p-frames and Shift Invariant Subspaces of L
p
Akram Aldroubi, Qiyu Sun and Wai-Shing Tang
May 16, 2000
Abstract
We investigate the frame properties and closedness for the shift invariant space Vp(Φ) = nXr i=1 X j∈Zd di(j)φi(· − j) : (di(j))j∈Zd ∈ ℓp o , 1 ≤ p ≤ ∞.
We derive necessary and sufficient conditions for a set {φi(· − j) : 1 ≤ i ≤
r, j ∈ Zd} to constitute a p-frame for Vp(Φ), and to generate a closed shift
invariant subspace of Lp. A function in the Lp-closure of V
p(Φ) is not necessarily
generated by ℓp coefficients. Hence we often want Vp(Φ) itself to be closed, i.e.,
a Banach space. For p 6= 2, this issue is complicated, but we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of p-frames, Banach frames, and the closedness of the space they generate. Since the frame vectors may be dependent, the relation between a function f ∈ Vp(Φ) and the coefficients of its representations is neither obvious,
nor unique, in general. In Hilbert spaces, it is well known that the coefficients sequence obtained by using the Duffin-Schaeffer representation has minimal size. Moreover, the ℓ2-norm of the coefficients sequence is equivalent to the norm of
f. For the case of p-frames, we are in the context of normed linear spaces, but
∗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.
we are still able to give a characterization of p-frames in terms of the equivalence between the norm of f and an ℓp-norm related to its representations. A Banach frame does not have a dual Banach frame in general, however, for the shift invariant spaces Vp(Φ), dual Banach frames exist and can be constructed.
1
Introduction
A frame for a Hilbert space H is a set of vectors {gλ}λ∈Λ ⊂ H that can be used to
represent, in a stable way, any vector f ∈ H as a linear combination f = P
λ∈Λd(λ)gλ
with D = (d(λ)) ∈ ℓ2(Λ), where Λ is a countable index set. In particular, any vector
f ∈ H can be represented as
f = X
λ∈Λ
hf, ˜gλigλ, (1.1)
where {˜gλ}λ∈Λ ⊂ H is any dual frame. Thus, frames are generalization of Riesz
bases and they were introduced by Duffin and Schaeffer in the context of nonharmonic Fourier series [14]. In particular, any Riesz basis is also a frame, but the vectors {gλ}λ∈Λ ⊂ H of a frame may be dependent, therefore a frame is not necessarily a
Riesz basis. The relatively recent interest in frame theory is due to its use in wavelet theory, time frequency analysis, and sampling theory (see for example [2, 5, 6, 8, 9, 10, 11, 12, 15, 20, 21, 26, 27, 28, 29]). In these contexts, the frames {gλ}λ∈Λ ⊂ H
have some additional structure. Specifically, in wavelet theory {gλ}λ∈Λ are of the form
{φ ((x − nb)/ma)}m∈Z+,n∈Zd ⊂ L2(Rd), in time frequency analysis the frames are of
the form {eimbxφ(x − na)}(m,n)∈Z2d ⊂ L2(Rd), and in sampling theory they are of
the form {φ(x − na)}n∈Zd ⊂ L2(Rd), generating shift invariant spaces. While the
concept of Riesz basis has been generalized to Lp spaces and studied in the context
of shift invariant spaces [23, 24], the systematic study of frames in Banach spaces is relatively recent [9, 16, 19], and we are not aware of any investigation of frames in shift invariant subspaces of Lp (p 6= 2). Aside from their theoretical appeal, frames in Lp
spaces and other Banach function spaces are effective tools for modeling a variety of natural signals and images [13, 26]. They are also used in the numerical computation of integral and differential equations. Hence, one goal of this paper is to study frames for certain shift invariant subspaces of Lp which are generated by finitely many functions.
Specifically, we investigate frames of the form {φi(· − j) : j ∈ Zd, 1 ≤ i ≤ r} ⊂ Lp,
that we call p-frames, for the spaces Vp(Φ) generated by linear combinations of the
frame vectors using ℓp coefficient sequences. Under certain conditions on the decay
{φi(· − j) : 1 ≤ i ≤ r, j ∈ Zd} to constitute a p-frame, and to generate a shift
invariant, closed subspace of Lp. For the case p = 2, this problem has been considered
by [6, 7, 25, 28].
Let V0(Φ) be the space of finite linear combination of integer translates of φi, 1 ≤
i ≤ r, and let V0,p(Φ) be the Lp closure of V0(Φ). The space V0,p(Φ) has been studied
by [6, 7, 22, 23, 28]. Obviously, we have V0(Φ) ⊂ Vp(Φ) ⊂ V0,p(Φ). We remark that a
function f in V0,p(Φ) is not necessarily generated by ℓpcoefficients. So the closedness of
the shift invariant spaces Vp(Φ) is an important issue in some applications. Hence we
often want Vp(Φ) itself to be closed, i.e., a Banach space. In that case, Vp(Φ) = V0,p(Φ).
For the case p = 2, this problem has been considered by many investigators, see for example [4]. In the case of p 6= 2, this issue is more complicated since Lp is not a
Hilbert space and the Parseval identity is no longer applicable. But we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of p-frames, Banach frames (with respect to ℓp), and the closedness of the
space they generate.
Since the vectors {gλ}λ∈Λof a frame may be dependent, the relation between a
func-tion f = P
λ∈Λd(λ)gλ and the coefficients of its representation D = (d(λ))λ∈Λis not
obvi-ous. In Hilbert spaces, it is well known that the coefficients sequence (d(λ)) = (hf, ˜gλi)
obtained by using the Duffin-Schaeffer dual frame {˜gλ}λ∈Λhas minimal size [14].
More-over, the ℓ2-norm of D is equivalent to the norm of f . For the case of p-frames, we are
in the context of Banach spaces, but we are still able to give a characterization of p-frames in terms of the equivalence between inf {Pr
i=1kDikℓp : f =Pri=1φi∗′Di} and
the Lp-norm of f . Actually, when {φ
i(· − j) : 1 ≤ i ≤ r, j ∈ Zd} is a p-frame for Vp(Φ),
we construct functions ψi, 1 ≤ i ≤ r, indepenedent of p with ψi ∈ V1(Φ) ⊂ Vp(Φ) such
that f = r X i=1 X j∈Zd hf, ψi(· − j)iφi(· − j) = r X i=1 X j∈Zd hf, φi(· − j)iψi(· − j) ∀ f ∈ Vp(Φ).
Moreover, Vp(Φ) = Vp(Ψ) and the family {ψi(· − j) : 1 ≤ i ≤ r, j ∈ Zd} is also a
p-frame for Vp(Φ). So the family {ψi(· − j) : 1 ≤ i ≤ r, j ∈ Zd} can be thought of as
The paper is organized as follows: Section 2 will introduce some notation, defini-tions and preliminaries. We present the main theorem and some of its corollaries in Section 3. Several technical lemmas are given in Section 4. In particular, Lemma 1 gives some equivalent relations to the condition (iii) of Theorem 1. Lemma 2 provides a localization technique in Fourier domain which is essential to the proofs of our main theorem, and Lemma 3 is crucial for two key estimates in the proofs of (i) =⇒ (iii) and (ii) =⇒ (iii) of Theorem 1. The localization technique in Lemma 2 and the estimate in Lemma 3 are not necessary if we restrict ourselves to the case p = 2. All the proofs are gathered in Section 5.
2
Notation, Definitions and Preliminaries
2.1
Periodic Distributions
We say that T is a 2π-periodic distribution if it is a tempered distribution on Rd
and T = T (· + 2jπ) for all j ∈ Zd. A 2π-periodic distribution can also be thought of
as a continuous linear functional on the space of all 2π-periodic C∞ functions on Rd.
For any 2π-periodic distribution T , there exist an integer N0 and a positive constant
C such that
|T (f )| ≤ C X
|α|≤N0
kDαf k
L∞ (2.1)
for any 2π-periodic C∞ function f on Rd. Note that e−ijξ is a 2π-periodic C∞function
on Rdfor any j ∈ Zd. We define T (eij·) as the j-th Fourier coefficient of T and formally
write T = Pj∈ZdT (eij·)e−ijξ. It follows from (2.1) that there exists a polynomial P
such that |T (eij·)| ≤ P (j) for all j ∈ Zd. Conversely for any sequence D = {d(j)} j∈Zd
dominated by some polynomial, F(D) =Pj∈Zdd(j)e−ij· is a 2π-periodic distribution.
For a 2π-periodic distribution T , we say that T is supported in A + 2πZd if T (f ) = 0
for any 2π-periodic C∞ function f supported in Rd\(A + 2πZd).
2.2
Sequence Spaces
For two sequences D1 = (d1(j))j∈Zd ∈ ℓp1 and D2 = (d2(j))j∈Zd ∈ ℓp2 with 1/p1 +
1/p2 ≥ 1, define their convolution D1 ∗ D2 as
D1∗ D2(j) =
X
j′
∈Zd
It is easy to check that D1 ∗ D2 ∈ ℓr with 1/r = 1/p1+ 1/p2− 1 for all D1 ∈ ℓp1 and
D2 ∈ ℓp2. Denote by WCp, 1 ≤ p ≤ ∞, the space of all 2π-periodic distributions
with their sequences of Fourier coefficients in ℓp. For p = 1, WC1 is simply the
Wiener class WC. For a 2π-periodic distribution T ∈ WCp, define kT kℓp∗ to be the
ℓp norm of its sequence of Fourier coefficients. For a vector T = (T
1, ..., Tr)T of
2π-periodic distributions, we say that T ∈ WCp if Ti ∈ WCp, 1 ≤ i ≤ r, and define
kT kℓp∗ =
Pr
i=1kTikℓp∗. For 2π-periodic distributions T1 and T2 with their sequences of
Fourier coefficients D1 and D2 respectively, define their product T1T2as the 2π-periodic
distribution with its sequence of Fourier coefficients given by D1∗D2 if it is well defined.
Thus the product of two 2π-periodic distributions T1 ∈ WCp1 and T2 ∈ WCp2 is well
defined and belongs to WCrwhen 1/r = 1/p1+1/p2−1 ≥ 0. In particular, the product
between a 2π-periodic distribution in WCp and a 2π-periodic function in the Wiener class WC still belongs to WCp.
2.3
Function Spaces
Let Lp = nf : kf kLp = ³ Z [0,1]d ³ X j∈Zd |f (x + j)|´pdx´1/p < ∞o if 1 ≤ p < ∞, L∞ = nf : kf k L∞ = sup x∈[0,1]d X j∈Zd |f (x + j)| < ∞o if p = ∞, and W =nf : kf kW = X j∈Zd sup x∈[0,1]d |f (x + j)| < ∞o.For F = (f1, ..., fr)T, we set kF kX = Pri=1kfikX and say that F ∈ X if kF kX < ∞,
where X = Lp, Lp or W. Here AT denotes the transpose of A. Obviously we have
W ⊂ L∞ ⊂ Lq ⊂ Lp ⊂ Lp, where 1 ≤ p ≤ q ≤ ∞ (for instance see [2, 24] for more
properties). For p = 1, we also have that L1 = L1. For any 1 ≤ p ≤ ∞, f ∈ Lp and
g ∈ L∞, by straightforward computation we have
° ° ° n Z Rdf (x)g(x − j)dx o j∈Zd ° ° ° ℓp ≤ kf kLpkgk 1−1/p L1 kgk 1/p L∞ (2.2)
Define the Fourier transform ˆf of an integrable function f by ˆf (ξ) =RRdf (x)e−ixξdx
and that of a vector-valued tempered distribution by the usual interpretation. Denote the inverse Fourier transform of f by F−1(f ). By the Riemann-Lebesgue Lemma, the
Fourier transform of an integrable function is continuous. Hence the Fourier transform of an Lp function, 1 ≤ p ≤ ∞, is continuous.
2.4
Semi-convolution
For any sequence D = (d(j))j∈Zd ∈ ℓp and f ∈ Lp, define their semi-convolution
f ∗′ D by f ∗′ D = P
j∈Zdd(j)f (· − j) (For p = ∞, the convergence of the series is
pointwise convergence, but not L∞ convergence). It is not difficult to check that f ∗′
is a continuous map from ℓp to Lp, and also from ℓ1 to Lp if f ∈ Lp, 1 ≤ p ≤ ∞, and
it is a continuous map from ℓ1 to W if f ∈ W. Indeed, we have
kf ∗′Dk Lp ≤ kDkℓpkf kLp, (2.3) kf ∗′Dk Lp ≤ kDkℓ1kf kLp, (2.4) and kf ∗′DkW ≤ kDkℓ1kf kW. (2.5)
2.5
Shift Invariant Space
For Φ = (φ1, ..., φr)T ∈ Lp, let Vp(Φ) = nXr i=1 φi∗′ Di : Di ∈ ℓp, 1 ≤ i ≤ r o .
It follows from (2.3) that Vp(Φ) is a shift invariant linear subspace of Lp for 1 ≤ p ≤ ∞.
Here the shift invariance of a linear space V means that g ∈ V implies g(· − j) ∈ V for all j ∈ Zd. The space V
p(Φ) is said to be the shift invariant space generated by Φ.
2.6
p-frames and Banach frames
Let 1 ≤ p ≤ ∞, let B be a normed linear space and B∗ be its dual. We say that a
countable collection {gλ : λ ∈ Λ} ⊂ B∗ is a p-frame for B if the map T defined by
T : B ∋ f 7−→ {hf, gλi}λ∈Λ ∈ ℓp(Λ),
is both bounded and bounded below, i.e., there exists a positive constant C such that
C−1kf kB ≤ ³ X λ∈Λ |hf, gλi|p ´1/p ≤ Ckf kB ∀ f ∈ B (2.6) for 1 ≤ p < ∞, and C−1kf kB ≤ sup λ∈Λ |hf, gλi| ≤ Ckf kB ∀ f ∈ B (2.7)
for p = ∞.
A Banach frame (with respect to ℓp) is a p-frame with a bounded left inverse R of
the operator T [16, p. 148].
For Hilbert spaces, (2.6) or (2.7) guarantee the existence of a reconstruction opera-tor R that allows the reconstruction of a function f ∈ B from the sequence {hf, gλi}λ∈Λ.
However, for Banach spaces the operator R does not exist in general. Thus, for Banach spaces, the existence of a reconstruction operator R is included as part of the definition of Banach frames (see [9, 16, 19]). However, in the definition of p-frame above, the existence of a reconstruction operator is not required. Instead for finitely generated shift invariant spaces Vp(Φ) considered in this paper, the p-frame condition (2.6), or
(2.7), is sufficient to prove the existence of a reconstruction operator R. Therefore a p-frame for Vp(Φ) is a Banach frame.
2.7
p-Riesz Basis
Let 1 ≤ p < ∞, B a normed linear space and Λ a countable index set. We say that a collection {gλ : λ ∈ Λ} ⊂ B is a p-Riesz basis in B if the map defined by
ℓp(Λ) ∋ (cλ)λ 7−→
X
λ∈Λ
cλgλ ∈ B
is both bounded and bounded below, i.e., there exists a positive constant C such that C−1kckℓp ≤ ° ° °X λ∈Λ cλgλ ° ° ° B ≤ Ckckℓp ∀ c = (cλ)λ∈Λ∈ ℓ p(Λ). (2.8)
For p = 2, this definition is consistent with the standard definition of a Riesz ba-sis for the closed span of its elements. Definition (2.8) implies that the space Vp =
n P
λ∈Λcλgλ : c ∈ ℓ
p(Λ)o is a complete subspace of B. Thus, V
p is a Banach space even
though B may not be complete. In fact, Definition (2.8) implies that ℓp(Λ) and V p are
isomorphic Banach spaces. Obviously, a p-Riesz basis is unconditional, i.e., the sum in the middle term of (2.8) is independent of the order in which the sum is performed.
2.8
Bracket Product
For any functions Φ = (φ1, . . . , φr)T and Ψ = (ψ1, . . . , ψs)T such that φbi(ξ)ψbi′(ξ) is
integrable for any 1 ≤ i ≤ r and 1 ≤ i′ ≤ s, define an r × s matrix
[Φ,b Ψ](ξ) =b ³ X j∈Zd b φi(ξ + 2jπ)ψbi′(ξ + 2jπ) ´ 1≤i≤r,1≤i′≤s.
Observe that for any φ, ψ ∈ L2, we have
X j∈Zd Z Rd|φ(x)ψ(x − j)|dx ≤ Z [0,1]d X k∈Zd |φ(x − k)| X j∈Zd |ψ(x − j)|dx ≤ kφkL2kψkL2
Thus for any Φ, Ψ ∈ L2, Poisson’s summation formula implies that all entries of
[Φ,b Ψ](ξ) belong to the Wiener class, and are continuous.b
2.9
Rank of a r × ∞ Matrix
For w(j) = (w1(j), . . . , wr(j))T ∈ Cr satisfying Pri=1
P
j∈Zd|wi(j)|2 < ∞, define an
r × ∞ matrix G by
G = (w(j))j∈Zd. (2.9)
For the matrix G in (2.9), define its column rank as the maximal number of linearly independent columns (as vectors in Cr) of G, and its row rank as the maximal number
of linearly independent rows (as vectors in ℓ2) of G. It is obvious that the column rank
and the row rank of G are equal, which is denoted by rank G. For the r × ∞ matrix G in (2.9), define the r × r matrix GGT =³ P
j∈Zdwi(j)wi′(j) ´ 1≤i,i′ ≤r. It can be shown that rank G = rank GGT. (2.10)
Recall that [Φ,b Φ](ξ) is continuous for any Φ ∈ Lb 2. Then by (2.10),
ΩN =
n
ξ ∈ Rd : rank (Φ(ξ + 2kπ))b k∈Zd > N
o
(2.11)
is an open set for any N ≥ 0 and Φ ∈ L2.
3
Main Results
In this paper, we shall prove
Theorem 1 Let Φ = (φ1, ..., φr)T ∈ L∞ if 1 < p < ∞, and Φ ∈ W if p = 1, ∞. Then
the following statements are equivalent to each other.
(ii) {φi(· − j) : j ∈ Zd, 1 ≤ i ≤ r} is a p-frame for Vp(Φ), i.e., there exists a positive
constant A (depending on Φ and p) such that
A−1kf kLp ≤ r X i=1 ° ° °³Z Rd f (x)φi(x − j)dx ´ j∈Zd ° ° ° ℓp ≤ Akf kLp ∀ f ∈ Vp(Φ). (3.1)
(iii) There exists a positive constant C such that
C−1[Φ,b Φ](ξ) ≤ [b Φ,b Φ](ξ)[b Φ,b Φ](ξ)b T ≤ C[Φ,b Φ](ξ) ∀ ξ ∈ [−π, π]b d.
(iv) There exists a positive constant B (depending on Φ and p) such that
B−1kf kLp ≤ inf f =Pri=1φi∗′Di r X i=1 kDikℓp ≤ Bkf kLp ∀ f ∈ Vp(Φ). (3.2)
(v) There exists Ψ = (ψ1, . . . , ψr)T ∈ L∞ if 1 < p < ∞, and Ψ ∈ W if p = 1, ∞,
such that f = r X i=1 X j∈Zd hf, ψi(· − j)iφi(· − j) = r X i=1 X j∈Zd hf, φi(· − j)iψi(· − j) ∀ f ∈ Vp(Φ). (3.3)
Remark 1 From (v) of Theorem 1 it follows that Vp(Ψ) = Vp(Φ). This together with
the implication (v) =⇒ (ii) in Theorem 1 leads to the conclusion that {ψi(· − j) : 1 ≤
i ≤ r, j ∈ Zd} is also a p-frame for V
p(Ψ) = Vp(Φ). Thus, {ψi(·−j) : 1 ≤ i ≤ r, j ∈ Zd}
can be thought of as a “dual p-frame” of the family {φi(· − j) : 1 ≤ i ≤ r, j ∈ Zd}.
Hence a p-frame for Vp(Φ) is a Banach frame (for definitions of Banach frames see
[9, 16, 19]).
Remark 2 The functions ψi, 1 ≤ i ≤ r, in (v) of Theorem 1 belong to V1(Φ) ⊂ Vp(Φ)
and are independent of p, 1 ≤ p ≤ ∞. In fact, from the proof of (v) in Theorem 1, we have ψi = r X i′ =1 X j∈Zd cii′(j)φi′(· − j)
for some ℓ1 sequences {c
Note that the condition (iii) in Theorem 1 is independent of p. Therefore, the p-frame property ((ii) and (iv) in Theorem 1) and closedness property ((i) in Theorem 1) of Vp(Φ) is independent of p.
Corollary 1 Let Φ = (φ1, ..., φr)T ∈ W, and 1 ≤ p0 ≤ ∞.
(i) If {φi(· − j) : j ∈ Zd, 1 ≤ i ≤ r} is a p0-frame for Vp0(Φ), then {φi(· − j) : j ∈
Zd, 1 ≤ i ≤ r} is a p-frame for V
p(Φ) for any 1 ≤ p ≤ ∞.
(ii) If Vp0(Φ) is closed in L
p0, then V
p(Φ) is closed in Lp for any 1 ≤ p ≤ ∞.
(iii) If (3.2) holds for p0, then (3.2) holds for any 1 ≤ p ≤ ∞.
Note that the p-Riesz basis for the shift invariant spaces Vp(Φ) can be characterized
in the following way (see for instance [24]).
Proposition 1 Let 1 ≤ p < ∞ and Φ be as in Theorem 1. Then {φi(· − j) : j ∈
Zd, 1 ≤ i ≤ r} is a p-Riesz basis for V
p(Φ) if and only if there exists a positive constant
C such that
C−1Ir ≤ [Φ,b Φ](ξ) ≤ CIb r ∀ ξ ∈ [−π, π]d, (3.4)
where Ir denotes the r × r identity matrix.
Then as a consequence of Theorem 1 and Proposition 1, we obtain the expected result below.
Corollary 2 Let Φ = (φ1, ..., φr)T ∈ L∞ if 1 < p < ∞, and Φ ∈ W if p = 1. If
{φi(· − j) : j ∈ Zd, 1 ≤ i ≤ r} is a p-Riesz basis for Vp(Φ), then {φi(· − j) : j ∈
Zd, 1 ≤ i ≤ r} is a p-frame for V p(Φ).
As further consequences of Theorem 1 and Proposition 1, a p-frame for Vp(Φ) is a
p-Riesz basis for Vp(Φ) if we impose more restrictions on Φ. Actually the additional
condition on Φ is that the matrix [Φ,b Φ](ξ) is invertible for some ξ. Obviously, thisb condition always holds if r = 1 (except for the trivial case Φ = 0).
Corollary 3 Let Φ = (φ1, ..., φr)T ∈ L∞ if 1 < p < ∞, and Φ ∈ W if p = 1. Assume
that {φi(· − j) : j ∈ Zd, 1 ≤ i ≤ r} is a p-frame for Vp(Φ).
(i) If there exists a ξ0 ∈ [−π, π]d such that the r × r matrix [Φ,b Φ](ξb 0) is invertible,
(ii) If r = 1 and φ1 6= 0, then {φ1(· − j) : j ∈ Zd} is a p-Riesz basis for Vp(φ1).
Part (ii) of Corollary 3 states that if φ1 is regular and r = 1, then it is not possible
to construct frames that are not Riesz bases. For example if |φ1| ≤ M < ∞ and
has compact support, or if φ1 is continuous and decays faster than |x|−2 at infinity,
then any frame of Vp(φ1) is also a Riesz basis of Vp(φ1). Thus, for r = 1, frames of
Vp(φ1) that are not Riesz bases must be generated by φ1 that are not regular, e.g., if φ1
has compact support, then φ1 must have an infinite singularity, or if φ1 is continuous
with global support, then φ must have slow decay. For example for p = 2 and r = 1, Benedetto and Li construct frames that are not Riesz bases [6]. However, in their construction the generator φ1 has slow decay at infinity.
In view of Corollary 2, we may consider the converse problem: Given a p-frame {φi(· − j) : 1 ≤ i ≤ r, j ∈ Zd} for Vp(Φ), can we find ˜φi, 1 ≤ i ≤ s, such that the closed
span of { ˜φi(· − j) : 1 ≤ i ≤ s, j ∈ Zd} using ℓp coefficient sequences is Vp(Φ) and such
that { ˜φi(· − j) : 1 ≤ i ≤ s, j ∈ Zd} is a p-Riesz basis for Vp(Φ).
Conjecture Let Φ = (φ1, ..., φr)T ∈ L∞ if 1 < p < ∞, and Φ ∈ W if p = 1.
Assume that {φi(· − j) : 1 ≤ i ≤ r, j ∈ Zd} is a p-frame for Vp(Φ). Then there exist
s ≥ 1, and ˜Φ = ( ˜φ1, . . . , ˜φs)T ∈ L∞ if 1 < p < ∞, and ˜Φ ∈ W if p = 1, such that
{ ˜φi(· − j) : 1 ≤ i ≤ s, j ∈ Zd} is a p-Riesz basis for Vp( ˜Φ) and Vp( ˜Φ) = Vp(Φ).
The assertion in the above conjecture is true under the additional assumption that φi, 1 ≤ i ≤ r, are compactly supported bounded functions on R (see [22] for 1 < p < ∞,
and [3] for 1 ≤ p < ∞ with a completely different proof). We feel strongly that the assertion of the above conjecture for higher dimensions would not be true.
Remark 3 We remark that the condition (iii) in Theorem 1 is different from the following condition about quasi-stability in [7]:
(∗) There exists a positive constant C such that
C−1Ir ≤ [Φ,b Φ](ξ) ≤ CIb r ∀ ξ ∈ Ω, where Ω = {ξ ∈ [−π, π]d: X j∈Zd ˆ φi(ξ +2πj) ˆφi(ξ + 2πj) 6= 0, for some 1 ≤ i ≤ r}.
Hence the p-frame property in this paper is different from the quasi-stability in [7]. For instance, let φ1, φ2 be two compactly supported L2 functions with their shifts being
orthonormal, and let φ3 = φ1(· − 1) − φ2. Define Φ = (φ1, φ2, φ3)T. Then
[Φ,b Φ](ξ) =b 1 0 e−iξ 0 1 −1 eiξ −1 2 , ξ ∈ Rd.
By direct computation, for each ξ ∈ Rd, the eigenvalues of [Φ,b Φ](ξ) are 0, 1 and 3.b
Therefore Condition (iii) of Theorem 1 holds for Φ, but Condition (∗) is not true for Φ since [Φ,b Φ](ξ) is not invertible for any ξ ∈ [−π, π]b d.
Remark 4 We note that from the proof of Theorem 1 together with some modifica-tions, we only need to assume that Φ ∈ Lp, 1 ≤ p ≤ ∞, for the equivalence between
(i) and (iv) in Theorem 1; Φ ∈ L2 ∩ Lp if 1 ≤ p < ∞ and Φ ∈ W if p = ∞ for the
equivalence between (i) and (iii); and Φ ∈ L∞ if 1 < p ≤ ∞ and Φ ∈ W if p = 1 for
the equivalence between (ii) and (iii).
4
Technical Lemmas
The first lemma will give some equivalent relations of the condition (iii) of Theorem 1. Corresponding results involving Riesz bases can be found in [1, 7, 17, 18, 24, 30].
Lemma 1 Let Φ = (φ1, . . . , φr)T ∈ L2. Then the following statements are equivalent
to each other.
(i) rank ³Φ(ξ + 2jπ)b ´
j∈Zd is a constant function on R
d.
(ii) rank [Φ,b Φ](ξ) is a constant function on Rb d.
(iii) There exists a positive constant C independent of ξ such that
C−1[Φ,b Φ](ξ) ≤ [b Φ,b Φ](ξ)[b Φ,b Φ](ξ)b T ≤ C[Φ,b Φ](ξ) ∀ ξ ∈ [−π, π]b d.
Proof of Lemma 1 Obviously the equivalence of (i) and (ii) follows from (2.10), where G = (Φ(ξ + 2jπ))b j∈Zd. Now we start to prove the equivalence of (ii) and (iii).
. . . ≥ λr(ξ). Note that [Φ,b Φ](ξ) is a positive semi-definite matrix. Thus λb i(ξ) ≥ 0 for
all i = 1, . . . , r and there exists an r × r matrix A(ξ) such that A(ξ)TA(ξ) = I r, the
r × r identity matrix, and
A(ξ)T[Φ,b Φ](ξ)A(ξ) = diag(λb
1(ξ), ..., λr(ξ)). (4.1)
Recall that [Φ,b Φ](ξ) is in the Wiener class, since Φ ∈ Lb 2. Then λ
i(ξ) are continuous
and 2π-periodic for all 1 ≤ i ≤ r. Denote the rank of [Φ,b Φ](ξ) by kb 1(ξ).
Assume that (ii) holds. Then k1(ξ) is a constant, which is denoted by k1. Thus
λi(ξ) > 0 for all ξ ∈ Rd and 1 ≤ i ≤ k1, and
λi(ξ) ≡ 0 ∀ ξ ∈ Rd and k1+ 1 ≤ i ≤ r. (4.2)
Hence it follows from the continuity and periodicity of λi(ξ) that there exists a positive
constant C such that
C−1 ≤ λi(ξ) ≤ C ∀ ξ ∈ Rd and 1 ≤ i ≤ k1. (4.3)
Combining (4.1) – (4.3), we get (iii).
Now assume that (iii) holds. Then by (4.1),
C−1λi(ξ) ≤ λi(ξ)2 ≤ Cλi(ξ) ∀ 1 ≤ i ≤ r and ξ ∈ Rd.
Thus either λi(ξ) = 0 or C−1 ≤ λi(ξ) ≤ C. Hence rank[Φ,b Φ](ξ) is a constant by theb
continuity and periodicity of λi(ξ). This completes the proof of (ii) ⇐⇒ (iii). 2
The next lemma and the technique introduced in its proof are crucial for our sub-sequent discussion. In Fourier domain, it allows us to replace locally the generator Φb of size r by a local generator Ψb1,λ of size k0.
Lemma 2 Let Φ ∈ L2 satisfy rank³Φ(ξ + 2kπ)b ´
k∈Zd = k0 ≥ 1 for all ξ ∈ R
d. Then
there exist a finite index set Λ, ηλ ∈ [−π, π]d, 0 < δλ < 1/4, nonsingular 2π-periodic r×
r matrix Pλ(ξ) with all entries in the Wiener class and Kλ ⊂ Zdwith cardinality(Kλ) =
k0 for all λ ∈ Λ, having the following properties:
(i)
∪λ∈ΛB(ηλ, δλ/2) ⊃ [−π, π]d, (4.4)
(ii) Pλ(ξ)Φ(ξ) =b à b Ψ1,λ(ξ) b Ψ2,λ(ξ) ! , ξ ∈ Rd and λ ∈ Λ, (4.5)
where Ψ1,λ and Ψ2,λ are functions from Rd to Ck0 and Cr−k0 respectively
satis-fying rank³Ψb1,λ(ξ + 2kπ) ´ k∈Kλ = k0 ∀ ξ ∈ B(ηλ, 2δλ) (4.6) and b Ψ2,λ(ξ) = 0 ∀ ξ ∈ B(ηλ, 8δλ/5) + 2πZd. (4.7)
Further there exist 2π-periodic C∞ functions hλ(ξ), λ ∈ Λ, on Rd such that
X
λ∈Λ
hλ(ξ) = 1 ∀ ξ ∈ Rd (4.8)
and
supp hλ(ξ) ⊂ B(ηλ, δλ/2) + 2πZd. (4.9)
Proof For any η0 ∈ [−π, π]d, there exist a nonsingular r × r matrix Pη0, a k0× k0
nonsingular matrix Aη0 and Kη0 ⊂ Z
d with cardinality(K η0) = k0 such that Pη0 ³ b Φ(η0+ 2kπ) ´ k∈Kη0 = Ã Aη0 0 ! . Write Pη0 ³ b Φ(ξ + 2kπ)´ k∈Kη0 = Ã Aη0 + R1(ξ) R2(ξ) ! .
By the continuity ofΦ, Rb 1(ξ) and R2(ξ) are continuous, and R1(η0) = 0 and R2(η0) =
0. Thus supξ∈B(η0,4δ)kR1(ξ)k + kR2(ξ)k is sufficiently small for any sufficiently small
positive δ. Let H(x) be a nonnegative C∞ function on Rd such that
H(x) =
(
1, if |x| ≤ 4/5,
0, if |x| ≥ 1. (4.10)
Then Aη0 + H((ξ − η0)/4δ)R1(ξ) is a k0× k0 nonsingular matrix for all ξ ∈ R
d when
δ is chosen sufficiently small. For ξ ∈ Rd, set αη0(ξ) = Aη0 + X j∈Zd H³ξ + 2jπ − η0 4δ ´ R1(ξ + 2jπ)
and βη0(ξ) = X j∈Zd H³ξ + 2jπ − η0 2δ ´ R2(ξ + 2jπ).
Then αη0(ξ) is 2π-periodic and nonsingular when δ is chosen sufficiently small. Note
that the ℓ1-norms of the sequences of the Fourier coefficients of α
η0(ξ) and βη0(ξ) are dominated by C + C X j∈Zd ¯ ¯ ¯F−1 ³ H(·/(4δ))Φb´(j)¯¯¯+ C X j∈Zd ¯ ¯ ¯F−1 ³ H(·/(2δ))Φb´(j)¯¯¯ ≤ CkΦkL1 < ∞,
where C is a positive constant. This shows that all entries of αη0(ξ) and βη0(ξ) belong
to the Wiener class.
Let Pη0(ξ) = Pη0 + Ã 0 0 −βη0(ξ)(αη0(ξ)) −1 0 ! Pη0, ξ ∈ R d.
Then Pη0(ξ) is a 2π-periodic nonsingular r × r matrix for any ξ ∈ R
d, and all entries of
Pη0(ξ) belong to the Wiener class. Note that αη0(ξ) = Aη0+ R1(ξ) and βη0(ξ) = R2(ξ)
for all ξ ∈ B(η0, 8δ/5). Thus
Pη0(ξ) ³ b Φ(ξ + 2kπ)´ k∈Kη0 = Ã Aη0 + R1(ξ) 0 ! ∀ ξ ∈ B(η0, 8δ/5), (4.11)
when 1/4 > δ > 0 is chosen sufficiently small.
Define Ψ1,η0 = (ψ1,η0,1, . . . , ψ1,η0,k0) T and Ψ 2,η0 = (ψ2,η0,1, . . . , ψ2,η0,r−k0) T by à b Ψ1,η0(ξ) b Ψ2,η0(ξ) ! = Pη0(ξ)Φ(ξ),b ξ ∈ R d. (4.12)
Recall that all entries of Pη0(ξ) belong to the Wiener class. Thus by (2.4), Ψ1,η0 ∈ L
1
and Ψ2,η0 ∈ L
p if Φ ∈ Lp. By (4.11) and (4.12), we have
b
Also for ξ ∈ B(η0, 8δ/5), rank (Aη0 + R1(ξ)) = k0 and
rank à Aη0 + R1(ξ) Ψb1,η0(ξ + 2k ′π) 0 Ψb2,η0(ξ + 2k ′π) ! k′∈Zd\K η0 = rank à b Ψ1,η0(ξ + 2kπ) b Ψ2,η0(ξ + 2kπ) ! k∈Zd = rank ³Φ(ξ + 2kπ)b ´ k∈Zd = k0. Thus Ψb2,η0(ξ + 2k ′π) = 0 for all ξ ∈ B(η
0, 8δ/5) and k′ ∈ Zd\Kη0. This together with
(4.13) lead to
b
Ψ2,η0(ξ) = 0 ∀ ξ ∈ B(η0, 8δ/5) + 2πZ
d.
Let δη0 be chosen such that 1 < δη0 < 1/4, Pη0(ξ) and Aη0+H((ξ −η0)/(4δη0))R1(ξ)
as defined above are nonsingular for all ξ ∈ Rd, and A
η0 + R1(ξ) is nonsingular for
ξ ∈ B(η0, 4δη0). For the family {B(η0, δη0/2) : η0 ∈ [−π, π]
d} of open balls covering
[−π, π]d, there exists a finite index set Λ such that ∪
λ∈ΛB(ηλ, δλ/2) ⊃ [−π, π]d by the
compactness of [−π, π]d. For such a finite covering, the corresponding P
λ(ξ), Ψ1,λ and
Ψ2,λ constructed above satisfy the desired properties in Lemma 2. The existence of
the desired 2π-periodic functions hλ(ξ), λ ∈ Λ, follows easily from (4.4). 2
The following result is about Lp and W estimates of φ near a point ξ
0 such that
ˆ
φ(ξ0 + 2kπ) = 0 for all k ∈ Zd. This lemma is crucial to obtaining (5.6) and (5.19),
the key estimates in the proofs of (i) =⇒ (iii) and (ii) =⇒ (iii). We remark that for p = 2, these estimates follow easily from L2 Fourier theory.
Lemma 3 Let φ ∈ Lp if 1 ≤ p < ∞, and φ ∈ W if p = ∞. Assume that P
j∈Zdφ(· −
j) = 0. Then for any function h on Rd satisfying
|h(x)| ≤ C(1+|x|)−d−1 and |h(x)−h(y)| ≤ C|x−y| (1 + min(|x|, |y|))−d−1, (4.14) we have lim n→∞2 −nd°° ° X j∈Zd h(2−nj)φ(· − j)°°° Lp = 0.
Note that any Lipschitz function with compact support and any Schwartz function satisfy (4.14).
Proof Here we only give the proof of the assertion for φ ∈ Lp, 1 ≤ p < ∞, in detail.
By the Lebesgue dominated convergence theorem, for any ǫ > 0 there exists N0 ≥ 2 such that ³ Z x∈[0,1]d ³ X |j|≥N0 |φ(x + j)|´pdx´1/p ≤ ǫ. Set φ1(x) = φ(x)χON0(x) + X |j|≥N0 φ(x + j)χ[0,1]d(x),
where χE denotes the characteristic function of a set E, and ON0 = ∪|j|<N0(j + [0, 1]
d). Thus Pj∈Zdφ1(· + j) =Pj∈Zdφ(· + j) = 0 and kφ1− φkLp ≤ 2 ³ Z [0,1]d ³ X |j|≥N0 |φ(x + j)|´pdx´1/p ≤ 2ǫ.
Therefore, using (2.4), (4.14), and the fact that supp φ1 ⊂ {x : |x| ≤ N0+ d}, we get
k2−nd X j∈Zd h(2−nj)(φ(x − j) − φ1(x − j))kLp ≤ 2−nd X j∈Zd |h(2−nj)| kφ − φ1kLp ≤ Cǫ, and 2−ndp Z [0,1]d ³ X k∈Zd ¯ ¯ ¯ X j∈Zd h(2−nj)φ1(x − j + k) ¯ ¯ ¯´pdx ≤ 2−ndpZ [0,1]d ³ X k∈Zd ¯ ¯ ¯ X j∈Zd (h(2−nj) − h(2−nk))φ 1(x − j + k) ¯ ¯ ¯ ´p dx ≤ 2−n(d+1)pC1(N0) Z [0,1]d ³ X k∈Zd (1 + 2−n|k|)−d−1 X |j−k|≤N0+d |φ1(x − j + k)| ´p dx ≤ 2−n(d+1)pC 2(N0)( X k∈Zd (1 + 2−n|k|)−d−1)pkφ 1kpLp ≤ 2−npC3(N0)(kφkLp+ 2ǫ)p ≤ (kφkLp+ 2ǫ)pǫp, when n > 1 pln ³ C3(N0)/ǫ ´
, where Ci(N0), i = 1, 2, 3, are positive constants depending
only on N0, d and the constant C in (4.14). Hence the assertion follows for 1 ≤ p < ∞
and φ ∈ Lp. 2
For p = ∞ the hypothesis φ ∈ W in Lemma 3 cannot be weakened to φ ∈ L∞, as
Example Let d = 1, φ(x) = χ[0,1](x) −P∞j=1χ2j+[2−j,2−j+1](x) and h(x) = (1 + x2)−1.
Obviously φ is in L∞ but not in W, and P
j∈Zφ(x − j) = 0. Set gn(x) = 2−n X k∈Z h(2−nk)φ(x − k). Thus we get gn(x + k) = 2−n(h(2−nk) − h(2−nk + 2l−n)), ∀ k ∈ Z, x ∈ [2−l, 2−l+1] and sup x∈[0,1] X k∈Z |gn(x + k)| ≥ 2−n X 0≤k≤2n−2 |h(2−nk) − h(2−nk + 1)| ≥ 1 10. This shows that kgnkL∞, n ≥ 1, does not tend to zero as n tends to infinity.
5
Proofs
In this section, we shall give the proof of Theorem 1. We divide the proof into the following steps: (v) =⇒ (iv) =⇒ (i) =⇒ (iii) =⇒ (v) and (iii) =⇒ (v) =⇒ (ii) =⇒ (iii). The proofs of (i) =⇒ (iii) and (ii) =⇒ (iii) are the most difficult and technical parts in our proof.
5.1
Proof of (v) =⇒(iv)
Let f =Pr i=1
P
j∈Zdhf, ψi(· − j)iφi(· − j). Then using (2.2) we have
inf f =Pri=1φi∗′Di r X i=1 kDikℓp ≤ k{ r X i=1 hf, ψi(· − j)i}j∈Zdkℓp ≤ Bkf kLp
which is the right hand side of inequality (3.2) .
To prove the left hand side of inequality (3.2), we simply note that if f = Pr
i=1Di ∗ ′ φi ∈ Vp(Φ), we have, using (2.3), kf kLp = k r X i=1 Di∗′φikLp ≤ C r X i=1 kDikℓp.
Taking the infimum on both sides of the inequality above, we obtain the left hand side of inequality (3.2). 2
5.2
Proof of (iv) =⇒(i)
The implication (iv) =⇒ (i) follows from standard functional analytic arguments. We include it here for the sake of completeness, which is a consequence of the following general result.
Theorem 2 Let (X, k·kX) and (Y, k·kY) be two Banach spaces, and let T be a bounded
linear operator from X to Y . If there is a positive constant C such that
C−1kyk
Y ≤ inf
y=T xkxkX ≤ CkykY ∀ y ∈ Ran(T ),
then the range Ran(T ) of T is closed.
Clearly Theorem 2 in turn follows from the following lemma.
Lemma 4 Let (X, k · kX) be a Banach space, (Y, k · kY) a normed linear space, and T
a bounded linear operator from X to Y . Define
|||y||| = inf
y=T xkxkX ∀ y ∈ Ran(T ).
Then (Ran(T ), ||| · |||) is a Banach space.
Proof It is routine to check that ||| · ||| is a norm on Ran(T ). Let yn, n ≥ 1, be
a Cauchy sequence in Ran(T ). Without loss of generality, we assume that |||yn+1 −
yn||| < 2−n. By the definition of the norm ||| · |||, there exists xn ∈ X, n ≥ 1, such
that T xn = yn+1 − yn and kxnkX ≤ 2−n for all n ≥ 1. Since X is complete and
P∞
n=1kxnkX < ∞, we have z = P∞n=1xn ∈ X and y1 + T z ∈ Ran(T ). Note that
|||T x||| ≤ kxkX for any x ∈ X. Hence
|||yn− y1 − T z||| = ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯T³ ∞ X k=n xk´¯¯¯ ¯ ¯ ¯ ¯ ¯ ¯≤ ∞ X k=n kxkkX → 0 as n → ∞.
This leads to the assertion. 2
5.3
Proof of (i)=⇒(iii)
Let k0 = minξ∈Rdrank
³ b
Φ(ξ+2kπ)´
k∈Zdand let Ωk0 = {ξ : rank
³ b
Φ(ξ+2kπ)´
k∈Zd > k0}.
Then Ωk0 6= R
d. By Lemma 1, it suffices to prove that Ω
that Ωk0 6= ∅. Then it suffices to construct a function G in the L
p closure of V p(Φ)
such that G cannot be written asPri=1φi∗′Di for some Di ∈ ℓp, 1 ≤ i ≤ r. Recall that
Ωk0 is open set by (2.11). Then the boundary ∂Ωk0 of Ωk0 is nonempty. Also for any
ξ0 ∈ ∂Ωk0, rank ³ b Φ(ξ0+ 2kπ) ´ k∈Zd = k0 and max ξ∈B(ξ0,δ) rank³Φ(ξ + 2kπ)b ´ k∈Zd > k0 ∀ δ > 0.
As in the proof of Lemma 2, there exist a r × r matrix Pξ0(ξ), δ0 > 0 and Kξ0 ⊂ Z
d
with cardinality(Kξ0) = k0 such that Pξ0(ξ) is a 2π-periodic nonsingular matrix with
its entries in the Wiener class, and Ψξ0 defined by
b Ψξ0(ξ) = Ã b Ψ1,ξ0(ξ) b Ψ2,ξ0(ξ) ! = Pξ0(ξ)Φ(ξ)b satisfies rank(Ψb1,ξ0(ξ + 2kπ))k∈Kξ0 = k0 ∀ ξ ∈ B(ξ0, 2δ0), (5.1) b Ψ2,ξ0(ξ0 + 2kπ) = 0 ∀ k ∈ Z d, (5.2) b Ψ2,ξ0(ξ + 2kπ) = 0 ∀ k ∈ Kξ0, ξ ∈ B(ξ0, 2δ0), (5.3) and b Ψ2,ξ0(ξ) 6≡ 0 on B(ξ0, δ) + 2πZ d (5.4)
for all 0 < δ < 2δ0. Since Pξ0(ξ) ∈ WC and Φ ∈ L
∞ (respectively Φ ∈ W), we
have Ψξ0 ∈ L
∞ (respectively Ψ
ξ0 ∈ W). This together with (5.2) and the Poisson
summation formula lead to
X
j∈Zd
e−iξ0(·+j)
Ψ2,ξ0(· + j) = 0. (5.5)
Let H be a nonnegative C∞ function satisfying (4.10), and let
Hn,ξ0(ξ) =
X
k∈Zd
H(2n(ξ + 2kπ − ξ0)), n ≥ 2.
Define an operator Tn,ξ0 from (ℓ
p)r−k0 to Lp by Tn,ξ0 : (D1, . . . , Dr−k0) T −→r−k 0 X i=1 (ψ2,ξ0,i∗ ′H n,ξ0) ∗ ′D i,
where Hn,ξ0 denotes the sequence of the Fourier coefficients of the 2π-periodic function
Hn,ξ0(ξ) and Ψ2,ξ0 = (ψ2,ξ0,1, . . . , ψ2,ξ0,r−k0)
T. By (2.3), (5.5) and Lemma 3 with taking
h = F−1H, we obtain
lim
n→∞kTn,ξ0k = 0. (5.6)
Let ˜H(x) = H(2x)−H(8x) and define ˜Hn,ξ0(ξ) =
P
k∈ZdH(2˜ n(ξ +2kπ−ξ0)), n ≥ 2.
Then for n ≥ 4, we have
˜
Hn,ξ0(ξ)Hn,ξ0(ξ) = ˜Hn,ξ0(ξ). (5.7)
Define an operator ˜Tn,ξ0 from (ℓ
p)r−k0 to Lp by ˜ Tn,ξ0 : (D1, . . . , Dr−k0) T −→r−kX0 i=1 (ψ2,ξ0,i∗ ′H˜ n,ξ0) ∗ ′ D i,
where ˜Hn,ξ0 denotes the sequence of Fourier coefficients of the 2π-periodic function
˜
Hn,ξ0(ξ). From (5.6), (5.7), and the fact that k ˜Hn,ξ0(ξ)kℓ1∗ ≤ C for some positive
constant C independent of n, it follows that k ˜Tn,ξ0k ≤ k ˜Hn,ξ0(ξ)kℓ1∗kTn,ξ0k, and
lim
n→∞k ˜Tn,ξ0k = 0. (5.8)
By (5.4) and (5.6), there exists a subsequence nl, l ≥ 1 such that nl+1 ≥ nl+ 8,
k ˜Tnl,ξ0k 6= 0 and 5 X k=0 kTnl+k,ξ0k ≤ 2 −l. (5.9) Let Dnl ∈ (ℓ
p)r−k0 be chosen such that
kDnlkℓp = 1 and k ˜Tnl,ξ0DnlkLp ≥ k ˜Tnl,ξ0k/2, (5.10)
and let Dnl(ξ) be the Fourier series having Dnl as its sequence of Fourier coefficients.
For sufficiently large l1, define Gs, s ≥ l1, by
b Gs(ξ) = s X l=l1 l(Hnl,ξ0(ξ) − Hnl+4,ξ0(ξ))Dnl(ξ) TΨb 2,ξ0(ξ) = s X l=l1 l(Hnl,ξ0(ξ) − Hnl+4,ξ0(ξ))(0, Dnl(ξ) T)P ξ0(ξ)Φ(ξ)b
and G by b G(ξ) = ∞ X l=l1 l(Hnl,ξ0(ξ) − Hnl+4,ξ0(ξ))Dnl(ξ) TΨb 2,ξ0(ξ).
In fact, we only need 2−l1 ≤ δ
0.
Now it remains to prove that G is in the Lp closure of V
p(Φ) and that G cannot be
written as Pri=1φi∗′Di for some Di ∈ ℓp, 1 ≤ i ≤ r. From the construction of Gs and
G, Gs∈ Vp(Φ) for all s ≥ l1, and
kGs− GkLp ≤ ∞ X l=s+1 l°°°F−1 ³³ Hnl,ξ0(ξ) − Hnl+4,ξ0(ξ) ´ Dnl(ξ) TΨb 2,ξ0(ξ) ´°°° Lp ≤ ∞ X l=s+1 l(kTnl+4,ξ0k + kTnl,ξ0k) → 0 as s → ∞,
where we have used (5.9). This shows that G is in the Lp closure of V p(Φ).
Finally we prove that G 6∈ Vp(Φ). On the contrary, suppose that
b
G(ξ) = A(ξ)TΦ(ξ)b (5.11)
for some vector-valued 2π-periodic distribution A(ξ) ∈ WCp. Note that suppGbs(ξ) ⊂
B(ξ0, 2−l1) + 2πZd for all s ≥ l1 and so is suppG(ξ). Hence we may assume that A(ξ)b
in (5.11) is supported in B(ξ0, δ0) + 2πZd when l1 is chosen sufficiently large. Write
A(ξ)T(P ξ0(ξ))
−1 = (A
1(ξ)T, A2(ξ)T). Then we may write (5.11) as
A1(ξ)TΨb1,ξ0(ξ) = ³ − A2(ξ)T + ∞ X l=l1 l³Hnl,ξ0(ξ) − Hnl+4,ξ0(ξ) ´ Dnl(ξ) T´Ψb 2,ξ0(ξ). (5.12)
Since A1(ξ) is a 2π-periodic distribution in WCp and supp A1(ξ) ⊂ B(ξ0, δ0) + 2πZd,
it follows from (5.1), (5.3) and (5.12) that A1(ξ) ≡ 0. Substituting this in (5.12),
A2(ξ)TΨb2,ξ0(ξ) = ∞ X l=l1 l³Hnl,ξ0(ξ) − Hnl+4,ξ0(ξ) ´ Dnl(ξ) TΨb 2,ξ0(ξ). (5.13)
By direct computation, and using the fact that nl+1 ≥ nl+ 8, we have
˜ Hnl,ξ0(ξ) ³ Hnl′,ξ0(ξ) − Hnl′+4,ξ0(ξ) ´ = ( ˜ Hnl,ξ0(ξ) l ′ = l 0 l′ 6= l.
Multiplying ˜Hnl,ξ0(ξ) on both sides of (5.13), ˜ Hnl,ξ0(ξ)A2(ξ) TΨb 2,ξ0(ξ) = l ˜Hnl,ξ0(ξ)Dnl(ξ) TΨb 2,ξ0(ξ). (5.14) By (5.10) and k ˜Tnl,ξ0k 6= 0, we get ° ° °F−1³l ˜Hnl,ξ0(ξ)Dnl(ξ) TΨb 2,ξ0(ξ) ´°°° Lp ≥ lk ˜Tnl,ξ0k/2 and ° ° °F−1³H˜nl,ξ0(ξ)A2(ξ) TΨb 2,ξ0(ξ) ´°°° Lp ≤ k ˜Tnl,ξ0k kA2(ξ)kℓp∗,
which contradicts (5.14). This completes the proof of (i) =⇒ (iii). 2
5.4
Proof of (iii) =⇒ (v)
Let hλ(ξ), Pλ(ξ) and Ψb1,λ be as in Lemma 2. Define
Bλ(ξ) = Hλ(ξ)Pλ(ξ)T Ã [Ψb1,λ,Ψb1,λ](ξ)−1 0 0 I ! Pλ(ξ), (5.15)
where Hλ(ξ) is a 2π-periodic C∞ function such that Hλ(ξ) = 1 on supp hλ, and Hλ is
supported in B(ηλ, δλ) + 2πZd. Then Bλ(ξ) ∈ WC. Define Ψ = (ψ1, . . . , ψr)T by
b
Ψ(ξ) = X
λ∈Λ
hλ(ξ)Bλ(ξ)Φ(ξ).b (5.16)
Then Ψ ∈ L∞ if 1 < p < ∞ and Ψ ∈ W if p = 1, ∞. Now we start to prove (3.3) for
such a Ψ. For any f ∈ Vp(Φ), define
g = r X i=1 X j∈Zd hf, ψi(· − j)iφi(· − j).
Then it suffices to prove that f = g. By the definition of the space Vp(Φ), there exists
a 2π-periodic distribution A(ξ) ∈ WCp such that ˆf (ξ) = A(ξ)TΦ(ξ). Therefore, byb
(5.15), (5.16) and Lemma 2, we get
ˆ
g(ξ) = A(ξ)T[Φ,b Ψ](ξ)b Φ(ξ)b
= X
λ∈Λ
= X λ∈Λ hλ(ξ) n A(ξ)TPλ(ξ)−1× Ã [Ψb1,λ,Ψb1,λ](ξ) 0 0 0 ! Ã [Ψb1,λ,Ψb1,λ](ξ)−1 0 0 I ! Ã b Ψ1,λ(ξ) 0 !) = X λ∈Λ hλ(ξ)A(ξ)TPλ(ξ)−1 Ã b Ψ1,λ(ξ) 0 ! = A(ξ)TΦ(ξ) = ˆb f (ξ).
Similarly we can prove that f =Pri=1Pj∈Zdhf, φi(· − j)iψi(· − j). This completes the
proof of (iii)=⇒(v). 2
5.5
Proof of (v) =⇒(ii)
Let f =Pri=1Pj∈Zdhf, φi(· − j)iψi(· − j). Then using (2.3) we get
kf kLp ≤ Ck{
r
X
i=1
hf, φi(· − j)i}j∈Zdkℓp
which is the left hand side of inequality in (ii).
The right hand side of (ii) is a direct consequence of (2.2). 2
5.6
Proof of (ii) =⇒(iii)
Let k0 = minξ∈Rdrank
³ b
Φ(ξ+2kπ)´
k∈Zdand let Ωk0 = {ξ : rank
³ b
Φ(ξ+2kπ)´
k∈Zd > k0}.
By Lemma 1, it suffices to prove that Ωk0 = ∅. Suppose on the contrary that Ωk0 6= ∅.
Let ξ0 ∈ ∂Ωk0, Ψ1,ξ0, Ψ2,ξ0, Pξ0(ξ), Hn,ξ0(ξ), ˜Hn,ξ0(ξ) and δ0 be as in the proof of (i) =⇒
(iii). Let n0 be chosen such that 2−n0 < δ0 and
αn(ξ) = [Ψb1,ξ0,Ψb1,ξ0](ξ0) + Hn,ξ0(ξ)
³
[Ψb1,ξ0,Ψb1,ξ0](ξ) − [Ψb1,ξ0,Ψb1,ξ0](ξ0)
´
is nonsingular for all n ≥ n0. The existence of n0follows from (5.1) and the continuity of
[Ψb1,ξ0,Ψb1,ξ0](ξ). Given any 2π-periodic distribution F (ξ) in WC
p, define g n, n ≥ n0+ 1, by b gn(ξ) = ˜Hn,ξ0(ξ) ³ − F (ξ)T[Ψb 2,ξ0,Ψb1,ξ0](ξ)(αn(ξ)) −1, F (ξ)T´ Ã b Ψ1,ξ0(ξ) b Ψ2,ξ0(ξ) ! .
Then gn∈ Vp(Φ), and [gbn,Ψb1,ξ0](ξ) = H˜n,ξ0(ξ) ³ − F (ξ)T[Ψb 2,ξ0,Ψb1,ξ0](ξ)(αn(ξ)) −1, F (ξ)T´ Ã [Ψb1,ξ0,Ψb1,ξ0](ξ) [Ψb2,ξ0,Ψb1,ξ0](ξ) ! = 0, (5.17)
by direct computation, where we have used (5.7) and the fact that αn(ξ) = [Ψb1,ξ0,Ψb1,ξ0](ξ)
on the support of Hfn,ξ0. Thus by (2.2), (5.7), (5.17) and
Pξ0(ξ)Φ(ξ) =b à b Ψ1,ξ0(ξ) b Ψ2,ξ0(ξ) ! , we get k[bgn,Φ](ξ)kb ℓp∗ = ° ° ° ° ° " b gn, à b Ψ1,ξ0 b Ψ2,ξ0 !# (ξ)Pξ0(ξ)−T ° ° ° ° ° ℓp∗ ≤ C ° ° ° ° ° " b gn, à b Ψ1,ξ0 b Ψ2,ξ0 !# (ξ) ° ° ° ° ° ℓp∗ ≤ Ck[gbn, Hn,ξ0Ψb2,ξ0](ξ)kℓp∗ ≤ CkgnkLpkF−1(Hn,ξ 0(ξ)Ψb2,ξ0(ξ))k 1/p L∞kF−1(Hn,ξ 0(ξ)Ψb2,ξ0(ξ))k 1−1/p L1 . (5.18)
It follows from (2.3), (2.4), (5.5) and Lemma 3 that
lim n→∞kF −1(H n,ξ0(ξ)Ψb2,ξ0(ξ))k 1/p L∞kF−1(Hn,ξ 0(ξ)Ψb2,ξ0(ξ))k 1−1/p L1 = 0.
This together with (5.18) leads to the existence of ǫn, n ≥ n0, such that
k[ˆgn,Φ](ξ)kb ℓ∗p ≤ ǫnkgnkLp (5.19)
and limn→∞ǫn= 0. On the other hand, by the assumption (ii) we have
k[bgn,Φ](ξ)kb ℓp∗ = ° ° ° n Z Rdgn(x)Φ(x − j)dx o j∈Zd ° ° ° ℓp ≥ CkgnkLp. (5.20)
Combining (5.19) and (5.20), there exists an integer n1 ≥ n0+ 1 such that
Thus for any 2π-periodic distribution F (ξ) in WCp and n ≥ n1,
˜ Hn,ξ0(ξ)F (ξ) T[Ψb 2,ξ0,Ψb1,ξ0](ξ)(αn(ξ)) −1Ψb 1,ξ0(ξ) = ˜Hn,ξ0(ξ)F (ξ) TΨb 2,ξ0(ξ). Hence ˜ Hn,ξ0(ξ)[Ψb2,ξ0,Ψb1,ξ0](ξ)(αn(ξ)) −1Ψb 1,ξ0(ξ) = ˜Hn,ξ0(ξ)Ψb2,ξ0(ξ). (5.21)
This together with (5.1) and (5.3) lead to ˜
Hn,ξ0(ξ)[Ψb2,ξ0,Ψb1,ξ0](ξ)(αn(ξ))
−1 = 0 ∀ ξ ∈ B(ξ
0, 2−n1)
and hence by 2π-periodicity, ˜
Hn,ξ0(ξ)[Ψb2,ξ0,Ψb1,ξ0](ξ)(αn(ξ))
−1Ψb
1,ξ0(ξ) = 0 ∀ ξ ∈ B(ξ0, 2
−n1) + 2πZd.
Substituting this into (5.21), and using the fact that (5.21) is valid for all n ≥ n1 we
obtain
b
Ψ2,ξ0(ξ) ≡ 0 on B(ξ0, 2
−n1−2
) + 2πZd, which contradicts (5.4). This completes the proof of (ii)=⇒(iii). 2
Acknowledgment We would like to thank the anonymous referees and Professor Hans Feichtinger for their helpful comments.
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