The function jdL(x)j being even one can suppose that x 2
0 ]. One can notice that cL = O(p
L). For all > 0, we have for all x 2 =2 + ], jdL(x)j 2 j =2; jcLjsin 2L;1( =2 + ) j which implies i).
We have obviously ii) since jdL(x)j 2 cL Z x ( ;t) 2L;1dt C 0 jx; j 2L:
iii) is a consequence of i) and of jdL(x)j 2+
jdL(x+ )j 2 = 1.
The following proposition will permit us to make an appropriate choice of the real that occurs in (3.1).
Proposition 3.6.
If 0 = 1=4then for allL1theQMFAL 0(
12),
as dened by (3:1) satises A. Cohen's criterion.
Proof. Let us show that for all (12)
2 ; =2 =2]
2 and for all
L 1 we have jAL 0( 12) j > 0. As jdL( 2) j > 0 it is sucient to show that j 0(2 1)dL(1) + 0(2 1) ~dL(1)e ;i2 2 j>0: We have j 0(2 1)dL(1) + 0(2 1) ~dL(1)e ;i2 2 j p 2 2 (j 0(2 1) j;j 0(2 1) j): At last, since j 0(2 1) j 3=4 and j 0(2 1) j 2 + j 0(2 1) j 2 = 1, it follows that j 0(2 1) j>j 0(2 1) j.
The following lemma will permit us to make an appropriate choice of the trigonometric polynomialq(x) that occurs in (3.1).
Lemma 3.7.
For all reals and satisfying 2]01 and 2]0 =6there existsfqL(x)gL
1 a sequence of trigonometric polynomials in one
variable with values in 01] and with real coecients, having the fol-lowing properties:
i) kq
Lk
1 = 1,
ii) qL(2x) L for all x20 ]; =32 =3],
iii) qL(2x) converges uniformly to 1 on =3 + 2 =3; ],
iv) there exists a real C 1 such that q
L(2x) C 2L
jxj
2L for all
Proof. Consider T an even, -periodic, C function with values on
01] such that
a)T(0) = 0 and for all x20 =3], 0 T
0(x)< (where =3< p
),
b) for all x2 =3 + =2],T(x) = 1,
c) for allx2 =2 ], T(x) =T( ;x).
Let KN(x) be the Fejer kernel, KN(x) is the trigonometric poly-nomial KN(x) = 2 (N1+ 1) N X k=0 eikx 2 = 2 (N1+ 1) sin 2 (N + 1) 2 x sin2 x 2 :
For every function f 2L
202 ],
KN f(x) = Z
;
KN(x;y)f(y)dy
will be the convolution product ofKN and f. LetQN(x) =KN T(x);
KN T(0) and
RN(x) = QN
kQNk 1
(x):
Since T is even and -periodic the trigonometric polynomialRN is with real coecients and -periodic.
The sequencesfRNgandfR 0
Ngconverge uniformly to the functions
T and T0. Thus it follows from a) that:
There existsC 1 such that for allx, for allN,jRN(x)j Cjxj. For allN N
0 and for all x
20 ]; =32 =3],jRN(x)j p
. At last, one can extract a sequence fRNLgL
1 satisfying N1 N 0 and kT ;RNLk 1 e ;L. We will take q L(2x) =jRNL(x)j 2L.
De nition 3.8.
AL(12) will be a QMF of the type (3:1) such that= 1=4 and q(x) =qL(x), where qL(x)is the trigonometric polynomial we have constructed in the Lemma 3:7.
Theorem 3.9.
The QMF's fAL( 12)gL
1 generate non separable
orthonormal compactly supported wavelet bases forL2( R
2) of arbitrarily
high regularity. Moreover these wavelets are not near to the separable I. Daubechies wavelets with the same number of vanishing moments since liminfL!1
kAL;DLk 1
1=4.
Proof. It follows from the Proposition 3.5, the Lemma 3.7 and the
inequality jAL( 12) j jdL( 1) jjdL( 2) j+ s qL(21) 2 jd~L( 1) jjdL( 2) j
that, for L big enough the QMF AL(12) satises the conditions a)
and b) of the Theorem 3.3. This QMF also satises the condition c) of the same theorem since its coecients are reals.
Let us show that liminfL!1
kAL;DLk 1 1=4. We have 1 4qL(21) jdL( 1) jjdL( 2) j;j 1=4(21) jjd~L( 1) jjdL( 2) j jAL( 12) ;DL( 12) j 1 4qL(21) jdL( 1) jjdL( 2) j+j 1=4(2 1) jjd~L( 1) jjdL( 2) j:
It follows from the Propositions 3.5.iii) and from the Lemma 3.7.iii) that for all (12)
2 =3 + =2]; =2 =2 lim L!1 jAL( 12) ;DL( 12) j= 14 therefore liminfL !1 kAL;DLk 1 1 4 :
4. Conclusion.
Some of the techniques we have used to construct non separable, dyadic, compactly supported, orthonormal, wavelet bases forL2(
R 2) of
arbitrarily high regularity, may be adapted to other types of wavelet bases.
In 2] we have constructed non separable, dyadic, compactly sup-ported, biorthogonal wavelet bases for L2(
R
2) of arbitrarily high
regu-larity by perturbing separable biorthogonal lters.
We have found recently a method for constructing QMF's that generate compactly supported, orthonormal wavelet bases for L2(
R 2) of dilation matrix R= 1 ;1 1 1
(R is a rotation of =4 and a dilation ofp
2). This method is inspired from the Theorem 2.2 Let (x) and(x) two trigonometric polynomials in one variable such that (0) = 1 andj (x)j
2+ j(x)j
2+1. Letm(x) be
a monodimensional QMF (i.e. m(0) = 1 andjm(x)j 2+
jm(x+ )j 2 = 1)
and ~m(x) its conjugate lter ( ~m(x) = ;e
;ixm(x+ )). If P(
12) is
one of the trigonometric polynomials
u(12) = (21)m(2) +(21) ~m(2) v(12) = (1+2)m(2) +(1+2) ~m(2) w(12) = (1 ; 2)m(2) +(1 ; 2) ~m(2) then we have ( P(00) = 1 jP( 12) j 2+ jP( 1+ 2+ ) j 2:
This means that when P(12) satises A. Cohen's criterion it
gener-ates a compactly supported, orthonormal wavelet basis for L2( R
2) of
dilation matrixR. We do not know yet whether the regularity of such wavelets could be made arbitrarily high.
Acknowledgements.
I would like to thank A. Cohen, P. G. Lemarie-Rieusset and Y. Meyer for fruitful discussions.References.
1] Ayache, A., Construction de bases orthonormees d'ondelettes deL2( R
2) non separables, a support compact et de regularite arbitrairement gran-de. C. R. Acad. Sci. Paris 325Serie I (1997), 17-20.
2] Ayache, A., Bases multivariees d'ondelettes, orthonormales, a support compact et de regularite arbitraire. Ph. D. Thesis. Ceremade, Univer-sity of Paris Dauphine, 1997.
3] Cohen, A., Ondelettes, analyses multiresolution et ltres miroir en qua-drature. Ann. Inst. H. Poincare, Analyse non lineaire 7 (1990), 439-459.
4] Cohen, A., Ondelettes et traitement numerique du signal. Ph. D. Thesis. Ceremade. University Paris Dauphine, 1990.
5] Cohen, A., Daubechies, I., Non seperable bidimensional wavelet bases.
Revista Mat. Iberoamericana9(1993), 51-137.
6] Daubechies, I., Orthonormal bases of compactly supported wavelets.
Comm. Pure Appl. Math. 41 (1988) 909-996.
7] Grochenig, K., Analyse multiechelle et bases d'ondelettes. C. R. Acad. Sci. Paris305 Serie I (1987), 13-17.
8] Jia, R. Q., Characterization of smoothness of multivariate renable func-tions in sobolev spaces. Preprint, 1995.
9] Meyer, Y., Ondelettes, fonctions splines et analyses graduees. Cahiers du Ceremade 8703 (1986).
10] Meyer, Y.,Ondelettes et operateurs. Vol. I. Hermann, 1990.
Recibido: 12 de mayo de 1.997
Antoine Ayache CEREMADE Place du Marechal de Lattre de Tassigny 75775 Paris Cedex 16, FRANCE
