A Class of M -Dilation Scaling Functions
with Regularity Growing Proportionally to Filter Support Width
Xianliang Shi1
Department of Mathematics, Hangzhou University
Hangzhou, Zhejiang 310028, P. R. China
and
Qiyu Sun2
Center for Mathematical Sciences, Zhejiang University
Hangzhou, Zhejiang 310027, P.R. China
September 1995
ABSTRACT In this paper, a class of M -dilation scaling functions with regularity growing proportionally to lter support width are constructed. This answers a question proposed by Daubechies in p.338 of her book Ten Lectures on
Wavelets(1992).
AMS Subject Classication: 42C15.
1 Current
address: Department of Mathematics, Texas A&M University, College Station, TX
77843-3368, USA
2 This author is partilally supported by the National Natural Science Foundation of China and
the Zhejiang Provincial Science Foundation of China.
Typeset by AMS-TEX
2
1. Introduction
Let M 2 be a xed integer. A multiresolution analysis for dilation M consists
of a sequence of closed subspaces Vj of L2 (R) that satisfy the following conditions
(see C], D], M]):
i) Vj Vj+1 8j 2 Z ii) j2Z Vj = L2 (R)
iii) \j2Z Vj = f0g
iv) f 2 Vj () f (2;j ) 2 V0 v) there exists a function in V0 such that f(; n) n 2 Z g is an orthonormal
basis of V0 .
The function is called an M -dilation scaling function. It is easy to see that satises the renement equation
(1)
(x) =
X
n2Z
cn (Mx ; n)
where the sequence fcn g satises
X
n2Z
cn = M:
In this paper we shall only deal with compactly supported M -dilation scaling
functions. In this case the sequence fcn g must have nite length. The function
(2)
H ( ) = M1
X
n2Z
cn ein
is called a symbol corresponding to the renement equation (1).
The lter support width W () of an M -dilation scaling function is dened
as the dierence of the largest and the smallest indices of the nonzero cn . The
regularity R() of is dened as the supremum of such that 2 C , where C denotes the Holder class of index .
In her book D, p.338], Daubechies remarks that:
At present, I know of no explicit scheme that provides an innite family of m0
(i.e., symbols H ), for dilation 3(i.e., M = 3), with regularity growing proportionally
to the lter support width.
To our knowledge, this question is still open. The purpose of this paper is to
construct a class of M -dilation scaling functions N for which there exists a constant
M independent of N such that
(3)
where M
R(N ) M W (N )
3. On the other hand it is already known that (see DL])
W (N ) R(N ):
These facts give an armative answer to Daubechies' question.
3
The regularity of has been studied in several papers, see for example BDS],
D], and HW]. In general, to study the regularity of we need to consider the
symbol (2) rst. By the Fourier transform, we see that all the symbols H satisfy
M
;1
X
(4)
l=0
jH ( + 2l=M )j2 = 1:
The solutions H of the equation (4) are determined by (see BDS], H])
(5)
;1
X
; sin2 (M=2) N N
M
jH ( )j = M 2 sin2 =2
2
where
s=0
N
2N
a(s) sin2s 2 + (sin M
)
R( )
2
M
;1 Y
N ; 1 + sj
1
a(s) =
2
s
j
sj
sin j=M
s1 ++sM ;1 =s j =1
and R P
is a real-valued trigonometric polynomial such that
;1
i) M
l=0 R( + 2l=M ) = 0
and
ii) the right hand side of (5) is nonnegative.
By the Riesz Lemma (see D], p.172), such symbol H exists. Let NH be a solution
of (5) with R = 0, and let N be the solution of (1) corresponding to the symbol
NH . In BDS], Bi, Dai and Sun prove the following estimates on the regularity of
N,
jR(N) ; 4lnlnNM j C
when M is odd, and
M
N
X
;
M ;1
+ ln N
4N ln sin 2M
+2
j C
jR(N) ;
4 ln M
when M is even. For the special cases M = 3 4 5 similar results are obtained by
Heller and Wells in HW]. This result shows that for these special N the regularity
does not grow proportionally to the lter support width when M is odd. To construct M -dilation scaling functions with regularity growing proportionally to the
lter support width we use the symbol HN determined by
X
; M + 1)! jHN ( )j2 =
N (k0 kM ;1) (MN
k0! kM ;1 !
k0 ++kM ;1 =MN ;M +1
M
;1
Y
l=0
where N
;
2kl
sin M=2
M sin(=2 + l=M ) 1, and N (k0 kM ;1) is dened by
N (k0 kM ;1) = 0 1 #(E )
if k0 N ; 1
if k0 N
4
where E = fj : kj N g and #(E ) is the cadinality of E . Let N be the solution
of (1) corresponding to a symbol HN . Then we have the following
Theorem. Let M 3 and N 2 be any natural numbers. Then N is a
M -dilation scaling function and there exists a constant C independnet of N such
that
(M ; 1) ln(1 + M1;1 )
ln N ; C 1
)
N
;
(2 ;
2 ln M
4 ln M
1
(M ; 1) ln(1 + M )
ln N + C:
)
N
;
R(N ) ( 12 ;
2 ln M
4 ln M
Remark 1. Observe that W (N ) 2(M ; 1)MN . Therefore the regularity
R(N ) of N grows proportionally to the lter support width W (N ), i.e., (3)
holds.
Remark 2. Let D() = R()=W () be the rate of regularity and lter support
width of a scaling function . Then
(M ; 1) ln(1 + M1;1 )
ln N D(N ) 4M (M1 ; 1) (1 ;
)
;
C
ln M
N
and
when M is odd, and
ln N + C
D(N) 4NM
ln M N
;
ln sin M=(2M + 2));1
ln N
D(N) +
C
M ln M
N
when M is even. Therefore we get
D(N )=D(N) lnNN ( Mln ;M1 ; ln(1 + M 1; 1 )) ; C
when M is odd, and
D(N )=D(N)
ln M ; (M ; 1) ln(1 + M1;1 )
4(M ; 1) ln(sin M=(2M + 2));1 ; C ln N=N
when M is even. This shows that D(N ) of the M -dilation scaling function N is
larger than the one of N even when M is an even integer larger than 4.
2. Proof of the Theorem
To prove the theorem, we estimate HN ( ) rst. Let
2
2
h( ) = sin2 M=
2
M sin =2
5
and
;
BN ( ) = h( )
Then for all real valued we have
BN (; ) =
;N
jHN ( )j2:
; M + 1)! N (k0 kM ;1 k1) (kMN
0 !k1 ! kM ;1 !
k0 ++kM ;1 =MN ;M +1
X
k0 ;N
(h( ))
M
;1
Y
l=1
=BN ( )
(h( + 2l=M ))kl
and
BN ( ) 0:
Therefore by the Riesz Lemma (D], p.172) we obtain the existence of HN ( ) with
; ; eiM N
~
HN ( ) = M1 (1
; ei ) HN ( )
and
jH~ N ( )j2 = BN ( ):
From the denition of N (k0 kM ;1 ) and from
M
;1
X
l=0
h( + 2l=M ) = 1
we get
N (k0 k1 kM ;1 )+N (k1 k2 kM ;1 k0)+ +N (kM ;1 k0 kM ;2 ) = 1
and
M
;1
X
=
l=0
jHN ( + 2l=M )j2
X
k0 ++k
;
M ;1 =MN ;M +1
N (k0 k1 kM ;1 ) + N (k1 k2 kM ;1 k0)
M
;1 ;
Y
(
MN
;
M
+
1)!
+ + N (kM ;1 k0 kM ;2) k !k ! k !
h( + 2l=M ) kl
0 1
M ;1
l=0
=
;1
X
;M
=1:
l=0
MN ;M +1
h( + 2l=M )
Therefore (4) holds for HN ( ). Recall that HN ( ) 6= 0 when j j =M . Hence
the solution N of (1) corresponding to the symbol HN ( ) is an M -dilation scaling
function by an elementary argument (D], p.182, Theorem 6.3.1 with K = ; ]).
6
To estimate the regularity of N , we need some estimates
on BN ( ). From the
p
n
;
n
Stirling formula, which says that n! is equivalent to n e n, from 1=(M ; 1) N (k0 kM ;1 ) 1 when k0 N and h(2=(M ; 1)) = 1=M 2, we get
BN ( ) (MN ; M + 1)! k 0 ! k M ;1 !
k0 ++kM ;1 =MN ;M +1k0 N
X
(h( ))k0;N
=
M
;1
Y
l=1
(h( + 2l=M ))kl
(MN ; M + 1)!
(k0 + N )!((N ; 1)(M ; 1) ; k0)! 0k0 (N ;1)(M ;1)
X
(h( ))k0 (1 ; h( ))(N ;1)(M ;1);k0
(MN ; M + 1)! CM N (1 + 1=(M ; 1))(M ;1)N N ;1=2
N !((
N ; 1)(M ; 1))!
and
(MN ; M + 1)! BN (2=(M ; 1)) M 1; 1
N !k1! kM ;1 !
k1 ++kM ;1 =(M ;1)(N ;1)
X
M
;1
Y
l=1
(h(2=(M ; 1) + 2l=M ))kl
(MN ; M + 1)! ;1 ; 1 (N ;1)(M ;1)
M ; 1 N !((N ; 1)(M ; 1))!
M2
1
CM N (1 + 1=M )(M ;1)N N ;1=2 :
Therefore we get
+ 1=(M ; 1)) )N ; ln N ; C
R(N ) ( 12 ; (M ; 1) ln(1
2 ln M
4 ln M
by an argument as in D, p.217]. Observe that 2M=(M ; 1) = 2=(M ; 1) + 2.
By an argument similar that in p.220 of D] we obtain
+ 1=M ) )N ; ln N ; C:
R(N ) ( 21 ; (M ; 1)2 ln(1
ln M
4 ln M
This completes the proof of the Theorem.
References
C] C. K. Chui, An Introduction to Wavelets, Academic Press, 1992.
BDS] N. Bi, X. Dai and Q. Sun, Construction of compactly supported M -band
wavelets, Research Report 9518, Center for Mathematical Sciences, Zhejiang University, 1995.
7
D] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference
Series in Applied Mathematics 61, Philadephia, 1992.
DL] I. Daubechies and J. Lagarias, Two-scale dierence equation I: existence
and global regularity of solution, SIAM J. Math. Anal., 22(1991), 1388-1410.
H] P. N. Heller, Rank M wavelets with N vanishing moments, SIAM J. Matrix
Anal. Appl., 16(1995), 502-519.
HW] P. N. Heller and R. O. Wells, Jr., The spectral theory of multiresolution
operators and applications, Technical Report AD930120, Aware Inc. 1993, to appear in Wavelets: Theory, Algorithms, and Applications, edited by C. K. Chui, L.
Montefusco, and L. Puccio, Academic Press.
M] Y. Meyer, Ondelettes et Operateurs,I: Ondelettes, Hermann, Paris, 1990.