In this paper, we consider the p-frame property of the space Vp( ) with being compactly supported function in. Lp \Lp=(p?1). Moreover, for the one-dimensional ...
Connection between p-frames and p-Riesz bases in locally nite SIS of Lp(R) Akram Aldroubi, Qiyu Sun and Wai-Shing Tang
ABSTRACT
Let 1 p 1 and = ( 1 ; : : : ; r )T be a vector-valued compactly supported Lp function on Rd. De ne
Vp () =
o
nP r P i=1 j 2Zd i (
d j ) i ( ? j ) : (di (j ))j2Zd 2 `p ; 1 i r :
In this paper, we consider the p-frame property of the space Vp () with being compactly supported function in Lp \ Lp=(p?1) . Moreover, for the one-dimensional case, we show that if f i ( ? j ) : 1 i r; j 2 Zg is a p-frame for Vp (), then there exist a positive integer s r and compactly supported functions 1 ; : : : ; s 2 Vp () such that f i ( ? j ) : 1 i s; j 2 Zg is a p-Riesz basis of Vp (), where = ( 1 ; : : : ; s )T . Keywords: Frame, Riesz basis, Shift-invariant spaces
1. INTRODUCTION AND RESULTS
Given 1 p 1 and a vector-valued compactly supported Lp function = ( 1 ; : : : ; r )T on Rd, de ne an operator 0 on (`p )r , the direct sum of r copies of `p , by
0 : (`p )r 3 D = (d1 (j ); : : : ; dr (j ))T
7?! j 2Zd
r X X i=1 j 2Zd
di (j ) i ( ? j ):
Here and henceforth, AT denotes the transpose of a matrix (or vector) A. We say that 0 D is the semiconvolution between and D, and that the operator 0 is a semi-convolution operator. It is easy to check that for any 1 p 1 and any vector-valued compactly supported Lp function on Rd, there exists a positive constant C such that k 0 Dkp C kDkp 8 D 2 (`p )r : (1.1) Let Vp () be the image of the semi-convolution operator 0 on (`p )r . Then Vp () is a linear subspace of Lp := Lp (Rd ) by (1.1). A linear space V of distributions on Rd is said to be shift-invariant if g 2 V implies g( ? j ) 2 V for all j 2 Zd . Obviously Vp () is a shift-invariant subspace of Lp for any 1 p 1. The space Vp () is said to be the shift-invariant space generated by , or 1 ; : : : ; r . The family of functions 1 ; : : : ; r is said to be a generator of the space Vp (). For shift-invariant spaces, there is a long list of publications concerning various problems.1,3,4,6{8,11 Let 1 p 1. We say that a countable collection fg : 2 g Lp is a p-Riesz basis in Lp if there exists a positive constant C such that
X
C ?1 kckp
2
c g
p C kckp
8 c = (c )2 2 `p ():
Let Vp be the linear span of g using `p () coecient sequences. Then there is a unique and stable representation P for any f 2 Vp , i.e., f = 2 c g , when fg : 2 g is a p-Riesz basis in Lp . De ne the Fourier transform
A. Aldroubi: Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA. Research was partially supported by NSF Grant DMS-9805483. Q. Sun and W.-S. Tang: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore. Research was partially supported by Wavelets Strategic Research Programme, National University of Singapore, under a grant from the National Science and Technology Board and the Ministry of Education, Singapore.
R
f^ of an integrable function f by f^( ) = Rd f (x)e?ix dx. For the shift-invariant space Vp (), the following characterization of a p-Riesz basis of that space is well known.8 Proposition 1.1. Let 1 p 1, and let = ( 1 ; : : : ; r )T be a vector-valued compactly supported Lp function on Rd. Then f i ( ? j ) : 1 i r; j 2 Zd g is a p-Riesz basis of Vp () if and only if
b + 2j ) rank ( = r 8 2 [?; ]d : j 2Zd
(1.2)
It is known12 that if f i ( ? j ) : 1 i r; j 2 Zd g is a p-Riesz basis of Vp (), then there is a stable way to represent functions in Vp (). In particular, there exist compactly supported C 1 functions ~i and sequences (Ppi (j ))j2Zd ; 1 i r, such that pi (j ) = 0 for all but nitely many j 2 Zd , the trigonometric polynomials ?ij ; 1 i r, do not vanish on Rd, and j 2Zd pi (j )e
f=
r X X X i=1 j 2Zd
k2Zd
ri (k)hf; ~i ( ? j + k)i i ( ? j ) for all f 2 Vp ();
(1.3)
?P P P where j2Zd ri (j )e?ij = j2Zd pi (j )e?ij ?1 : The functions ~i = k2Zd ri (k) ~i ( + k); 1 i r, are said to be dual functions. The dual functions ~i ; 1 i r, are C 1 functions with exponential decay at in nity. (We remark that the construction of the dual functions ~i ; 1 i r, under the linear independence assumption for were given by Ben-Artzi, Ron and Zhao2,13). A stable representation of the form (1.3) is crucial for some applications. To give such a stable representation for all functions in Vp (), it is not necessary to assume that f i ( ? j ) : 1 i r; j 2 Zg is a p-Riesz basis of Vp (). In fact, a p-frame for Vp () is enough for that purpose. Let 1 p 1, B be a normed linear space and B be its dual. We say that an index family fg : 2 g B is a p-frame for B if there exists a positive constant C such that
C ?1 kf kB fhf; g ig2 p C kf kB 8 f 2 B: The p-frame property of a nitely generated shift-invariant space with p 6= 2 was studied by the authors.1 In
this paper, we shall re ne the results there for the case when the generators are compactly supported. In order to do so, we need two function spaces Lp and L~p . For any 1 p 1, let n
X
Lp = f : kf kLp =
and
L~p = ff : kf kL~p =
k2Zd X
k2Zd
jf ( + k)j
p;[0;1]d < 1
o
kf ( + k)kp;[0;1]d < 1g;
where kf kp;E is the Lp norm of a measurable function f on the set E . It is easy to check that L~1 = L1 = L1 and Lpc L~p Lp Lp 8 1 p 1; (1.4)
where Lpc is the space of all compactly supported Lp functions. Further the spaces L~p and Lp are invariant under linear combinations of shifts using `1 coecients, i.e., if 2 L~p (resp. Lp ) and D = (d(j ))j2Zd 2 `1, then P p ~p j 2Zd d(j ) ( ? j ) 2 L (resp. L ). In particular,
k 0 DkL~p kDk1k kL~p and k 0 DkLp kDk1k kLp :
(1.5)
In this paper, we prove the following result about p-frame property of the space Vp (). Theorem 1.2. Let 1 p 1, and let = ( 1 ; : : : ; r )T be a vector-valued compactly supported Lp \ Lp=(p?1) function on Rd. Then the following statements are equivalent: (i) Vp () is closed in Lp .
(ii) f i ( ? j ) : j 2 Zd ; 1 i rg is a p-frame for Vp (), i.e., there exists a positive constant A (depending on and p) such that r Z
X A?1 kf kp
i=1
?
f (x) i (x ? j )dx j2Zd
p Akf kp 8 f 2 Vp (): d
R
d b + 2j ) (iii) The rank of the matrix ( j 2Zd is independent of 2 [?; ] . (iv) There exists a positive constant B (depending on and p) such that
B ?1 kf kp
r X Prinf f = i=1 i 0 Di i=1
kDi kp B kf kp
8 f 2 Vp ():
(v) There exists = ( 1 ; : : : ; r )T 2 L~p \ L~p=(p?1) such that
f=
r X X i=1 j 2Zd
hf; i ( ? j )i i ( ? j ) =
r X X i=1 j 2Zd
hf; i ( ? j )i i ( ? j )
8 f 2 Vp ():
(1.6)
1 We Under the assumption that 2 L1 c , the assertion of Theorem 1.2 has been proved by the authors. remark that the functions i , 1 i r, in (v) of Theorem 1.2 belong to V1 () Vp () and are independent of p, 1 p 1, but not compactly supported in general. In fact, i
=
r X X i0 =1 j 2Zd
cii0 (j ) i0 ( ? j )
for some `1 sequences fcii0 (j ) : j 2 Zd g, 1 i; i0 r, independent of p. By Proposition 1.1 and Theorem 1.2, for compactly supported generators i 2 Lp \ Lp=(p?1) ; 1 i r, if f i ( ? j ) : 1 i r; j 2 Zd g is a p-Riesz basis of Vp () then it is p-frame for Vp (). The converse is obviously false. From (v) of Theorem 1.2, the p-frame property for f i ( ? j ) : 1 i r; j 2 Zd g is sucient, and almost necessary, for a stable representation of the form (1.3) for all functions in Vp (). The conditions in Theorem 1.2 is not so easy to check. For generator having explicit expression, we can reduce the p-frame properties of the space Vp () to checking the rank of some matrix with trigonometric polynomial entries. For a vector-valued compactly supported Lp function = ( 1 ; : : : ; r )T , let
V0 () =
r X nX i=1 j 2Zd
o
di (j ) i ( ? j ) : di (j ) = 0 for all but finitely many j 2 Zd ; 1 i r ;
and V[0;1]d () be the restriction of all functions in V0 () to [0; 1]d. Since is compactly supported, V[0;1]d () is a nite dimensional linear space. Let f 1 ; : : : ; r0 g be a basis of V[0;1]d () and set = ( 1 ; : : : ; r0 )T : The vector-valued function belongs to Lp , and can be constructed explicitly if has explicit expression. Recall that i ; 1 i r0 , are supported in [0; 1]d, and linearly independent on [0; 1]d. Then there exist sequences Ds = fds (j )gj2Zd ; 1 s r0 , such that ds (j ) = 0 for all but nitely many j 2 Zd and 1 s r0 , and =
r0 X X j 2Zd s=1
ds (j ) s ( ? j ):
(1.7)
Theorem 1.3. Let 1 p 1, = ( 1 ; : : : ; r )T be a vector-valued compactly supported Lp \ Lp=(p?1) function, for Vp ( ) if and only if and let Ds = (ds (j ))j2Zd be as in (1.7). Then f i ( ? j ) : 1 i r; j 2 Zd g is a p-frame P the rank of the matrix (D1 ( ); : : : ; Dr0 ( )) is independent of 2 Rd, where Ds ( ) = j2Zd ds (j )e?ij ; 1 s r0 .
For the one-dimensional case, we shall prove that, from the viewpoint of the nitely generated shift-invariant space, the existence of a p-frame and that of a p-Riesz basis of that space are essentially the same. Theorem 1.4. Let 1 p 1, and let = ( 1 ; : : : ; r )T be a vector-valued compactly supported Lp \ Lp=(p?1) function on R. Assume that f i (?j ) : 1 i r; j 2 Zg is a p-frame for Vp (). Then there exist a positive integer s r and a compactly supported function = ( 1 ; : : : ; s )T in Lp \Lp=(p?1) such that f i (?j ) : 1 i s; j 2 Zg is a p-Riesz basis of Vp ( ), and Vp ( ) = Vp (). For 1 < p < 1, the theorem 1.4 follows by combining our Theorem 1.2 and Theorem 1 by Jia.6 However, we give a simple proof below.
2. PROOFS
In this section, we gather the proofs of Theorems 1.2, 1.3 and 1.4.
2.1. Proof of Theorem 1.2
We shall not give the detail of the proof of Theorem 1.2, because we may follow the same procedure as the one in our previous paper1 with the following modi cations, the estimate
(
hf; g( ? j )i)j2Zd
p C kf kpkgk11?1=pkgk1L=p1 8 f 2 Lp
replaced by
(
(2.1) hf; g( ? j )i)j2Zd
p kf kpkgkL~ p= p? 8 f 2 Lp ; and Lemma 3 in the proof of (ii)=)(iii) there by P Lemma 2.1. Let 1 p 1 and g 2 L~p . Assume that j2Zd g( ? j ) = 0: Then for any function h on Rd satisfying
(
1)
jh(x)j C (1 + jxj)?d?1 and jh(x) ? h(y)j C jx ? yj (1 + min(jxj; jyj))?d?1 ;
we have
X
lim 2?nd
n!1
j 2Zd
(2.2)
h(2?nj )g( ? j )
L~p = 0:
(2.3)
Then it remains to prove (2.1) and Lemma 2.1. Let us rst prove (2.1). For 1 p < 1, we have
p
(
hf; g( ? j )i)j2Zd
p
X X Z
j 2Zd k2Zd [0;1] X X j 2Zd k2Zd X X
jf (x + k)jjg(x ? j + k)jdx d
p
kf ( + k)kp;[0;1]d kg( ? j + k)kp=(p?1);[0;1]d
p
kf ( + k)kpp;[0;1]d kg( ? j + k)kp=(p?1);[0;1]d
j 2Zd k2Zd f pp g pL~p=(p?1) :
k kkk
X
k2Zd
kg( ? j + k)kp=(p?1);[0;1]d
p?1
This proves (2.1) for 1 p < 1. The proof of (2.1) for the case p = 1 can be done in a similar way. We omit the details here. Secondly, we prove Lemma 2.1. In fact, we use the same procedure as the one used to prove Lemma 3 of our previous paper.1 Note that there exists a positive constant C independent of n such that 2?nd
X
j 2Zd
jh(2?nj )j C 8 n 1:
This together with (1.5) leads to
X
2?nd
j 2Zd
h(2?nj )g( ? j )
L~p C kgkL~p 8 g 2 L~p and n 1;
where C is a positive constant independent of n and g. Thus we may assume that g has compact support. Let N0 be the positive constant such that g(x) = 0 for all jxj N0 ? 1. For 1 p 1, we obtain X
X
k2Zd j 2Zd X
X ( 2?nd
k2Zd j 2Zd X X 2?nd
2?nd
=
h(2?nj )g( ? j + k)
p;[0;1]d
h(2?n j ) ? h(2?nk))g( ? j + k)
p;[0;1]d
k2Zd jj ?kjN0
jh(2?n j ) ? h(2?nk)jkgkp ! 0 as n ! 1:
This proves (2.3) for 1 p 1, and hence Lemma 2.1.
2.2. Proof of Theorem 1.3
Proof. We say that a vector-valued compactly supported distribution F = (f1 ; : : : ; fr )T has linearly independent shifts if di (j ) = 0 for all 1 i r and j 2 Zd is a necessary and sucient condition such that P r P 0 d i=1 j 2Zd di (j )fi (? j ) 0. Recall that i ; 1 i r , are linearly independent and supported in [0; 1] . Then b + 2k )) has linear independent shifts. By the characterization via Fourier transform,9,10 the matrix ( ( k2Zd d has full rank for any 2 R , i.e., d b + 2k )) rank( ( k2Zd = r 8 2 R :
(2.4)
Taking Fourier transform at both sides of (1.7) leads to b ) = (
which implies
b + 2k ) (
Combining (2.4) and (2.5), we have
b + 2k ) rank (
k2Z
k2Z
s=1
Ds ( ) bs ( );
r0 X
b + 2k ) = (D1 ( ); : : : ; Dr0 ( )) (
= rank D1 ( ); : : : ; Dr0 ( ) d
k2Z
:
8 2 [?; ]d :
(2.5) (2.6)
Hence Theorem 1.3 follows from (2.6) and Theorem 1.2.
2.3. Proof of Theorem 1.4
Proof. For any vector-valued compactly supported L2 function = ( 1 ; : : : ; r )T , de ne its auto-correlation matrix by X T b ) = b + 2j )( b + 2j ) [b ; ]( ( (2.7) j 2Z b ) is an r r matrix with trigonometric polynomial entries. By Smith normal form,5 there exist Then [b ; ]( integers k1 and k2 , and r r matrices E ( ); F ( ) and D( ) with trigonometric polynomial entries such that D( ) is diagonal, E ( )E (? )T = e?ik Ir and F ( )F (? )T = e?ik Ir (2.8) and b ) = E ( )D( )F ( ); [b ; ]( (2.9) 1
2
where Ir is the r r unit matrix. b ) = F ( )( b ). Then i ; 1 i r, are compactly supported, and De ne = ( 1 ; : : : ; r )T by ( b ) = F ( )[ b ; ]( b )F (? )T = e?ik F ( )E ( )D( ) [ b ; ]( 2
(2.10)
b ) has constant rank s( r) for any 2 R, which follows from (iii) of by (2.7), (2.8) and (2.9). Recall that [b ; ]( Theorem 1.2. Then D( ) = D10( ) 00 (2.11)
for some s s diagonal matrix D1 ( ) with trigonometric polynomial diagonal entries. Furthermore D1 ( ) is b ) T = [ b ; ]( b ? ) lead to nonsingular for all 2 R. Substituting (2.11) into (2.10), and using [ b ; ](
e?ik F ( )E ( )D( ) = 2
R( ) 0 0
0
(2.12)
for some s s matrix R with trigonometric polynomial entries, and with R( )T = R(? ). Recall that F ( ) and E ( ) are nonsingular for any 2 R. Then the matrix R( ) has full rank s for any 2 R. De ne 1 = ( 1 ; : : : ; s )T and 2 = ( s+1 ; : : : ; r )T . Then 2 = 0 (2.13) and [ b 1 ; b 1]( ) = R( ) by (2.10) and (2.12). It follows from Proposition 1.1 that f i ( ? j ) : 1 i s; j 2 Zg is a p-Riesz basis of Vp ( 1 ). By the de nition, i ; 1 i r, are nite linear combinations of the integer shifts of i ; 1 i r. Thus
Vp ( ) Vp ():
(2.14)
b ) = eik F (? )T ( b ), which implies that i ; 1 i r, are nite By (2.8), F (? )T F ( ) = e?ik Ir . Thus ( linear combinations of the integer shifts of i ; 1 i r. Hence 2
2
Vp () Vp ( ):
(2.15)
Combining (2.14) and (2.15) leads to Vp ( ) = Vp (). Obviously Vp ( ) = Vp ( 1 ) since 2 = 0 by (2.13). This prove that f i ( ? j ) : 1 i s; j 2 Zg is a p-Riesz basis of Vp ().
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