CONVERGENCE OF CASCADE ALGORITHMS IN ... - Semantic Scholar
CONVERGENCE OF CASCADE ALGORITHMS IN SOBOLEV SPACES FOR PERTURBED REFINEMENT MASKS Di-Rong Chen1 Department of Applied Mathematics Beijing Univ. of Aeronautics and Astronautics Beijing 100083, The People's Republic of China Gerlind Plonka Department of Mathematics University of Duisburg D-47048 Duisburg, Germany
Abstract. In this paper the convergence of the cascade algorithm in a Sobolev
space is considered if the re nement mask is perturbed. It is proved that the cascade algorithm corresponding to a slightly perturbed mask converges. Moreover, the perturbation of the resulting limit function is estimated in terms of that of the masks.
x1.
Introduction
In this paper we are concerned with the following problem: Given a compactly supported multivariate re nable function , how does perturbation of its nite re nement mask a ect the convergence of the cascade algorithm? Further, if the cascade algorithm for the perturbed mask also converges, how the resulting limit function is related with ? We say that a compactly supported function is M -re nable if it satis es a re nement equation X = a( ) (M ? ); (1:1) 2ZZs
where the nitely supported sequence a = (a( )) 2ZZ s is called the re nement mask. The s s matrix M is called a dilation matrix. We suppose that its entries are 1991 Mathematics Subject Classi cation. 39B12, 41A25, 65Q05. Key words and phrases. cascade algorithm, Sobolev space, joint spectral radius, perturbation of re nable functions. 1 Supported in part by NSF of China 1
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2
D.R. CHEN AND G. PLONKA
integers and that klim M ?k = 0. Throughout the paper we assume that M is !1 isotropic. This means that there is an invertible matrix such that M ?1 = diag(1 ; : : : ; s )
(1:2)
with j1j = = js j = m1=s = %(M ), where m := jdet M j and where %(M ) is the spectral radius of M: Let the Fourier transform fb of a function f 2 L1(IRs) be de ned by
fb(!) :=
Z
IR
?ix! dx; f ( x ) e s
! 2 IRs ;
where x ! denotes the inner product of two vectors x and ! in IRs. The Fourier transform is naturally extended to the space of all compactly supported distributions. We can rewrite the equation (1.1) as
b(M T !) = Ha (!) b(!); where the re nement mask symbol X a( )e?i! ; Ha (!) = m1 2ZZs
! 2 IRs ;
(1:3)
! 2 IRs
is a (multivariate) trigonometric polynomial. Looking at the re nement equation (1.1) as a functional equation, one can give necessary and sucient conditions for the mask a to ensure existence, uniqueness and regularity of the solution a (see e.g. [1,5]). Provided that X
2ZZs
a( ) = m;
(1:4)
there exists a unique compactly supported distribution a with ba(0) = 1 satisfying (1.1) (see e.g. [1]). Throughout the paper we assume that the condition (1.4) holds for the re nement masks considered. Before posing the problem more explicitly, we need to verify some notations. For 1 p 1, the norm of Lp(IRs) is denoted by kkp. Let Wp(IRs) denote Lp(IRs) for 1 p < 1 and the space Cu(IRs) of uniformly continuous and bounded functions on IRs equipped with norm jj jj1 for p = 1. Let ZZ+ be the set of nonnegative integers and n
ZZs+ := (1 ; : : : ; s ) 2 ZZs : i 0
o
8 i = 1; : : : ; s :
For any = (1 ; : : : ; s ) 2 ZZs+, let jj := 1 + + s, ! := 1! s! and x := x1 1 xs s . Further, n
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
3
denotes the linear span of fx : jj ng. For multi-integers = (1 ; : : : ; s ) and ? = (1 ; : : : ; s), we say if i i for all i = 1; : : : ; s. For ; we use to denote (?!)!! . For n 2 ZZ+, the Sobolev space Wpn(IRs) is the set of all tempered distributions f such that D f 2 Wp(IRs) for jj n, where D = D11 : : : Dss and Dj := @x@ j (j = 1; : : : ; s). Wpn(IRs) is a Banach space with the norm
jjf jjW
n p
(IR ) := s
X
jjn
jjD f jjpW (IR ) p
s
1=p
:
Let E be a complete set of representatives of distinct cosets of the quotient group ZZs=M ZZs . Thus, each element 2 ZZs can be uniquely represented as = " + M ; " 2 E and 2 ZZs: It is known that the cardinality of E is equal to m = jdet M j: Without loss of generality we can assume that 0 2 E . Let `p(ZZs) be the space of complex-valued sequences = (( )) 2ZZ s such that jjjjp < 1, where
jjjjp :=
X
2
ZZs
j( )jp
1=p
;
1p 0, satisfying the sum rules of order 1 (see Section 3 for the notion of sum rules of order n), then the cascade algorithm associated with b also converges on W0 in Lp-norm and we have
kQka 0 ? Qkb 0kp C ka ? bk1 ;
k 1:
(1:7)
Here the constant C is depend on the re nement mask a under consideration as well as on p; 1 p 1. However, it is independent of the perturbed mask b and of k. In this paper we want to generalize the above result to cascade algorithms converging in Sobolev spaces. In Section 2, we recall from [3] a characterization of convergence of cascade algorithms in Wpn(IRs) in terms of joint spectral radius. In Section 3 we construct a special initial function satisfying a useful stability property. Moreover, an implicit relation between the boundedness and convergence of a cascade algorithm in di erent Sobolev spaces is established. Section 4 is devoted to a generalization of (1.7) to Sobolev spaces.
x2.
Joint spectral radii
In the study of convergence of the cascade algorithm, the joint spectral radius of linear operators is an useful tool. The uniform joint spectral radius was employed in [5] to investigate the regularity of re nable functions. For 1 p < 1, the p-joint spectral radius was introduced and applied to the study of Lp-convergence of cascade algorithms by Jia [12]. We cite from [12] the de nition of p-norm joint spectral radius for the convenience of the reader. Let V be a nite-dimensional space with norm jj jj. For a linear operator A on V de ne jjAjj := max jjAvjj : jjvjj = 1 : Let A be a nite collection of linear operators on a nite-dimensional vector space V . For a positive integer k we denote by Ak the Cartesian power of A:
Ak = (A1 ; : : : ; Ak ) : A1 ; : : : ; Ak 2 A :
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
5
For 1 p < 1, let
jjAk jjp =
X
(A1 ;:::;Ak )2A
k
jjA1 Ak jjp
!1=p
;
and for p = 1, de ne
jjAk jj1 = max jjA1 Ak jj : (A1 ; : : : ; Ak ) 2 Ak : For 1 p 1, the p-norm joint spectral radius of A is de ned to be
p(A) := klim jjAk jj1p=k : !1
(2:1)
This limit indeed exists and does not depend on the choice of norm on V . In fact we have (2:2) jjAk jj1p=k : lim jjAk jj1p=k = kinf 1 k!1 Further, let for v 2 V and 1 p < 1
jjAkvjjp =
X
(A1 ;:::;Ak )2Ak
jjA1 Ak vjjp
!1=p
;
and for p = 1,
jjAk vjj1 = max jjA1 Ak vjj : (A1 ; : : : ; Ak ) 2 Ak : For a re nement mask a, we can construct m operators A"; " 2 E; on `0(ZZs) de ned by the biin nite matrices respectively
A" ( ; ) = a(" + M ? );
; 2 ZZs:
(2:3)
We shall establish a close relation between the iteration of cascade operator and the operators A" ; " 2 E: To this end, we need some nitely supported sequences ak deduced from the re nement mask a, where a1 = a and
ak ( ) =
X
ak?1 ( )a( ? M );
2ZZs
2 ZZs; k 2:
(2:4)
By induction it can easily be veri ed that the cascade operator Qa is of the form
Qka 0 =
X
2
ZZs
ak ( ) 0 (M k ? );
k = 1; 2; : : : :
(2:5)
6
D.R. CHEN AND G. PLONKA
On the other hand, there is a relation between ak and the matrices A"; " 2 E: Let 2 ZZs and k be a positive number. Then there are "1 ; : : : ; "k 2 E and 2 ZZs such that = "1 + M"2 + + M k?1 "k + M k and we have (see [10], Lemma 2.1) ak ( ? ) = A"k A"1 ( ; ) 8 2 ZZs: (2:6) Furthermore, one can construct a nite set K ZZs such that `(K ) is an invariant subspace under A" for any " 2 E , where `(K ) denotes the subspace of `0(ZZs) consisting of all sequences with support on K . To this end, let be a nite set of ZZs such that supp a := f : a( ) 6= 0g . Suppose G := [?1; 1]s: Let H := G [ ( ? E ), where the set ? E consists of all points ! ? " with ! 2 and " 2 E . Now let 1 X (2:7) K := ZZs \ M ?k H: k=1 P s ?k In other words, an element 2 ZZ belongs to K if and only if = 1 k=1 M hk for some sequence of elements hk 2 H . It is not dicult to see that `(K ) is invariant under A" ; " 2 E , i.e. for v 2 `(K ) we have A"v 2 `(K ) (see [10], Lemma 2.3).
In order to give a characterization for convergence of the cascade algorithm in
Wpn(IRs)-norm we introduce the subspace n Vn := v = (v( )) 2ZZs 2 `0(ZZs) :
X
2
ZZs
o
v( ) = 0; jj n :
(2:8)
Theorem 2.1. ([3]) Let a 2 `0(ZZs ) and let Wn be given in (1.6). The cascade s n algorithm associated with a converges on Wn in Wp (IR )-norm (1 p 1) if and only if the following conditions are satis ed: (1) Vnis invariant under A" ;" 2 E ; (2) p A"jVn\`(K) : " 2 E < m?n=s+1=p; where K is given in (2.7).
We have seen that the set K is important in the above characterization. The following result tells us its importance when we consider the action of operators A" ; " 2 E , on the sequences with supports contained in any xed nite set K1 ZZs: Lemma 2.2. Let K be de ned by (2.7). Then for any nite set K1 ZZs , there is a positive integer j such that A"j A"1 v 2 `(K ) 8v 2 `(K1 ) and "1; : : : ; "j 2 E: (2:9) Consequently, for any integer k > j A"k A"1 v 2 `(K ) 8v 2 `(K1 ) and "1; : : : ; "k 2 E: (2:10) Proof. For any v 2 `(K1 ) we have supp A"v M ?1 (K1 + ? E ); " 2 E . Iterative application yields for any integer j > 0 supp A"j A"1 v ? ZZs \ M ?j (K1 + ? E ) + M ?j+1 ( ? E ) + M ?1 ( ? E ) ;
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
7
where is the support of a. Since M is isotropic, there is a constant c being independent of j such that kM ?j !k cm?j=s k!k 8! 2 IRs and j = 1; 2; : : : : (2:11) with m = j det M j (see e.g. [14], Lemma 6.1). Therefore, applying (2.11) to 2 K1, we can nd an integer j such that M ?j 2 M ?1 ( ? E ) and (2.9) holds. Since `(K ) is an invariant subspace under A" for any " 2 E , (2.10) follows for any k > j .
For two sequences u 2 `p(ZZs) and v 2 `0(ZZs), the discrete convolution u v 2 `p(ZZs) is de ned by (u v)( ) =
X
2ZZs
u( ? )v( );
2 ZZs:
It follows from equality (2.6) that, for any v 2 `0(ZZs); (ak v)( ) = A"k A"1 v( ); and consequently,
jjak vjjpp =
X
"1 ;:::;"k 2E
jjA" A"1 vjjpp = kAk vkpp:
(2:12)
k
With the help of Lemma 2.2 we get the following estimate of jjak vjjp in terms of the joint radius p(fA" jV \`(K) : " 2 E g). Theorem 2.3. Let V be an invariant subspace under A" for any " 2 E . Suppose s that K1 ZZ is a nite set. Further, let K be given in (2.7) and let j be an integer satisfying (2.9). Then for any > 0, there is a positive constant c such that
kak vkp cjjvjjp p(fA" jV \`(K) : " 2 E g) +
k
8 k > j and v 2 `(K1 ):
(2:13) Proof. For any nite set K1 ZZs, by Lemma 2.2, there is an integer j such that (2.10) holds for any k j . Clearly, jjA"j A"1 vjjp c1jjvjjp for some positive constant c1 and any "1 ; : : : ; "j 2 E . Further, for k > j , we obtain by (2.12) that
kak vkpp =
X
A"k
"1 ;:::;"k 2E c1 mj jjvjjpp
p
A"1 v
X
"j+1 ;:::;"k 2E = c1 mj jjvjjppjjAk?j jjpp;
A"k
p
jV \`(K) A" +1 jV \`(K)
j
where A = fA"jV \`(K) : " 2 E g: For a xed > 0 we now can nd a constant c satisfying (2.13) by the de nition (2.1) of p(fA" jV \`(K) : " 2 E g) and the equality (2.12). The proof is complete.
8
D.R. CHEN AND G. PLONKA
x3.
Differential and difference operator We now turn our attention to the norms jjQka 0jjWpn (IRs). The goal is to estimate them in terms of sequence norms deduced from ak . In particular, we shall show in this section, that boundedness of (Qka 0;n)k1 (where 0;n is a suitably chosen initial function in Wn) implies convergence of the cascade algorithm on Wn?1 in Wpn?1(IRs)-norm. Let f be a di erentiable function on IRs and let D := (D1 ; : : : ; Ds)T with Dj = @ @xj . Then D(f (M k ))(x) = M k Df (M k x); x 2 IRs:
Hence we have
D(f (M k ))(x) = diag (1k ; : : : ; sk ) Df (M k x) with given in (1.2). More generally, let qj (D) := j D, where j denotes the j th row of , and for any = (1 ; : : : ; s)T 2 ZZs+, let q (D) := q1(D)1 : : : qs (d)s . Using the chain rule for di erentiation, we have for any f 2 Wpn (IRs )
qj (D)(f (M k ))(x) = jk qj (D) f (M k x);
j = 1; : : : ; s;
and hence
q(D)(f (M k ))(x) = diag (11 k ; : : : ; ss k ) q (D)f (M k x); x 2 IRs; (see [23]). It is easily seen that the operators q (D) may be expressed as
q (D) =
X
j j=jj
c; D ;
where c; are determined by . Since is invertible, there exists a positive number satisfying, for any f 2 Wpn(IRs),
?1
X
jj=n
jD f (x)j
X
jj=n
jq (D)f (x)j
X
jj=n
jD f (x)j;
x 2 IRs:
Therefore for any f 2 Wpn(IRs)
?2mnk=s
2mnk=s
X
jj=n
X
jj=n
jD f (M k x)j
jD f (M k x)j;
X
jj=n
jD (f (M k ))(x)j
x 2 IRs and k = 1; 2; : : : ;
where we have used that j1 j = = jsj = m1=s. The second inequality has been also proved in [14]. Applying the above inequalities we obtain:
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
9
Lemma 3.1. There is a positive number c such that for any f 2 Wpn(IRs) P
k j=n jjD (f (M ))jjp cm(n=s?1=p)k ; c?1m(n=s?1=p)k jP jj=n jjD f jjp
k = 1; 2; : : : :
In these inequalities, the factor m?k=p is due to the change of variables M k x ! x in the norms. For our considerations, we want to use a special initial function 0 which is a tensor product of univariate B-splines. For k 2 ZZ+; let Nk be the univariate forward B-spline of degree k with the knots 1; 2; : : : ; k + 1, iteratively given by
Nk = Nk?1 N0 =
Z 1
0
Nk?1 ( ? t)dt; t 2 IR;
where N0 := [0;1) is the characteristic function of [0,1). Furthermore, if = (1 ; : : : ; s ) 2 ZZs+; let N(x) := N1 (x1 ) Ns (xs ), where x = (x1 ; : : : ; xs)T 2 IRs . Let ej be the j th coordinate unit vector of IRs; j = 1; 2; : : : ; s. Recall that, for any j = 1; 2; : : : ; s and a function f de ned on IRs the di erence operator j is given by j f = f () ? f ( ? ej ): Analogously, let the di erence operator j be de ned for sequences 2 `(ZZs ); by j = () ? (? ej ). Further, for any = (1 ; : : : ; s) 2 ZZs+, denote 1 1 s s by . Observe that for any pair of and 2 ZZs+ with
D N = N? :
(3:1)
A second important property of N in this context is the stability of its shifts. This means that, for any 2 ZZs+, there is a positive number , which is independent of , satisfying
?1jjjjp
X
2ZZs
jj( )N ( ? )jjp jjjjp
8 2 `p(ZZs):
(3:2)
The functions N are appropriate candidates for the initial function in the cascade algorithm. In fact, 0;n 2 Wn for any 1 p 1, where
0;n = N(n+1;:::;n+1):
(3:3)
Theorems3.2. There exists a constant > 0 which is independent of 2 `p(ZZs) and k 2 ZZ+ and satis es P j j =n jjD g jjp (n=s?1=p)k ; ? 1 ( n=s ? 1 =p ) k P m (3:4) m jj=n jj jjp
10
D.R. CHEN AND G. PLONKA P
where g is associated with by g = 2ZZs ( ) 0;n (M k ? ): In particular, if the sequence (Qka 0;n)k1 is bounded in Wpn(IRs), then there is a constant c being independent of k such that X
jj=n
jjak jjp cm(?n=s+1=p)k
8k = 1; 2; : : : :
(3:5)
Proof. For = ak , the corresponding function g equals to Qka 0;n by (2.5). If the sequence (Qka 0;n)k1 is bounded in Wpn(IRs), then (3.5) directly follows from the rst inequality in (3.4) by setting = ak . Let's now prove (3.4). Putting f = g(M ?k ) we obtain by (3.1)
D f =
X
2ZZs
( ) N ( ? ) =
X
2ZZs
( )N ( ? );
where = (n + 1 ? 1 ; : : : ; n + 1 ? s): Consequently,
D g = D f (M k ) =
X
2ZZs
( )N (M k ? ):
Therefore the inequalities in (3.4) are true by Lemma 3.1 and the stability (3.2) of N . Before going further, we want to cite the following result: Result 3.3. Let a 2 `0(ZZs) be a given re nement mask. If the cascade algorithm associated with the mask a converges in the Sobolev space Wpn(IRs) (1 p < 1), then (1) The space Vn is invariant under A" for all " 2 E , i.e., for v 2 Vn it follows that A"v 2 Vn. (2) The re nement mask a satis es the sum rules of order n + 1, i.e., for any p 2 n X
2
ZZs
p(M + ")a(M + ") =
X
2
ZZs
p(M )a(M )
8" 2 E:
(3:6)
Moreover, the assertions (1) and (2) are equivalent. The equivalence of (1) and (2) in Result 3.3 has already been shown in [14], Theorem 5.2 (see also [11] Theorem 3.4.12). The necessity of (1) for convergence of the cascade algorithm in Wpn(IRs)-norm follows from Theorem 2.1. If a sequence a 2 `0(ZZs) satis es (1.4) and (3.6), we say that a satis es the sum rules of order n + 1. We can now show the following relation.
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
11
Lemma 3.4. Assume that (3.5) is true for a given re nement mask a. Then Vn?1 in (2.8) is an invariant subspace under A" for all " 2 E: Proof. For " 2 E and 2 ZZs+, we de ne a polynomial p"; 2 jj by p";(x) =
X
2ZZs
a(M + ")(M ?1 (x ? ") ? ) :
By Result 3.3, Vn?1 is invariant under A" 8 " 2 E if and only if
8 "1; "2 2 E and jj n ? 1:
p"1; = p"2 ;
(3:7)
For n = 1, (3.7) has been proved in [10]. We shall prove (3.7) by induction on n. Assume that (3.7) holds for n = n0. If it is not true for n = n0 + 1; then there are "1 ; "2 2 E and 2 ZZs+ with jj = n0 such that p"1; 6= p"2;. We shall show that this leads to a contradiction. For any 2 ZZs+ and k = 1; 2; : : : ; let hk; 2 `(ZZs) be de ned by
hk;( ) =
X
2ZZs
ak ( ? M ) ;
2 ZZs;
where ak are given in (2.4). Observe that
h1;(M + ") = p"; (M + ")
8 2 ZZs and 8 " 2 E:
Thus, the induction assumption (3.7) for n = n0 implies now that
h1;( ) = p"; ( )
8 ; jj n0 ? 1; 8 " 2 E and 2 ZZs :
Consequently, for any 2 ZZs+ with j j = n = n0 + 1; we have h1; = 0; < : Moreover, by the assumption that p"1; 6= p"2; for some "1 ; "2 2 E and some jj = n0, we have a 2 ZZs+ with j j = n and an 2 ZZs such that h1;( ) 6= 0:
(3:8))
On the other hand, the induction relation of ak tells us that for 2 ZZs
hk;( ) =
X X
2ZZs 2ZZs X
a( ? M ? M)ak?1 ()( + ? ) X
(?1)j?j h1; ( ) ak?1 ()? 2ZZ s X X ? + h1; ( ) X ak?1 ( ): (?1)j?j h1; ( ) a = k ? 1 ( ) 0 satisfying that for any b 2 `( ) with jja ? bjj1 < we have X jjA"k A"1 ? B"k B"1 jjp < m(?n=s+1=p?t)kp: "1 ;:::;"k 2E
Note that Vn \ `(K ) is an invariant subspace of any A" and B"; " 2 E: Consequently, X jjA"k jVn\`(K) A"1 jVn\`(K) ? B"k jVn\`(K) B"1 jVn\`(K)jjp "1 ;:::;"k 2E
=
X
"1 ;:::;"k 2E
?
A"k
?
p
A"1 jV \`(K) ? B" B"1 jV \`(K)
< m(?n=s+1=p?t)kp: n
k
n
It follows from the triangle inequality that X jjB"k jVn\`(K) B"1 jVn\`(K)jjp < m(?n=s+1=p?t1)kp; "1 ;:::;"k 2E
(4:1)
where the positive number t1 is de ned by 2m(?n=s+1=p?t)kp = m(?n=s+1=p?t1)kp. Equality (2.2) tells us now p(fB"jVn\`(K) : " 2 E g) m(?n=s+1=p?t1) < m?n=s+1=p; and the assertion follows from Theorem 2.1. Our goal is now, to estimate the perturbation of the limit function in terms of the perturbation of the mask, i.e., we want to show that kQka 0 ? Qkb 0kWpn (IRs) c ka ? bk1 ; k = 1; 2; : : : ; where a; b meet the assumptions of Theorem 4.1. It turns out that the initial function 0 needs to satisfy a stability condition as given in Theorem 3.2. Indeed, choosing g = Qka 0;n ? Qkb 0;n, (3.4) implies that X X kD Qka 0;n ? D Qkb 0;nkp c m(n=s?1=p)k kak ? bk kp: jj=n
jj=n
Hence, taking into account the result of Theorem 3.5, we only have to estimate the norm kak ? bk kp for jj = n. In order to obtain this estimate we rst need:
14
D.R. CHEN AND G. PLONKA
Lemma 4.2. Assume that a and b satisfy sum rules of order n + 1. Then for any v 2 Vn?1 we have (B" ? A" )v 2 Vn 8" 2 E: Proof. We claim that, for any a satisfying sum rules of order n +1 and any p 2 n, there is a polynomial q 2 n?1 such that X p(? )a(" + M ? ) = p(M ?1 (" ? )) + q(" ? ) 8" 2 E and 8 2 ZZs : 2ZZs
(4:2)
In fact, it follows from Taylor's formula that p(M ?1 (" ? )) D ?1 (M ? + ")) : ( ? M p(? ) = ! jjn X
Therefore
X
2
ZZs
=
p(? )a(" + M ? )
D p(M ?1 (" ? )) X (?M ?1 (M ? + ")) a(" + M ? ): ! 2ZZs jjn X
P
Note that a satis es (1.4) and (3.6), i.e., we have 2ZZs a(" + M ? ) = 1 and X
2ZZs
(?M ?1 (M ? ? ")) a(" + M ? ) =
X
2ZZs
(? ) a(M ); jj n:
Hence, we obtain X
2ZZs
p(? )a(" + M ? )
=p(M ?1 (" ? )) +
D p(M ?1 (" ? )) (? ) a(M ): ! 0<jjn 2ZZs X
X
This proves (4.2). Applying (4.2) to b we get a polynomial g 2 n?1 such that X
2ZZs
p(? )(b(" + M ? ) ? a(" + M ? )) = g(" ? )
8" 2 E and 8 2 ZZs:
It follows from v 2 Vn?1 that, for any p 2 n, X
2
ZZs
p(? )(B" ? A" )v( ) =
The proof is complete.
X
2
ZZs
g(" ? )v( ) = 0
8" 2 E:
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
15
Lemma 4.3. Suppose that ZZs is a nites set and that the cascade algorithm corresponding to a 2 `( ) converges in Wpn(IR )-norm. Further, let b 2 `( ) satisfy the sum rules of order n+1 and ka?bk < , where is chosen such that the assertion of Theorem 4.1 holds. Then there is a positive number c such that we have
jjak ? bk jjp cjja ? bjj1m(?n=s+1=p)k
8jj = n and k = 1; 2; : : : ;
where c is independent of b and k. Proof. Let K be given in (2.7). By (2.12) and the equality
B"k B"1 ? A"k A"1 =
k X j =1
B"k B"j+1 (B"j ? A"j )A"j?1 A"1
we obtain
jj(bk ? ak ) vjjp 1=p X X = jB" B"1 v( ) ? A" A"1 v( )jp
"1 ;:::;"k 2E 2K k X X X
j =1 "1 ;:::;"k 2E 2K
k
k
jB" B" +1 (B" ? A" )A" ?1 A"1 v( )jp j
j
j
k
1=p
j
:
Thus,
jj(bk ? ak ) vjjp
k X
X
j =1 "1 ;:::;"k 2E
1=p p jjB"k B"j+1 (B"j ? A"j )A"j?1 A"1 vjj :
(4:3) P Let jj = n. Note that jjaj?1 jjpp = "1;:::;"j?12E jjA"j?1 A"1 jjp: Hence, by (3.5) in Theorem 3.2, there is a constant c1 > 0 such that for any j X
"1 ;:::;"j?1 2E
jjA" ?1 A"1 jjp c1m(?n=s+1=p)(j?1)p: j
(4:4)
On the other hand, from 2 Vn?1 and Lemma 4.2 it follows that (B"j ? A"j )A"j?1 A"1 2 Vn
8"1; : : : ; "j 2 E:
Moreover, by Theorem 4.1 we already know that the cascade algorithm corresponding to b converges in Wpn(IRs)-norm and by (4.1) there are a positive number t1 and a constant c2 such that X
jjB" B" +1 (B" ? A" )A" ?1 A"1 jjp k
j
"j+1 :::"k 2E c2m(?n=s+1=p?t1)(k?j)pjj(B"
j
j
j
j
? A" )A" ?1 A"1 jjp j
j
8 k > j:
16
D.R. CHEN AND G. PLONKA
This together with (4.4) implies X
X
jjB" B" +1 (B" ? A" )A" ?1 A"1 jjp k
j
j
j
j
"1 :::;"j?1 2E "j+1 :::"k 2E c3m(?n=s+1=p?t1)(k?j)pm(?n=s+1=p)(j?1)pjja ? bjjp; 1
where c3 is some constant which is independent of b and k. It follows from (4.3) that
jj(bk ? ak ) jjp c13=pjja ? bjj1m(?n=s+1=p)k
k X j =1
m?(k?j)t1 ;
k = 1; 2; : : : :
The proof is complete. We are now ready to present the main theorem of this section. Theorem 4.4. Let be a nite set in ZZs. Assume sthat the cascade algorithm corresponding to a 2 `( ) converges on Wn in Wpn(IR )-norm. Then there exists a positive constant such that, for any b 2 `( ) satisfying the sum rules of order n + 1 with ka ? bk1 < , the cascade algorithm corresponding to b converges in Wpn(IRs)-norm. Moreover, there exists a constant c, which is independent of b and k, such that
kQka 0;n ? Qkb 0;nkW
n p
(IRs )
c ka ? bk1 k = 1; 2; : : : :
(4:5)
where 0;n is given in (3.3). Consequently, we nd for the limit functions
k a ? bkW
n p
(IRs )
cka ? bk1 :
(4:6)
Proof. By Theorem 4.1 we know that for b 2 `( ) satisfying the sum rules of order n +1 and with ka ? bk < for some suitable the cascade algorithm corresponding to mask b converges on Wn in Wpn(IRs)-norm. Therefore, b 2 Wpn(IRs). Since 0;n 2 Wn the cascade algorithm converges on 0;n for a and b, i.e., we have
lim jjQka 0;n ? ajjWpn (IRs) = klim jjQk ? jj n s = 0: !1 b 0;n b Wp (IR )
k!1
The inequality (4.6) follows now from (4.5). In order to prove (4.5) we appeal to Theorem 3.2. Put = ak ? bk in (3.4). This corresponds to g = Qka 0;n ? Qkb 0;n. Then the second inequality in (3.4) yields for some constant c1 and for k = 1; 2; : : : X
jj=n
jjD (Qka 0;n ? Qkb 0;n)jjp c1m(n=s?1=p)k
X
jj=n
jjak ? bk jjp:
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
17
Together with Lemma 4.3, it in turn implies X
jj=n
jjD (Qka 0;n ? Qkb 0;n)jjp c2jja ? bjj1 ;
k = 1; 2; : : : ;
(4:7)
where c2 is some positive number being independent of b and k. As shown in Theorem 3.5, the cascade algorithm corresponding to a also con0 s n 0 verges on Wn in Wp (IR )-norm with n0 < n. Replacing n with n0 in (4.7) and then taking the sum of the resulting inequalities we obtain (4.5). The proof is complete.
The proof of the estimate (4.5) is strongly based on the stability property of 0;n in Theorem 3.2. Thus, we obtain the following corollary. Corollarys 4.5. Let be a nite set in ZZs. Suppose that a is a re nable function in Wpn(IR ) corresponding to mask a 2 `( ) and the shifts of a are stable. Then there are positive constants and c such that, for any b 2 `( ) satisfying sum rules of order n + 1 and jja ? bjj1 < ; the re nable distribution b is in Wpn(IRs ) and satis es (4.6). Proof. By the stability of the shifts of a, the cascade algorithm corresponding to a converges on Wn in Wpn (IRs )-norm. This conclusion has been established in [16] for p = 2. The method works for general p 1. Now, using Theorem 3.2, the proof is analogous to that of Theorem 4.4. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
References A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision, Memoirs Amer. Math. Soc. 93 (1991), 1-186. D. R. Chen, Algebraic properties of subdivision operators with matrix mask and their applications, J. Approx. Theory 97 (1999), 294-310. D. R. Chen, R. Q. Jia and S. D. Riemenschneider, Vector subdivision schemes in Sobolev spaces, manuscript (1998). I. Daubechies and Y. Huang, How does truncation of the mask a ect a re nable function?, Constr. Approx. 11 (1995), 365-380. I. Daubechies and J. Lagarias, Two-scale di erence equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388-1410. T. N. T. Goodman and S. L. Lee, Convergence of cascade algorithms, in \Mathematical Methods for Curves and Surfaces II," Morten Daehlen, Tom Lyche and Larry L. Schumaker (eds), Vanderbilt University Press, Nashville, 1998, 191-212. B. Han, Subdivision schemes, biorthogonal wavelets and image compression, PhD theses (1998). B. Han, Error estimate of a subdivision scheme with a truncated re nement mask, manuscript (1997). B. Han and T.A. Hogan, How is a vector pyramid scheme a ected by perturbation in the mask?, in \Approximation Theory IX," Charles K. Chui, Larry, L. Schumaker (eds.), Vanderbilt University Press, Nashville, 1998, 97-104. B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal. (1998), 1177 - 1199. K. Jetter and G. Plonka, A survey on L2 -approximation order from shift-invariant spaces, manuscript (1999).
18
D.R. CHEN AND G. PLONKA
12. R. Q. Jia, Subdivision schemes in Lp spaces, Advances in Computational Mathematics 3 (1995), 309-341. 13. R. Q. Jia, The Toeplitz theorem and its applications to approximation theory and linear PDE's, Trans. Amer. Math. Soc. 347 (1995), 2585-2594. 14. R. Q. Jia, Approximation properties of multivariate wavelets, Math. Comp. 67 (1998), 647665. 15. R. Q. Jia, Shift-invariant subspaces and linear operator equations, Israel J. of Math. 103 (1998), 259-288. 16. R. Q. Jia, Q. T. Jiang and S. L. Lee, Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets, manuscript. 17. R.Q. Jia, S.D. Riemenschneider and D.X. Zhou, Approximation by multiple re nable functions, Canadian J. Math. 49 (1998), 944-962. 18. R.Q. Jia, S.D. Riemenschneider and D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998), 1533-1563. 19. Q. Jiang, Multivariate matrix re nable functions with arbitrary matrix dilation, Trans. Amer. Math. Soc. 351 (1999), 2407-2438. 20. C. A. Micchelli and T. Sauer, Regularity of multiwavelets, Advances in Comp. Math. 7 (1997), 455-545. 21. C. A. Micchelli and T. Sauer, On vector subdivision, Math. Z. 229 (1998), 621-674. 22. Q. Sun, Convergence and boundedness of cascade algorithm in Besov spaces and TriebelLizorkin spaces, manuscript (1998). 23. D.-X. Zhou, Norms concerning subdivision sequences and their applications in wavelets, manuscript (1998).
Abstract. In this paper the convergence of the cascade algorithm in a Sobolev
space is considered if the re nement mask is perturbed. It is proved that the cascade algorithm corresponding to a slightly perturbed mask converges. Moreover, the perturbation of the resulting limit function is estimated in terms of that of the masks.
x1.
Introduction
In this paper we are concerned with the following problem: Given a compactly supported multivariate re nable function , how does perturbation of its nite re nement mask a ect the convergence of the cascade algorithm? Further, if the cascade algorithm for the perturbed mask also converges, how the resulting limit function is related with ? We say that a compactly supported function is M -re nable if it satis es a re nement equation X = a( ) (M ? ); (1:1) 2ZZs
where the nitely supported sequence a = (a( )) 2ZZ s is called the re nement mask. The s s matrix M is called a dilation matrix. We suppose that its entries are 1991 Mathematics Subject Classi cation. 39B12, 41A25, 65Q05. Key words and phrases. cascade algorithm, Sobolev space, joint spectral radius, perturbation of re nable functions. 1 Supported in part by NSF of China 1
Typeset by AMS-TEX
2
D.R. CHEN AND G. PLONKA
integers and that klim M ?k = 0. Throughout the paper we assume that M is !1 isotropic. This means that there is an invertible matrix such that M ?1 = diag(1 ; : : : ; s )
(1:2)
with j1j = = js j = m1=s = %(M ), where m := jdet M j and where %(M ) is the spectral radius of M: Let the Fourier transform fb of a function f 2 L1(IRs) be de ned by
fb(!) :=
Z
IR
?ix! dx; f ( x ) e s
! 2 IRs ;
where x ! denotes the inner product of two vectors x and ! in IRs. The Fourier transform is naturally extended to the space of all compactly supported distributions. We can rewrite the equation (1.1) as
b(M T !) = Ha (!) b(!); where the re nement mask symbol X a( )e?i! ; Ha (!) = m1 2ZZs
! 2 IRs ;
(1:3)
! 2 IRs
is a (multivariate) trigonometric polynomial. Looking at the re nement equation (1.1) as a functional equation, one can give necessary and sucient conditions for the mask a to ensure existence, uniqueness and regularity of the solution a (see e.g. [1,5]). Provided that X
2ZZs
a( ) = m;
(1:4)
there exists a unique compactly supported distribution a with ba(0) = 1 satisfying (1.1) (see e.g. [1]). Throughout the paper we assume that the condition (1.4) holds for the re nement masks considered. Before posing the problem more explicitly, we need to verify some notations. For 1 p 1, the norm of Lp(IRs) is denoted by kkp. Let Wp(IRs) denote Lp(IRs) for 1 p < 1 and the space Cu(IRs) of uniformly continuous and bounded functions on IRs equipped with norm jj jj1 for p = 1. Let ZZ+ be the set of nonnegative integers and n
ZZs+ := (1 ; : : : ; s ) 2 ZZs : i 0
o
8 i = 1; : : : ; s :
For any = (1 ; : : : ; s ) 2 ZZs+, let jj := 1 + + s, ! := 1! s! and x := x1 1 xs s . Further, n
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
3
denotes the linear span of fx : jj ng. For multi-integers = (1 ; : : : ; s ) and ? = (1 ; : : : ; s), we say if i i for all i = 1; : : : ; s. For ; we use to denote (?!)!! . For n 2 ZZ+, the Sobolev space Wpn(IRs) is the set of all tempered distributions f such that D f 2 Wp(IRs) for jj n, where D = D11 : : : Dss and Dj := @x@ j (j = 1; : : : ; s). Wpn(IRs) is a Banach space with the norm
jjf jjW
n p
(IR ) := s
X
jjn
jjD f jjpW (IR ) p
s
1=p
:
Let E be a complete set of representatives of distinct cosets of the quotient group ZZs=M ZZs . Thus, each element 2 ZZs can be uniquely represented as = " + M ; " 2 E and 2 ZZs: It is known that the cardinality of E is equal to m = jdet M j: Without loss of generality we can assume that 0 2 E . Let `p(ZZs) be the space of complex-valued sequences = (( )) 2ZZ s such that jjjjp < 1, where
jjjjp :=
X
2
ZZs
j( )jp
1=p
;
1p 0, satisfying the sum rules of order 1 (see Section 3 for the notion of sum rules of order n), then the cascade algorithm associated with b also converges on W0 in Lp-norm and we have
kQka 0 ? Qkb 0kp C ka ? bk1 ;
k 1:
(1:7)
Here the constant C is depend on the re nement mask a under consideration as well as on p; 1 p 1. However, it is independent of the perturbed mask b and of k. In this paper we want to generalize the above result to cascade algorithms converging in Sobolev spaces. In Section 2, we recall from [3] a characterization of convergence of cascade algorithms in Wpn(IRs) in terms of joint spectral radius. In Section 3 we construct a special initial function satisfying a useful stability property. Moreover, an implicit relation between the boundedness and convergence of a cascade algorithm in di erent Sobolev spaces is established. Section 4 is devoted to a generalization of (1.7) to Sobolev spaces.
x2.
Joint spectral radii
In the study of convergence of the cascade algorithm, the joint spectral radius of linear operators is an useful tool. The uniform joint spectral radius was employed in [5] to investigate the regularity of re nable functions. For 1 p < 1, the p-joint spectral radius was introduced and applied to the study of Lp-convergence of cascade algorithms by Jia [12]. We cite from [12] the de nition of p-norm joint spectral radius for the convenience of the reader. Let V be a nite-dimensional space with norm jj jj. For a linear operator A on V de ne jjAjj := max jjAvjj : jjvjj = 1 : Let A be a nite collection of linear operators on a nite-dimensional vector space V . For a positive integer k we denote by Ak the Cartesian power of A:
Ak = (A1 ; : : : ; Ak ) : A1 ; : : : ; Ak 2 A :
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
5
For 1 p < 1, let
jjAk jjp =
X
(A1 ;:::;Ak )2A
k
jjA1 Ak jjp
!1=p
;
and for p = 1, de ne
jjAk jj1 = max jjA1 Ak jj : (A1 ; : : : ; Ak ) 2 Ak : For 1 p 1, the p-norm joint spectral radius of A is de ned to be
p(A) := klim jjAk jj1p=k : !1
(2:1)
This limit indeed exists and does not depend on the choice of norm on V . In fact we have (2:2) jjAk jj1p=k : lim jjAk jj1p=k = kinf 1 k!1 Further, let for v 2 V and 1 p < 1
jjAkvjjp =
X
(A1 ;:::;Ak )2Ak
jjA1 Ak vjjp
!1=p
;
and for p = 1,
jjAk vjj1 = max jjA1 Ak vjj : (A1 ; : : : ; Ak ) 2 Ak : For a re nement mask a, we can construct m operators A"; " 2 E; on `0(ZZs) de ned by the biin nite matrices respectively
A" ( ; ) = a(" + M ? );
; 2 ZZs:
(2:3)
We shall establish a close relation between the iteration of cascade operator and the operators A" ; " 2 E: To this end, we need some nitely supported sequences ak deduced from the re nement mask a, where a1 = a and
ak ( ) =
X
ak?1 ( )a( ? M );
2ZZs
2 ZZs; k 2:
(2:4)
By induction it can easily be veri ed that the cascade operator Qa is of the form
Qka 0 =
X
2
ZZs
ak ( ) 0 (M k ? );
k = 1; 2; : : : :
(2:5)
6
D.R. CHEN AND G. PLONKA
On the other hand, there is a relation between ak and the matrices A"; " 2 E: Let 2 ZZs and k be a positive number. Then there are "1 ; : : : ; "k 2 E and 2 ZZs such that = "1 + M"2 + + M k?1 "k + M k and we have (see [10], Lemma 2.1) ak ( ? ) = A"k A"1 ( ; ) 8 2 ZZs: (2:6) Furthermore, one can construct a nite set K ZZs such that `(K ) is an invariant subspace under A" for any " 2 E , where `(K ) denotes the subspace of `0(ZZs) consisting of all sequences with support on K . To this end, let be a nite set of ZZs such that supp a := f : a( ) 6= 0g . Suppose G := [?1; 1]s: Let H := G [ ( ? E ), where the set ? E consists of all points ! ? " with ! 2 and " 2 E . Now let 1 X (2:7) K := ZZs \ M ?k H: k=1 P s ?k In other words, an element 2 ZZ belongs to K if and only if = 1 k=1 M hk for some sequence of elements hk 2 H . It is not dicult to see that `(K ) is invariant under A" ; " 2 E , i.e. for v 2 `(K ) we have A"v 2 `(K ) (see [10], Lemma 2.3).
In order to give a characterization for convergence of the cascade algorithm in
Wpn(IRs)-norm we introduce the subspace n Vn := v = (v( )) 2ZZs 2 `0(ZZs) :
X
2
ZZs
o
v( ) = 0; jj n :
(2:8)
Theorem 2.1. ([3]) Let a 2 `0(ZZs ) and let Wn be given in (1.6). The cascade s n algorithm associated with a converges on Wn in Wp (IR )-norm (1 p 1) if and only if the following conditions are satis ed: (1) Vnis invariant under A" ;" 2 E ; (2) p A"jVn\`(K) : " 2 E < m?n=s+1=p; where K is given in (2.7).
We have seen that the set K is important in the above characterization. The following result tells us its importance when we consider the action of operators A" ; " 2 E , on the sequences with supports contained in any xed nite set K1 ZZs: Lemma 2.2. Let K be de ned by (2.7). Then for any nite set K1 ZZs , there is a positive integer j such that A"j A"1 v 2 `(K ) 8v 2 `(K1 ) and "1; : : : ; "j 2 E: (2:9) Consequently, for any integer k > j A"k A"1 v 2 `(K ) 8v 2 `(K1 ) and "1; : : : ; "k 2 E: (2:10) Proof. For any v 2 `(K1 ) we have supp A"v M ?1 (K1 + ? E ); " 2 E . Iterative application yields for any integer j > 0 supp A"j A"1 v ? ZZs \ M ?j (K1 + ? E ) + M ?j+1 ( ? E ) + M ?1 ( ? E ) ;
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
7
where is the support of a. Since M is isotropic, there is a constant c being independent of j such that kM ?j !k cm?j=s k!k 8! 2 IRs and j = 1; 2; : : : : (2:11) with m = j det M j (see e.g. [14], Lemma 6.1). Therefore, applying (2.11) to 2 K1, we can nd an integer j such that M ?j 2 M ?1 ( ? E ) and (2.9) holds. Since `(K ) is an invariant subspace under A" for any " 2 E , (2.10) follows for any k > j .
For two sequences u 2 `p(ZZs) and v 2 `0(ZZs), the discrete convolution u v 2 `p(ZZs) is de ned by (u v)( ) =
X
2ZZs
u( ? )v( );
2 ZZs:
It follows from equality (2.6) that, for any v 2 `0(ZZs); (ak v)( ) = A"k A"1 v( ); and consequently,
jjak vjjpp =
X
"1 ;:::;"k 2E
jjA" A"1 vjjpp = kAk vkpp:
(2:12)
k
With the help of Lemma 2.2 we get the following estimate of jjak vjjp in terms of the joint radius p(fA" jV \`(K) : " 2 E g). Theorem 2.3. Let V be an invariant subspace under A" for any " 2 E . Suppose s that K1 ZZ is a nite set. Further, let K be given in (2.7) and let j be an integer satisfying (2.9). Then for any > 0, there is a positive constant c such that
kak vkp cjjvjjp p(fA" jV \`(K) : " 2 E g) +
k
8 k > j and v 2 `(K1 ):
(2:13) Proof. For any nite set K1 ZZs, by Lemma 2.2, there is an integer j such that (2.10) holds for any k j . Clearly, jjA"j A"1 vjjp c1jjvjjp for some positive constant c1 and any "1 ; : : : ; "j 2 E . Further, for k > j , we obtain by (2.12) that
kak vkpp =
X
A"k
"1 ;:::;"k 2E c1 mj jjvjjpp
p
A"1 v
X
"j+1 ;:::;"k 2E = c1 mj jjvjjppjjAk?j jjpp;
A"k
p
jV \`(K) A" +1 jV \`(K)
j
where A = fA"jV \`(K) : " 2 E g: For a xed > 0 we now can nd a constant c satisfying (2.13) by the de nition (2.1) of p(fA" jV \`(K) : " 2 E g) and the equality (2.12). The proof is complete.
8
D.R. CHEN AND G. PLONKA
x3.
Differential and difference operator We now turn our attention to the norms jjQka 0jjWpn (IRs). The goal is to estimate them in terms of sequence norms deduced from ak . In particular, we shall show in this section, that boundedness of (Qka 0;n)k1 (where 0;n is a suitably chosen initial function in Wn) implies convergence of the cascade algorithm on Wn?1 in Wpn?1(IRs)-norm. Let f be a di erentiable function on IRs and let D := (D1 ; : : : ; Ds)T with Dj = @ @xj . Then D(f (M k ))(x) = M k Df (M k x); x 2 IRs:
Hence we have
D(f (M k ))(x) = diag (1k ; : : : ; sk ) Df (M k x) with given in (1.2). More generally, let qj (D) := j D, where j denotes the j th row of , and for any = (1 ; : : : ; s)T 2 ZZs+, let q (D) := q1(D)1 : : : qs (d)s . Using the chain rule for di erentiation, we have for any f 2 Wpn (IRs )
qj (D)(f (M k ))(x) = jk qj (D) f (M k x);
j = 1; : : : ; s;
and hence
q(D)(f (M k ))(x) = diag (11 k ; : : : ; ss k ) q (D)f (M k x); x 2 IRs; (see [23]). It is easily seen that the operators q (D) may be expressed as
q (D) =
X
j j=jj
c; D ;
where c; are determined by . Since is invertible, there exists a positive number satisfying, for any f 2 Wpn(IRs),
?1
X
jj=n
jD f (x)j
X
jj=n
jq (D)f (x)j
X
jj=n
jD f (x)j;
x 2 IRs:
Therefore for any f 2 Wpn(IRs)
?2mnk=s
2mnk=s
X
jj=n
X
jj=n
jD f (M k x)j
jD f (M k x)j;
X
jj=n
jD (f (M k ))(x)j
x 2 IRs and k = 1; 2; : : : ;
where we have used that j1 j = = jsj = m1=s. The second inequality has been also proved in [14]. Applying the above inequalities we obtain:
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
9
Lemma 3.1. There is a positive number c such that for any f 2 Wpn(IRs) P
k j=n jjD (f (M ))jjp cm(n=s?1=p)k ; c?1m(n=s?1=p)k jP jj=n jjD f jjp
k = 1; 2; : : : :
In these inequalities, the factor m?k=p is due to the change of variables M k x ! x in the norms. For our considerations, we want to use a special initial function 0 which is a tensor product of univariate B-splines. For k 2 ZZ+; let Nk be the univariate forward B-spline of degree k with the knots 1; 2; : : : ; k + 1, iteratively given by
Nk = Nk?1 N0 =
Z 1
0
Nk?1 ( ? t)dt; t 2 IR;
where N0 := [0;1) is the characteristic function of [0,1). Furthermore, if = (1 ; : : : ; s ) 2 ZZs+; let N(x) := N1 (x1 ) Ns (xs ), where x = (x1 ; : : : ; xs)T 2 IRs . Let ej be the j th coordinate unit vector of IRs; j = 1; 2; : : : ; s. Recall that, for any j = 1; 2; : : : ; s and a function f de ned on IRs the di erence operator j is given by j f = f () ? f ( ? ej ): Analogously, let the di erence operator j be de ned for sequences 2 `(ZZs ); by j = () ? (? ej ). Further, for any = (1 ; : : : ; s) 2 ZZs+, denote 1 1 s s by . Observe that for any pair of and 2 ZZs+ with
D N = N? :
(3:1)
A second important property of N in this context is the stability of its shifts. This means that, for any 2 ZZs+, there is a positive number , which is independent of , satisfying
?1jjjjp
X
2ZZs
jj( )N ( ? )jjp jjjjp
8 2 `p(ZZs):
(3:2)
The functions N are appropriate candidates for the initial function in the cascade algorithm. In fact, 0;n 2 Wn for any 1 p 1, where
0;n = N(n+1;:::;n+1):
(3:3)
Theorems3.2. There exists a constant > 0 which is independent of 2 `p(ZZs) and k 2 ZZ+ and satis es P j j =n jjD g jjp (n=s?1=p)k ; ? 1 ( n=s ? 1 =p ) k P m (3:4) m jj=n jj jjp
10
D.R. CHEN AND G. PLONKA P
where g is associated with by g = 2ZZs ( ) 0;n (M k ? ): In particular, if the sequence (Qka 0;n)k1 is bounded in Wpn(IRs), then there is a constant c being independent of k such that X
jj=n
jjak jjp cm(?n=s+1=p)k
8k = 1; 2; : : : :
(3:5)
Proof. For = ak , the corresponding function g equals to Qka 0;n by (2.5). If the sequence (Qka 0;n)k1 is bounded in Wpn(IRs), then (3.5) directly follows from the rst inequality in (3.4) by setting = ak . Let's now prove (3.4). Putting f = g(M ?k ) we obtain by (3.1)
D f =
X
2ZZs
( ) N ( ? ) =
X
2ZZs
( )N ( ? );
where = (n + 1 ? 1 ; : : : ; n + 1 ? s): Consequently,
D g = D f (M k ) =
X
2ZZs
( )N (M k ? ):
Therefore the inequalities in (3.4) are true by Lemma 3.1 and the stability (3.2) of N . Before going further, we want to cite the following result: Result 3.3. Let a 2 `0(ZZs) be a given re nement mask. If the cascade algorithm associated with the mask a converges in the Sobolev space Wpn(IRs) (1 p < 1), then (1) The space Vn is invariant under A" for all " 2 E , i.e., for v 2 Vn it follows that A"v 2 Vn. (2) The re nement mask a satis es the sum rules of order n + 1, i.e., for any p 2 n X
2
ZZs
p(M + ")a(M + ") =
X
2
ZZs
p(M )a(M )
8" 2 E:
(3:6)
Moreover, the assertions (1) and (2) are equivalent. The equivalence of (1) and (2) in Result 3.3 has already been shown in [14], Theorem 5.2 (see also [11] Theorem 3.4.12). The necessity of (1) for convergence of the cascade algorithm in Wpn(IRs)-norm follows from Theorem 2.1. If a sequence a 2 `0(ZZs) satis es (1.4) and (3.6), we say that a satis es the sum rules of order n + 1. We can now show the following relation.
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
11
Lemma 3.4. Assume that (3.5) is true for a given re nement mask a. Then Vn?1 in (2.8) is an invariant subspace under A" for all " 2 E: Proof. For " 2 E and 2 ZZs+, we de ne a polynomial p"; 2 jj by p";(x) =
X
2ZZs
a(M + ")(M ?1 (x ? ") ? ) :
By Result 3.3, Vn?1 is invariant under A" 8 " 2 E if and only if
8 "1; "2 2 E and jj n ? 1:
p"1; = p"2 ;
(3:7)
For n = 1, (3.7) has been proved in [10]. We shall prove (3.7) by induction on n. Assume that (3.7) holds for n = n0. If it is not true for n = n0 + 1; then there are "1 ; "2 2 E and 2 ZZs+ with jj = n0 such that p"1; 6= p"2;. We shall show that this leads to a contradiction. For any 2 ZZs+ and k = 1; 2; : : : ; let hk; 2 `(ZZs) be de ned by
hk;( ) =
X
2ZZs
ak ( ? M ) ;
2 ZZs;
where ak are given in (2.4). Observe that
h1;(M + ") = p"; (M + ")
8 2 ZZs and 8 " 2 E:
Thus, the induction assumption (3.7) for n = n0 implies now that
h1;( ) = p"; ( )
8 ; jj n0 ? 1; 8 " 2 E and 2 ZZs :
Consequently, for any 2 ZZs+ with j j = n = n0 + 1; we have h1; = 0; < : Moreover, by the assumption that p"1; 6= p"2; for some "1 ; "2 2 E and some jj = n0, we have a 2 ZZs+ with j j = n and an 2 ZZs such that h1;( ) 6= 0:
(3:8))
On the other hand, the induction relation of ak tells us that for 2 ZZs
hk;( ) =
X X
2ZZs 2ZZs X
a( ? M ? M)ak?1 ()( + ? ) X
(?1)j?j h1; ( ) ak?1 ()? 2ZZ s X X ? + h1; ( ) X ak?1 ( ): (?1)j?j h1; ( ) a = k ? 1 ( ) 0 satisfying that for any b 2 `( ) with jja ? bjj1 < we have X jjA"k A"1 ? B"k B"1 jjp < m(?n=s+1=p?t)kp: "1 ;:::;"k 2E
Note that Vn \ `(K ) is an invariant subspace of any A" and B"; " 2 E: Consequently, X jjA"k jVn\`(K) A"1 jVn\`(K) ? B"k jVn\`(K) B"1 jVn\`(K)jjp "1 ;:::;"k 2E
=
X
"1 ;:::;"k 2E
?
A"k
?
p
A"1 jV \`(K) ? B" B"1 jV \`(K)
< m(?n=s+1=p?t)kp: n
k
n
It follows from the triangle inequality that X jjB"k jVn\`(K) B"1 jVn\`(K)jjp < m(?n=s+1=p?t1)kp; "1 ;:::;"k 2E
(4:1)
where the positive number t1 is de ned by 2m(?n=s+1=p?t)kp = m(?n=s+1=p?t1)kp. Equality (2.2) tells us now p(fB"jVn\`(K) : " 2 E g) m(?n=s+1=p?t1) < m?n=s+1=p; and the assertion follows from Theorem 2.1. Our goal is now, to estimate the perturbation of the limit function in terms of the perturbation of the mask, i.e., we want to show that kQka 0 ? Qkb 0kWpn (IRs) c ka ? bk1 ; k = 1; 2; : : : ; where a; b meet the assumptions of Theorem 4.1. It turns out that the initial function 0 needs to satisfy a stability condition as given in Theorem 3.2. Indeed, choosing g = Qka 0;n ? Qkb 0;n, (3.4) implies that X X kD Qka 0;n ? D Qkb 0;nkp c m(n=s?1=p)k kak ? bk kp: jj=n
jj=n
Hence, taking into account the result of Theorem 3.5, we only have to estimate the norm kak ? bk kp for jj = n. In order to obtain this estimate we rst need:
14
D.R. CHEN AND G. PLONKA
Lemma 4.2. Assume that a and b satisfy sum rules of order n + 1. Then for any v 2 Vn?1 we have (B" ? A" )v 2 Vn 8" 2 E: Proof. We claim that, for any a satisfying sum rules of order n +1 and any p 2 n, there is a polynomial q 2 n?1 such that X p(? )a(" + M ? ) = p(M ?1 (" ? )) + q(" ? ) 8" 2 E and 8 2 ZZs : 2ZZs
(4:2)
In fact, it follows from Taylor's formula that p(M ?1 (" ? )) D ?1 (M ? + ")) : ( ? M p(? ) = ! jjn X
Therefore
X
2
ZZs
=
p(? )a(" + M ? )
D p(M ?1 (" ? )) X (?M ?1 (M ? + ")) a(" + M ? ): ! 2ZZs jjn X
P
Note that a satis es (1.4) and (3.6), i.e., we have 2ZZs a(" + M ? ) = 1 and X
2ZZs
(?M ?1 (M ? ? ")) a(" + M ? ) =
X
2ZZs
(? ) a(M ); jj n:
Hence, we obtain X
2ZZs
p(? )a(" + M ? )
=p(M ?1 (" ? )) +
D p(M ?1 (" ? )) (? ) a(M ): ! 0<jjn 2ZZs X
X
This proves (4.2). Applying (4.2) to b we get a polynomial g 2 n?1 such that X
2ZZs
p(? )(b(" + M ? ) ? a(" + M ? )) = g(" ? )
8" 2 E and 8 2 ZZs:
It follows from v 2 Vn?1 that, for any p 2 n, X
2
ZZs
p(? )(B" ? A" )v( ) =
The proof is complete.
X
2
ZZs
g(" ? )v( ) = 0
8" 2 E:
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
15
Lemma 4.3. Suppose that ZZs is a nites set and that the cascade algorithm corresponding to a 2 `( ) converges in Wpn(IR )-norm. Further, let b 2 `( ) satisfy the sum rules of order n+1 and ka?bk < , where is chosen such that the assertion of Theorem 4.1 holds. Then there is a positive number c such that we have
jjak ? bk jjp cjja ? bjj1m(?n=s+1=p)k
8jj = n and k = 1; 2; : : : ;
where c is independent of b and k. Proof. Let K be given in (2.7). By (2.12) and the equality
B"k B"1 ? A"k A"1 =
k X j =1
B"k B"j+1 (B"j ? A"j )A"j?1 A"1
we obtain
jj(bk ? ak ) vjjp 1=p X X = jB" B"1 v( ) ? A" A"1 v( )jp
"1 ;:::;"k 2E 2K k X X X
j =1 "1 ;:::;"k 2E 2K
k
k
jB" B" +1 (B" ? A" )A" ?1 A"1 v( )jp j
j
j
k
1=p
j
:
Thus,
jj(bk ? ak ) vjjp
k X
X
j =1 "1 ;:::;"k 2E
1=p p jjB"k B"j+1 (B"j ? A"j )A"j?1 A"1 vjj :
(4:3) P Let jj = n. Note that jjaj?1 jjpp = "1;:::;"j?12E jjA"j?1 A"1 jjp: Hence, by (3.5) in Theorem 3.2, there is a constant c1 > 0 such that for any j X
"1 ;:::;"j?1 2E
jjA" ?1 A"1 jjp c1m(?n=s+1=p)(j?1)p: j
(4:4)
On the other hand, from 2 Vn?1 and Lemma 4.2 it follows that (B"j ? A"j )A"j?1 A"1 2 Vn
8"1; : : : ; "j 2 E:
Moreover, by Theorem 4.1 we already know that the cascade algorithm corresponding to b converges in Wpn(IRs)-norm and by (4.1) there are a positive number t1 and a constant c2 such that X
jjB" B" +1 (B" ? A" )A" ?1 A"1 jjp k
j
"j+1 :::"k 2E c2m(?n=s+1=p?t1)(k?j)pjj(B"
j
j
j
j
? A" )A" ?1 A"1 jjp j
j
8 k > j:
16
D.R. CHEN AND G. PLONKA
This together with (4.4) implies X
X
jjB" B" +1 (B" ? A" )A" ?1 A"1 jjp k
j
j
j
j
"1 :::;"j?1 2E "j+1 :::"k 2E c3m(?n=s+1=p?t1)(k?j)pm(?n=s+1=p)(j?1)pjja ? bjjp; 1
where c3 is some constant which is independent of b and k. It follows from (4.3) that
jj(bk ? ak ) jjp c13=pjja ? bjj1m(?n=s+1=p)k
k X j =1
m?(k?j)t1 ;
k = 1; 2; : : : :
The proof is complete. We are now ready to present the main theorem of this section. Theorem 4.4. Let be a nite set in ZZs. Assume sthat the cascade algorithm corresponding to a 2 `( ) converges on Wn in Wpn(IR )-norm. Then there exists a positive constant such that, for any b 2 `( ) satisfying the sum rules of order n + 1 with ka ? bk1 < , the cascade algorithm corresponding to b converges in Wpn(IRs)-norm. Moreover, there exists a constant c, which is independent of b and k, such that
kQka 0;n ? Qkb 0;nkW
n p
(IRs )
c ka ? bk1 k = 1; 2; : : : :
(4:5)
where 0;n is given in (3.3). Consequently, we nd for the limit functions
k a ? bkW
n p
(IRs )
cka ? bk1 :
(4:6)
Proof. By Theorem 4.1 we know that for b 2 `( ) satisfying the sum rules of order n +1 and with ka ? bk < for some suitable the cascade algorithm corresponding to mask b converges on Wn in Wpn(IRs)-norm. Therefore, b 2 Wpn(IRs). Since 0;n 2 Wn the cascade algorithm converges on 0;n for a and b, i.e., we have
lim jjQka 0;n ? ajjWpn (IRs) = klim jjQk ? jj n s = 0: !1 b 0;n b Wp (IR )
k!1
The inequality (4.6) follows now from (4.5). In order to prove (4.5) we appeal to Theorem 3.2. Put = ak ? bk in (3.4). This corresponds to g = Qka 0;n ? Qkb 0;n. Then the second inequality in (3.4) yields for some constant c1 and for k = 1; 2; : : : X
jj=n
jjD (Qka 0;n ? Qkb 0;n)jjp c1m(n=s?1=p)k
X
jj=n
jjak ? bk jjp:
CONVERGENCE OF CASCADE ALGORITHMS FOR PERTURBED MASKS
17
Together with Lemma 4.3, it in turn implies X
jj=n
jjD (Qka 0;n ? Qkb 0;n)jjp c2jja ? bjj1 ;
k = 1; 2; : : : ;
(4:7)
where c2 is some positive number being independent of b and k. As shown in Theorem 3.5, the cascade algorithm corresponding to a also con0 s n 0 verges on Wn in Wp (IR )-norm with n0 < n. Replacing n with n0 in (4.7) and then taking the sum of the resulting inequalities we obtain (4.5). The proof is complete.
The proof of the estimate (4.5) is strongly based on the stability property of 0;n in Theorem 3.2. Thus, we obtain the following corollary. Corollarys 4.5. Let be a nite set in ZZs. Suppose that a is a re nable function in Wpn(IR ) corresponding to mask a 2 `( ) and the shifts of a are stable. Then there are positive constants and c such that, for any b 2 `( ) satisfying sum rules of order n + 1 and jja ? bjj1 < ; the re nable distribution b is in Wpn(IRs ) and satis es (4.6). Proof. By the stability of the shifts of a, the cascade algorithm corresponding to a converges on Wn in Wpn (IRs )-norm. This conclusion has been established in [16] for p = 2. The method works for general p 1. Now, using Theorem 3.2, the proof is analogous to that of Theorem 4.4. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
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D.R. CHEN AND G. PLONKA
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