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Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue spaces Lp,q(Rd+1)
Accepted Manuscript Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue Spaces Lp,q (Rd+1 )
Rui Li, Bei Liu, Rui Liu, Qingyue Zhang
PII: DOI: Reference:
S0022-247X(17)30390-6 http://dx.doi.org/10.1016/j.jmaa.2017.04.036 YJMAA 21323
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
20 September 2016
Please cite this article in press as: R. Li et al., Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue Spaces Lp,q (Rd+1 ), J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2017.04.036
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Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue Spaces Lp,q Rd+1 ∗ Rui Li a , Bei Liu a , Rui Liu b† and Qingyue Zhang a a b
College of Science, Tianjin University of Technology, Tianjin 300384, China
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
e-mails: [email protected], [email protected], [email protected], [email protected]
April 19, 2017
Abstract. In this paper, we study the nonuniform sampling and reconstruction problem in shift-invariant subspaces of mixed Lebesgue We first show that shift d+1spaces. p,q invariant subspaces in mixed Lebesgue spaces L R can be well-defined. Then we propose that the sampling problem in shift-invariant subspaces of mixed Lebesgue spaces is well-posed. At last, the nonuniform samples {f (xj , yk ) : k, j ∈ J} of a function f belonging to a shift-invariant subspace of mixed Lebesgue spaces are proposed, and we give a fast reconstruction algorithm that allows exact reconstruction of f as long as the sampling set X = {(xj , yk ) : k, j ∈ J} is sufficiently dense. Key words. mixed Lebesgue spaces; nonuniform sampling; shift-invariant subspaces. 2010 MR Subject Classification 94A20, 94A12, 42C15, 41A58
1
Introduction and motivation
Mixed Lebesgue spaces are a generalization of Lebesgue spaces that arise naturally when considering functions that depend on independent quantities with different properties [5, 6, 9, 13]. For example, a function which depends on spacial and time variables may belong to mixed Lebesgue spaces. If a function comes from mixed Lebesgue spaces, we can consider the integrability of each variable separately. This is different from Lebesgue spaces which mainly require the same level of control over all the variables of a function. The flexibility allows these mixed Lebesgue spaces to play an important role in the study of time-based partial differential equations. In this environment, we explore the sampling theorems in mixed Lebesgue spaces. The sampling theorem is one of the most powerful tools in signal analysis, and provides the theoretical basis of modern pulse code modulation communication system. The wellknown classical Shannon sampling theorem [14, 15] states that for any f ∈ L2 (R) with ∗
This work was supported partially by the National Natural Science Foundation of China (NSFC Grant 11201336, 11326094, 11371200, 11401435, 11526150, 11601383 and 11671214). Rui Liu is also supported by Hundred Young Academia Leaders Program of Nankai University. † Corresponding author.
1
suppfˆ ⊆ [−T, T ], f (x) =
n sin π(2T x − n) n = sinc(2T x − n), f f 2T π(2T x − n) 2T
n∈Z
n∈Z
where the series converges in L2 (R) and uniformly on compact sets. Here for any f ∈ L2 (R), ˆ f (ξ) = f (x)e−2πixξ dx, ξ ∈ R. R
This theorem tells us that we can recover the entirety of a function from well-chosen n : n ∈ Z} is uniform. However, in many realistic situasample. The sampling set { 2T tions, the data are known only on a nonuniform sampling set. For example, the loss of data packets during transmission through the internet from satellites can be viewed as a nonuniform sampling problem. The nonuniform sampling problem has been studied in recent years [7, 8, 12, 16, 17, 20, 22, 24]. The problems of uniform and nonuniform sampling and reconstruction also have been extended to more general shift-invariant spaces of the form 2 V (φ) = c(k)φ(x − k) : {c(k) : k ∈ Z} ∈ (Z) . k∈Z
Recently, many researchers studied the sampling theory on general shift-invariant spaces in Lp (Rd ), 1 ≤ p < +∞, the details can be seen in [1–4, 19, 23]. The uniform sampling problem for band-limited functions in mixed Lebesgue spaces has been studied by Rodolfo H. Torres and Erika L. Ward [18,21]. They established PlancherelPolya inequality for mixed Lebesgue spaces. In this paper, we will discuss the nonuniform sampling and reconstruction problem in shift-invariant spaces of mixed Lebesgue d+1 spaces. p,q We first show that shift-invariant spaces in mixed Lebesgue spaces L R can be well-defined. Then we prove that the sampling problem in shift-invariant spaces of mixed Lebesgue spaces is well-posed. In the last section, for given samples {f (xj , yk ) : k, j ∈ J} of a function f belonging to a shift-invariant space of mixed Lebesgue spaces, we provide a fast reconstruction algorithm that allows exact reconstruction of f as long as the sampling set X = {(xj , yk ) : k, j ∈ J} is sufficiently dense.
2
Definitions and preliminary results In this section, we begin with the definition of mixed Lebesgue spaces Lp,q (Rd+1 ).
Definition 2.1 Let 1 ≤ p, q < +∞. Then Lp,q = Lp,q (Rd+1 ) consists of all measurable functions f = f (x, y) defined on R × Rd such that
p p1 q f Lp,q = |f (x, y)|q dy dx < +∞. R
Rd
The corresponding sequence spaces are ⎧ ⎫ ⎛ ⎞p q ⎪ ⎪ ⎨ ⎬ ⎝ |c(k1 , k2 )|q ⎠ < +∞ . p,q = p,q (Zd+1 ) = c : cpp,q = ⎪ ⎪ ⎩ ⎭ k1 ∈Z k2 ∈Zd 2
Obviously, Lp,p (Rd+1 ) = Lp (Rd+1 ), p,p (Zd+1 ) = p (Zd+1 ) = p . In order to control the local behavior of a function, mixed Wiener amalgam spaces are introduced. The following is the definition of mixed Wiener amalgam spaces W (Lp,q )(Rd+1 ). Definition 2.2 A measurable function f belongs to mixed Wiener amalgam spaces W (Lp,q ) = W (Lp,q )(Rd+1 ), 1 ≤ p, q < ∞, if it satisfies f pW (Lp,q ) :=
⎡ sup ⎣
n∈Z x∈[0,1]
⎤p/q sup |f (x + n, y + l)|q ⎦
d l∈Zd y∈[0,1]
< ∞.
Let W0 (Lp,q ) (1 ≤ p, q < ∞) be the space of all continuous functions in W (Lp,q ). For 1 ≤ p < ∞, a function f belongs to Wiener amalgam spaces W (Lp ) = W (Lp )(Rd+1 ), if it satisfies ess supx∈[0,1]d+1 |f (x + k)|p < ∞. f pW (Lp ) := k∈Zd+1
For p = ∞, a measurable function f belongs to W (L∞ ) = W (L∞ )(Rd+1 ) if it satisfies f W (L∞ ) := sup ess supx∈[0,1]d+1 |f (x + k)| < ∞. k∈Zd+1
It is easy to see that W (Lp ) ⊂ W (Lp,p ). Let W0 (Lp ) (1 ≤ p ≤ ∞) be the space of all continuous functions in W (Lp ). For any c ∈ p , d ∈ 1 (1 ≤ p ≤ +∞), define the discrete convolution c(k)d(l − k). (c ∗ d)(l) = k∈Zd+1
For any c ∈ p (1 ≤ p ≤ +∞), f ∈ W (L1 ), define the semi-discrete convolution c(k)f (x − k). (c ∗sd f )(x) = k∈Zd+1
The following is the discrete version of Young’s inequality. Proposition 2.3 [10, Theorem 6.18] If c ∈ p , d ∈ 1 (1 ≤ p ≤ ∞), then c ∗ dp ≤ cp d1 . Given a function φ ∈ W (L1 ), the following lemma shows that φ∗sd maps p to Lp and the semi-discrete version of Young’s inequality is provided. Lemma 2.4 Let φ ∈ W (L1 ) and c ∈ p (1 ≤ p ≤ ∞). Then c ∗sd φLp ≤ cp φW (L1 ) .
3
(2.1)
Proof. For p = ∞, (2.1) holds obviously. In the case 1 ≤ p < ∞, by Fubini’s theorem, p p c(k)φ(x − k) dx c ∗sd φLp = Rd+1 k∈Zd+1 p c(k)φ(x + l − k) dx = [0,1]d+1 l∈Zd+1 k∈Zd+1 p c(k)φ(x + l − k) dx. = [0,1]d+1 l∈Zd+1 k∈Zd+1 Fix x for the moment, let dx be the sequence {dx (k) = φ(x + k) : k ∈ Zd+1 }. Then c(k)φ(x + l − k) = c(k)dx (l − k) = (c ∗ dx )(l). k∈Zd+1
k∈Zd+1
Proposition 2.3 leads to p c(k)φ(x + l − k) = c ∗ dx pp l∈Zd+1 k∈Zd+1 ⎞p ⎛ ≤ cpp dx p1 = cpp ⎝ |φ(x + k)|⎠ k∈Zd+1
≤
cpp φpW (L1 ) .
Therefore, one has c ∗sd φpLp ≤
[0,1]d+1
cpp φpW (L1 ) dx = cpp φpW (L1 ) .
One completes the proof. The following lemma provides a discrete version of Young’s inequality in discrete mixed spaces. Lemma 2.5 If c ∈ p,q , d ∈ 1 (1 ≤ p, q < ∞), then c ∗ dp,q ≤ cp,q d1 . Proof. Fix k1 ∈ Z for the moment, denote ck1 = {ck1 (l) = c(k1 , l) : l ∈ Zd } and dk1 = {dk1 (l) = d(k1 , l) : l ∈ Zd }, then c ∗ dpp,q
q ⎤p/q ⎣ = c(k1 , k2 )d(n1 − k1 , n2 − k2 ) ⎦ n1 ∈Z n2 ∈Zd k1 ∈Z k2 ∈Zd
⎡
4
p = (ck1 ∗ dn1 −k1 )(·) q n1 ∈Z k1 ∈Z ⎛ ⎞p ⎝ ≤ (ck1 ∗ dn1 −k1 )(·)q ⎠ . n1 ∈Z
k1 ∈Z
The discrete version of Young’s inequality (Proposition 2.3) leads to ⎛ ⎞p ⎝ ck1 q dn1 −k1 1 ⎠ . c ∗ dpp,q ≤ n1 ∈Z
k1 ∈Z
Let a = {a(n) = cn q : n ∈ Z} and b = {b(n) = dn 1 : n ∈ Z}. Using Proposition 2.3 again, one has ⎛ ⎞p ⎛ ⎞p ⎝ ck1 q dn1 −k1 1 ⎠ = a ∗ bpp ≤ app bp1 = ck1 pq ⎝ dk1 1 ⎠ . n1 ∈Z
k1 ∈Z
k1 ∈Z
Therefore, c ∗ dpp,q
≤
k1 ∈Z
= =
⎛ ck1 pq ⎝ ⎛ ⎝
k1 ∈Z
⎞p
dk1 1 ⎠
k1 ∈Z
⎞p/q ⎛
|c(k1 , k2 )|q ⎠
k1 ∈Z k2 ∈Zd cpp,q dp1 .
⎝
⎞p |d(k1 , k2 )|⎠
k1 ∈Z k2 ∈Zd
The following is H¨ older’s inequality for mixed Lebesgue spaces. Proposition 2.6 [21, Theorem 1.1.3] Let 1 ≤ p, q ≤ ∞ , with Then f gL1,1 ≤ f Lp,q f Lp ,q .
3
1 p
+ p1 = 1 and
1 q
+ q1 = 1.
Shift-invariant subspaces in Lp,q
For φ ∈ W (L1,1 ), shift-invariant subspaces in mixed Legesgue spaces Lp,q are defined as follows ⎫ ⎧ ⎬ ⎨ Vp,q (φ) = c(k1 , k2 )φ(· − k1 , · − k2 ) : c(k1 , k2 ) : k1 ∈ Z, k2 ∈ Zd ∈ p,q . ⎭ ⎩ d k1 ∈Z k2 ∈Z
It is obvious that the double sum pointwisely converges almost everywhere. In fact, c(k1 , k2 ) ∈ p,q tells that c(k1 , k2 ) ∈ ∞ . This with φ ∈ W (L1,1 ) leads to |c(k1 , k2 )φ(x − k1 , y − k2 )| ≤ c∞ |φ(x − k1 , y − k2 )| k1 ∈Z k2 ∈Zd
k1 ∈Z k2 ∈Zd
≤ c∞ φW (L1,1 ) (a.e.). 5
The following theorem shows that shift-invariant subspaces are well-defined in Lp,q . 1,1 Theorem 3.1 for any c ∈ p,q , the function ! Let 1 ≤ p, q < ∞ and φ ∈ W (L ). Then ! p,q f = k1 ∈Z k2 ∈Zd c(k1 , k2 )φ(· − k1 , · − k2 ) belongs to L . Furthermore,
f Lp,q ≤ cp,q φW (L1,1 ) . Proof. For fixed x ∈ R and k1 ∈ Z, let ck1 (k2 ) = {c(k1 , k2 ) : k2 ∈ Zd }, φx−k1 (y) = φ(x − k1 , y). Then c(k1 , k2 )φ(x − k1 , y − k2 ) = ck1 (k2 )φx−k1 (y − k2 ) = (ck1 ∗sd φx−k1 )(y). k2 ∈Zd
k2 ∈Zd
This and Lemma 2.4 lead to ⎡ ⎣ f pLp,q =
q ⎤ p q ⎦ c(k1 , k2 )φ(x − k1 , y − k2 ) dy dx R Rd k ∈Z k2 ∈Zd 1 p dx (c ∗ φ )(·) = k1 sd x−k1 R k ∈Z q 1 L ⎡ ⎤p ⎣ ≤ (ck1 ∗sd φx−k1 )(·)Lq ⎦ dx R
≤
⎡ ⎣
R
⎡ ⎣
= R
k1 ∈Z
⎤p ck1 q φx−k1 W (L1 ) ⎦ dx
k1 ∈Z
⎤p c(k1 , ·)q φ(x − k1 , ·)W (L1 ) ⎦ dx.
(3.1)
k1 ∈Z
Let d = {d(k1 ) = c(k1 , ·)q : k1 ∈ Z} and h(x) = φ(x, ·)W (L1 ) . Then by (3.1) and Lemma 2.4 again, one has p q c(k , ·) φ(x − k , ·) 1 ) dx 1 1 W (L R k ∈Z 1 p p d(k )h(x − k ) = 1 1 dx = d ∗sd hLp R k ∈Z 1 ≤ dpp hpW (L1 ) = cpp,q φpW (L1,1 ) . Hence, f Lp,q ≤ cp,q φW (L1,1 ) . In order to obtain the main theorem, we need the following results. Proposition 3.2 [11] Let φ ∈ W L1,1 . Then φ satisfies 2 " φ (ξ + 2πk) > 0, ξ ∈ Rd+1 , k∈Zd+1
6
if and only if there exists a function d(k1 , k2 )φ(· − k1 , · − k2 ) g= k1 ∈Z k2 ∈Zd
such that φ (· − α) , g = δ0,α , where d = {d(k1 , k2 ) : k1 ∈ Z, k2 ∈ Zd } ∈ 1 . Lemma 3.3 The function g in Proposition 3.2 belongs to W (L1,1 ). Proof. It is easy to see gW (L1,1 ) = sup sup |g (x + n, y + l)| n∈Z x∈[0,1] l∈Zd y∈[0,1]
d
sup sup d(k1 , k2 )φ(x + n − k1 , y + l − k2 ) = d n∈Z x∈[0,1] l∈Zd y∈[0,1] k1 ∈Z k2 ∈Zd ≤ sup sup |d(k1 , k2 )||φ(x + n − k1 , y + l − k2 )|
d
n∈Z x∈[0,1] l∈Zd y∈[0,1] k1 ∈Z k2 ∈Zd
≤
k1 ∈Z k2
|d(k1 , k2 )|
∈Zd
sup
n∈Z x∈[0,1]
l∈Zd
sup |φ(x + n − k1 , y + l − k2 )|
y∈[0,1]d
= d1 φW (L1,1 ) < ∞. Now, we show the main result in this section which gives the norm equivalences. As usual, for quantities X and Y , X ≈ Y and X Y mean that there exist constants c1 , c2 , c such that c1 X ≤ Y ≤ c2 X and X ≤ cY , respectively. Theorem 3.4 Let 1 < p, q < ∞. Assume that φ ∈ W (L1 ) satisfies 2 " φ (ξ + 2πk) > 0,
ξ ∈ Rd+1 .
k∈Zd+1
! ! Then for any c ∈ p,q , the function f = k1 ∈Z k2 ∈Zd c(k1 , k2 )φ(· − k1 , · − k2 ) belongs to Lp,q , and cp,q ≈ f Lp,q ≈ f W (Lp,q ) . Proof. We first consider cp,q ≈ f Lp,q . Theorem 3.1 leads to f Lp,q cp,q , thus one only needs to prove cp,q f Lp,q . By Proposition 3.2, there exists a function g such that φ (· − α) , g = δ0,α . Therefore,
c(k1 , k2 ) =
R
Rd
f (x, y)g (x − k1 , y − k2 )dxdy. 7
# $ Let b = b(k1 , k2 ) : k1 ∈ Z, k2 ∈ Zd ∈ p ,q with
1 p
+
1 p
= 1 and
1 q
+
1 q
= 1. Then
c(k1 , k2 )b(k1 , k2 ) | c, b | = k1 ∈Z,k2 ∈Zd = b(k1 , k2 ) f (x, y)g (x − k1 , y − k2 )dxdy R Rd k1 ∈Z,k2 ∈Zd = f (x, y) b(k1 , k2 )g (x − k1 , y − k2 )dxdy . R Rd k1 ∈Z,k2 ∈Zd Using Proposition 2.6, Lemma 3.3 and Theorem 3.1, one has | c, b | ≤ f Lp,q b(k1 , k2 )g (x − k1 , y − k2 ) k1 ∈Z,k2 ∈Zd ≤ f Lp,q bp ,q gW (L1,1 ) .
,q
Lp
Thus cp,q ≤ f Lp,q gW (L1,1 ) ,
(3.2)
that is, cp,q f Lp,q . Therefore, one obtains cp,q ≈ f Lp,q . Now, we prove f Lp,q ≈ f W (Lp,q ) . It is easy to see that f pLp,q
p
= R
=
Rd
1
⎛ ⎝
n∈Z 0
≤
q
|f (x, y)| dy q
⎞p/q
l∈Zd
⎡
sup ⎣
n∈Z x∈[0,1]
dx
[0,1]d
|f (x + n, y + l)|q dy ⎠
dx
⎤p/q sup |f (x + n, y + l)|q ⎦
l∈Zd y∈[0,1]
d
= f pW (Lp,q ) .
(3.3)
On the other hand, p p c(k1 , k2 )φ(· − k1 , · − k2 ) f W (Lp,q ) = k1 ∈Z k2 ∈Zd W (Lp,q ) q ⎤p/q ⎡ ⎣ = sup sup c(k1 , k2 )φ(x + n − k1 , y + l − k2 ) ⎦ d n∈Z x∈[0,1] l∈Zd y∈[0,1] k1 ∈Z k2 ∈Zd
8
≤
⎡
sup ⎣
n∈Z x∈[0,1]
⎡
⎛ sup ⎝
d l∈Zd y∈[0,1]
⎛
n∈Z
k1 ∈Z k2
|c(k1 , k2 )||φ(x + n − k1 , y + l − k2 )|⎠ ⎦
k1 ∈Z k2 ∈Zd
⎣ ⎝ ≤ |c(k1 , k2 )| l∈Zd
⎞q ⎤p/q
∈Zd
⎞q ⎤p/q sup
(x,y)∈[0,1]×[0,1]d
|φ(x + n − k1 , y + l − k2 )|⎠ ⎦
.
Denote d(n, l) = sup(x,y)∈[0,1]×[0,1]d |φ(x + n, y + l)|, then by Lemma 2.5, one obtains ⎞q ⎤p/q ⎡ ⎛ ⎣ ⎝ |c(k1 , k2 )|d(n − k1 , l − k2 )⎠ ⎦ = |c| ∗ dpp,q f pW (Lp,q ) = l∈Zd k1 ∈Z k2 ∈Zd p |c|p,q dp1 = cpp,q φpW (L1 ) n∈Z
≤
≤ f pLp,q gpW (L1,1 ) φpW (L1 ) ,
(3.4)
where the last inequality follows by (3.2) and |c| = {|c(k1 , k2 )| : k1 ∈ Z, k2 ∈ Zd }. Combining (3.3) and (3.4), one has f Lp,q ≈ f W (Lp,q ) . Therefore, one completes the proof.
4
Nonuniform sampling in shift-invariant subspaces In this section, we mainly discuss nonuniform sampling in shift-invariant subspaces.
Definition 4.1 A set X = {(xk , yj ) : k, j ∈ J} ⊂ Rd+1 is said to be strongly separated if inf (k,j)=(k ,j ) |(xk , yj ) − (xk , yj )| = δ1 > 0 and inf k=k |xk − xk | = δ2 > 0, where % |(xk , yj ) − (xk , yj )| = (xk − xk )2 + |yj − yj |2 , and J is a countable index set. The following theorem provides a sufficient condition under which the problem of nonuniform sampling in shift-invariant subspaces is well-defined. It tells us the stability of nonuniform sampling in shift-invariant subspaces. Theorem 4.2 If X = {(xk , yj ) : k, j ∈ J, xk ∈ R, yj ∈ Rd } is strongly separated and f ∈ W0 (Lp,q ). Then ⎡ ⎛ ⎞p/q ⎤1/p d/q 1/p 1 1 ⎥ ⎢ ⎝ q⎠ |f (xk , yj )| +1 +1 f W (Lp,q ) . ⎦ ≤ ⎣ δ1 δ2 k∈J
j∈J
Proof. For fixed xk , |f (xk , yj )|q =
|f (xk , yj )|q
l∈Zd yj ∈[l,l+1]d
j∈J
≤ ≤
sup
l∈Zd yj ∈[l,l+1]d
y∈[l,l+1]d
+1
sup
1
l∈Zd
=
δ1
1 +1 δ1 9
|f (xk , y)|q
d
d
y∈[l,l+1]d
sup
l∈Zd y∈[l,l+1]
d
|f (xk , y)|q |f (xk , y)|q .
Therefore, ⎛ ⎞p/q ⎡ ⎤p/q pd/q 1 ⎝ ⎣ |f (xk , yj )|q ⎠ ≤ +1 sup |f (xk , y)|q ⎦ δ1 d d y∈[l,l+1] k∈J
j∈J
k∈J
= ≤ = =
1 +1 δ1 1 +1 δ1 1 +1 δ1 1 +1 δ1
l∈Z
pd/q
⎡
⎣
n∈Z xk ∈[n,n+1]
pd/q n∈Z
pd/q pd/q
1 +1 δ2
l∈Zd
sup y∈[l,l+1]d
⎡
|f (xk , y)|q ⎦
⎣
sup x∈[n,n+1]
⎤p/q
⎤p/q sup
d l∈Zd y∈[l,l+1]
|f (x, y)|q ⎦
⎡ ⎤p/q 1 +1 sup ⎣ sup |f (x + n, y + l)|q ⎦ δ2 d x∈[0,1] d y∈[0,1]
n∈Z
l∈Z
1 + 1 f pW (Lp,q ) . δ2
Then one has ⎡ ⎛ ⎞p/q ⎤1/p d/q 1/p 1 1 ⎥ ⎢ ⎝ q⎠ |f (xk , yj )| +1 +1 f W (Lp,q ) . ⎦ ≤ ⎣ δ1 δ2 k∈J
j∈J
Let f be a continuous function. We define the oscillation (or modulus of continuity) of f by oscδ (f )(x) = sup|y|≤δ |f (x + y) − f (x)|. Lemma 4.3 Let φ ∈ W (L1 ), then oscδ (φ) ∈ W (L1 ). Proof. Without loss of generality, assume that δ ≤ 1. Then for j ∈ Zd+1 , sup x∈[0,1]d+1
|oscδ (φ)(x + j)| = ≤
sup |φ(x + j + y) − φ(x + j)|
sup
x∈[0,1]d+1 |y|≤δ
sup |φ(x + y + j)| +
sup
x∈[0,1]d+1 |y|≤δ
≤ 2
sup x∈[−1,2]d+1
sup
k∈[−1,2]d+1 ∩Zd+1
Summing over j ∈ Zd+1 , one obtains sup |oscδ (φ)(x + j)| ≤ 2 j∈Zd+1
x∈[0,1]d+1
x∈[0,1]d+1
|φ(x + j + k)|.
j∈Zd+1
sup
k∈[−1,2]d+1 ∩Zd+1
k∈[−1,2]d+1 ∩Zd+1
j∈Zd+1
= 2
≤ 22d+3 φW (L1 ) < +∞. 10
|φ(x + j)|
|φ(x + j)|
≤ 2
sup x∈[0,1]d+1
x∈[0,1]d+1
sup x∈[0,1]d+1
|φ(x + j + k)| |φ(x + j + k)|
Therefore, oscδ (φ) ∈ W (L1 ).
Lemma 4.4 Let φ ∈ W0 (L1 ). Then limδ→0 oscδ (φ)W (L1 ) = 0. Proof. By Lemma 4.3, one has oscδ (φ) ∈ W (L1 ). Given > 0, there exists an integer L0 > 0 such that (4.1) sup |oscδ (φ)(x + j)| < . 2 x∈[0,1]d+1 |j|≥L0
Moreover, since φ is continuous, there exists δ0 > 0 such that sup x∈[0,1]d+1
sup |φ(x + j + y) − φ(x + j)| ≤
|y|≤δ
for all |j| < L0 and all δ < δ0 . Thus sup |oscδ (φ)(x + j)| = |j|
d+1
sup
|j|
≤
|j|
≤
sup |φ(x + j + y) − φ(x + j)|
|y|≤δ
2(2L0 )d+1
. 2
Combining (4.1) and (4.2), one obtains oscδ (φ)W (L1 ) ≤ sup j∈Zd+1
d+1
2(2L0 )d+1
x∈[0,1]d+1
(4.2)
|oscδ (φ)(x + j)| < ,
∀ 0 < δ ≤ δ0
Thus limδ→0 oscδ (φ)W (L1 ) = 0.
Lemma 4.5 Let φ ∈ W0 (L1 ). If f ∈ Vp,q (φ), then the oscillation (or modulus of continuity) oscδ (f ) belongs to Lp,q . Moreover for all > 0, there exists δ0 > 0 such that oscδ (f )Lp,q ≤ f Lp,q uniformly for all f ∈ Vp,q (φ) and 0 < δ < δ0 . ! Proof. If f = k∈Zd+1 c(k)φ(· − k) ∈ V p,q (φ), then oscδ (f )(x) = ≤ ≤
sup |f (x + y) − f (x)|
|y|≤δ
sup |y|≤δ
k∈Zd+1
=
|c(k)||φ(x + y − k) − φ(x − k)|
k∈Zd+1
|c(k)| sup |φ(x − k + y) − φ(x − k)| |y|≤δ
|c(k)|oscδ (φ)(x − k)
k∈Zd+1
= [|c| ∗sd (oscδ )(φ)](x),
11
where |c| = {|c(k)| : k ∈ Zd+1 }. By Theorem 3.1, Theorem 3.4 and Lemma 4.3, there exists M > 0 such that oscδ (f )Lp,q
≤ |c| ∗sd (oscδ )(φ)Lp,q ≤ |c|p,q oscδ (φ)W (L1,1 ) = cp,q oscδ (φ)W (L1,1 ) ≤ M f Lp,q oscδ (φ)W (L1,1 ) ≤ M f Lp,q oscδ (φ)W (L1 ) .
(4.3)
Using Lemma 4.4, for any > 0, there exists δ0 > 0 such that , ∀ 0 < δ ≤ δ0 . oscδ (φ)W (L1 ) < M This with (4.3) leads to oscδ (f )Lp,q ≤ f Lp,q .
Definition 4.6 A set X = {(xj , yk ) : k, j ∈ J, xk ∈ R, yj ∈ Rd } is γ0 -dense in Rd+1 if Rd+1 = ∪j,k Bγ (xj , yk )
for every γ > γ0 ,
where Bγ (xj , yk ) is the open ball with center (xj , yk ) and radius γ, and J is a countable index set. A bounded uniform partition of unity {βj,k }j,k∈J associated to a strongly separated sampling set X = {(xj , yk ) : j, k ∈ J} is a set of functions such that (i). 0 ≤ βj,k ≤ 1, ∀ j, k ∈ J, (ii). suppβj,k ⊂ Bγ (xj , yk ), ! ! (iii). j∈J k∈J βj,k = 1. If f ∈ W0 (Lp,q ), we write QX f =
f (xj , yk )βj,k
j∈J k∈J
for the quasi-interpolant of the sequence cj,k = f (xj , yk ). In order to obtain the reconstruction algorithm, we need the following lemma. Lemma 4.7 Let P be any bounded projection from Lp,q onto Vp,q (φ) with φ ∈ W0 (L1 ). Then there exists γ0 = γ0 (p, q, P ) such that the operator I − P QX is a contraction on Vp,q (φ) for every strongly-separated γ-dense set X with γ ≤ γ0 . Proof. For f ∈ Vp,q (φ), one has f − P QX f Lp,q
= P f − P QX f Lp,q ≤ P op f − QX f Lp,q ≤ P op oscγ f Lp,q ≤ P op f Lp,q .
Here, the last inequality follows by Lemma 4.5. One can choose γ0 such that for any γ < γ0 , P op < 1. Hence, it leads to a contraction. Now we give a fast iterative algorithm to reconstruct f ∈ Vp,q (φ) from its samples {f (xj , yk ) : j, k ∈ J}. 12
Theorem 4.8 Let φ ∈ W0 (L1 ). Assume that P is a bounded projection from Lp,q onto Vp,q (φ). Then there exists a density γ > 0 (γ = γ(p, q, P )) such that any f ∈ Vp,q (φ) can be recovered from its samples {f (xj , yk ) : (xj , yk ) ∈ X} on any γ-dense set X = {(xj , yk ) : j, k ∈ J} by the following iterative algorithm: ( f 1 = P QX f (4.4) fn+1 = P QX (f − fn ) + fn . The iterates fn converges to f uniformly and in Lp,q norms. Furthermore, the convergence is geometric, that is, f − fn Lp,q ≤ M αn for some α = α(γ) < 1 and M < ∞. Proof. Let en = f − fn be the error after n iterations. By (4.4), en+1 = f − fn+1 = f − fn − P QX (f − fn ) = (I − P QX )en . By Lemma 4.7, one may choose γ so small that I − P QX op = α < 1. Then one obtains en+1 Lp,q ≤ αen Lp,q ≤ αn e0 Lp,q . Thus en Lp,q → 0, when n → ∞. The proof is completed.
References [1] Aldroubi, A., Feichtinger, H.: Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The Lp theory. Proc. Amer. Math. Soc. 126(9), 2677–2686 (1998) [2] Aldroubi, A., Gr¨ochenig, K.: Beurling-Landau-type theorems for nonuniform sampling in shift-invariant spaces. J. Fourier Anal. Appl. 6(1), 91–101 (2000) [3] Aldroubi, A., Gr¨ochenig, K.: Nonuniform sampling and reconstruction in shiftinvariant spaces. SIAM Rev. 43(4), 585–562 (2001) [4] Aldroubi, A., Unser, M.: Sampling procedure in function spaces and asymptotic equivalence with Shannon’s sampling theory. Numer. Funct. Anal. Optim. 15(1), 1–21 (1994) [5] Benedek, A., Panzone, R.: The space Lp with mixed norm. Duke Math. J. 28, 301–324 (1961) [6] Benedek, A., Calder´on, A.P., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Nat. Acad. Sci. USA 48, 356–365 (1962) [7] Benedetto, J.J., Ferreira, P.J.S.G.: Modern sampling theory, Mathematics and applications. Birkh¨ auser, Boston (2001) 13
[8] Heil, C.: Harmonic Analysis and Applications. Birkh¨auser, Boston (2006) [9] Fernandez, D.L.: Vector-valued singular integral operators on Lp -spaces with mixed norms and applications. Pac. J. Math. 129(2), 257–275 (1987) [10] Folland, G. B.: Real analysis. John Wiley & Sons, New York (1984) [11] Jia, R.Q., Micchelli, C.A.: Using the refinement equation for the construction of prewavelets II: Powers of two. P.J. Laurent, A. Le M´ehaut´e, L.L. Schumaker (Eds.), Curves and Surfaces, Academic Press, New York, 209–246 (1991) [12] Marvasti, F.: Nonuniform sampling, theory and practice. Kluwer Academic/Plenum Publishers, New York (2001) [13] Francia, J.L., Ruiz, F.J., Torrea, J.L.: Calder´on–Zygmund theory for operatorvalued kernels. Adv. Math. 62(1), 7–48 (1986) [14] Shannon, C.: Communication in the presence of noise. Proc IRE 37(1), 10–21 (1949) [15] Shannon, C.: Communication in the presence of noise. Proc IEEE, 447–457 (1998) [16] Sun, Q.: Nonuniform average sampling and reconstruction of signals with finite rate of innovation. SIAM J. Math. Anal. 38(5), 1389–1422 (2006) [17] Sun, W., Zhou, X.: Sampling theorem for multiwavelet subspaces. Chinese Sci. Bull. 44(14), 1283–1286 (1999) [18] Torres, R., Ward. E. Leibniz’s Rule, Sampling and wavelets on mixed Lebesgue spaces. J. Fourier Anal. Appl. 21(5), 1053–1076 (2015) [19] Vaidyanathan, P.P., Vrcelj, B.: Biorthogonal partners and applications. IEEE Trans. Signal Process. 49, 1013–1027 (2001) [20] Venkataramani, R., Bresler, Y.: Sampling theorems for uniform and periodic nonuniform MIMO sampling of multiband signals. IEEE Trans. Signal Process. 51(12), 3152–3163 (2003) [21] Ward, E.: New estimates in harmonic analysis for mixed Lebesgue spaces. Ph.D. Thesis, University of Kansas (2010) [22] Zhang, Q., Sun, W.: Reconstruction of splines from nonuniform samples. Acta Math. Appl. Sin., In Press [23] Zhang, Q., Sun, W.: Invariance of shift invariant spaces. Sci. China Seri. A 55(7), 1395–1401 (2012) [24] Zhou, X., Sun, W.: On the sampling theorem for wavelet subspaces. J. Fourier Anal. Appl. 5(4), 347–354 (1999)
14
Rui Li, Bei Liu, Rui Liu, Qingyue Zhang
PII: DOI: Reference:
S0022-247X(17)30390-6 http://dx.doi.org/10.1016/j.jmaa.2017.04.036 YJMAA 21323
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
20 September 2016
Please cite this article in press as: R. Li et al., Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue Spaces Lp,q (Rd+1 ), J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2017.04.036
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Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue Spaces Lp,q Rd+1 ∗ Rui Li a , Bei Liu a , Rui Liu b† and Qingyue Zhang a a b
College of Science, Tianjin University of Technology, Tianjin 300384, China
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
e-mails: [email protected], [email protected], [email protected], [email protected]
April 19, 2017
Abstract. In this paper, we study the nonuniform sampling and reconstruction problem in shift-invariant subspaces of mixed Lebesgue We first show that shift d+1spaces. p,q invariant subspaces in mixed Lebesgue spaces L R can be well-defined. Then we propose that the sampling problem in shift-invariant subspaces of mixed Lebesgue spaces is well-posed. At last, the nonuniform samples {f (xj , yk ) : k, j ∈ J} of a function f belonging to a shift-invariant subspace of mixed Lebesgue spaces are proposed, and we give a fast reconstruction algorithm that allows exact reconstruction of f as long as the sampling set X = {(xj , yk ) : k, j ∈ J} is sufficiently dense. Key words. mixed Lebesgue spaces; nonuniform sampling; shift-invariant subspaces. 2010 MR Subject Classification 94A20, 94A12, 42C15, 41A58
1
Introduction and motivation
Mixed Lebesgue spaces are a generalization of Lebesgue spaces that arise naturally when considering functions that depend on independent quantities with different properties [5, 6, 9, 13]. For example, a function which depends on spacial and time variables may belong to mixed Lebesgue spaces. If a function comes from mixed Lebesgue spaces, we can consider the integrability of each variable separately. This is different from Lebesgue spaces which mainly require the same level of control over all the variables of a function. The flexibility allows these mixed Lebesgue spaces to play an important role in the study of time-based partial differential equations. In this environment, we explore the sampling theorems in mixed Lebesgue spaces. The sampling theorem is one of the most powerful tools in signal analysis, and provides the theoretical basis of modern pulse code modulation communication system. The wellknown classical Shannon sampling theorem [14, 15] states that for any f ∈ L2 (R) with ∗
This work was supported partially by the National Natural Science Foundation of China (NSFC Grant 11201336, 11326094, 11371200, 11401435, 11526150, 11601383 and 11671214). Rui Liu is also supported by Hundred Young Academia Leaders Program of Nankai University. † Corresponding author.
1
suppfˆ ⊆ [−T, T ], f (x) =
n sin π(2T x − n) n = sinc(2T x − n), f f 2T π(2T x − n) 2T
n∈Z
n∈Z
where the series converges in L2 (R) and uniformly on compact sets. Here for any f ∈ L2 (R), ˆ f (ξ) = f (x)e−2πixξ dx, ξ ∈ R. R
This theorem tells us that we can recover the entirety of a function from well-chosen n : n ∈ Z} is uniform. However, in many realistic situasample. The sampling set { 2T tions, the data are known only on a nonuniform sampling set. For example, the loss of data packets during transmission through the internet from satellites can be viewed as a nonuniform sampling problem. The nonuniform sampling problem has been studied in recent years [7, 8, 12, 16, 17, 20, 22, 24]. The problems of uniform and nonuniform sampling and reconstruction also have been extended to more general shift-invariant spaces of the form 2 V (φ) = c(k)φ(x − k) : {c(k) : k ∈ Z} ∈ (Z) . k∈Z
Recently, many researchers studied the sampling theory on general shift-invariant spaces in Lp (Rd ), 1 ≤ p < +∞, the details can be seen in [1–4, 19, 23]. The uniform sampling problem for band-limited functions in mixed Lebesgue spaces has been studied by Rodolfo H. Torres and Erika L. Ward [18,21]. They established PlancherelPolya inequality for mixed Lebesgue spaces. In this paper, we will discuss the nonuniform sampling and reconstruction problem in shift-invariant spaces of mixed Lebesgue d+1 spaces. p,q We first show that shift-invariant spaces in mixed Lebesgue spaces L R can be well-defined. Then we prove that the sampling problem in shift-invariant spaces of mixed Lebesgue spaces is well-posed. In the last section, for given samples {f (xj , yk ) : k, j ∈ J} of a function f belonging to a shift-invariant space of mixed Lebesgue spaces, we provide a fast reconstruction algorithm that allows exact reconstruction of f as long as the sampling set X = {(xj , yk ) : k, j ∈ J} is sufficiently dense.
2
Definitions and preliminary results In this section, we begin with the definition of mixed Lebesgue spaces Lp,q (Rd+1 ).
Definition 2.1 Let 1 ≤ p, q < +∞. Then Lp,q = Lp,q (Rd+1 ) consists of all measurable functions f = f (x, y) defined on R × Rd such that
p p1 q f Lp,q = |f (x, y)|q dy dx < +∞. R
Rd
The corresponding sequence spaces are ⎧ ⎫ ⎛ ⎞p q ⎪ ⎪ ⎨ ⎬ ⎝ |c(k1 , k2 )|q ⎠ < +∞ . p,q = p,q (Zd+1 ) = c : cpp,q = ⎪ ⎪ ⎩ ⎭ k1 ∈Z k2 ∈Zd 2
Obviously, Lp,p (Rd+1 ) = Lp (Rd+1 ), p,p (Zd+1 ) = p (Zd+1 ) = p . In order to control the local behavior of a function, mixed Wiener amalgam spaces are introduced. The following is the definition of mixed Wiener amalgam spaces W (Lp,q )(Rd+1 ). Definition 2.2 A measurable function f belongs to mixed Wiener amalgam spaces W (Lp,q ) = W (Lp,q )(Rd+1 ), 1 ≤ p, q < ∞, if it satisfies f pW (Lp,q ) :=
⎡ sup ⎣
n∈Z x∈[0,1]
⎤p/q sup |f (x + n, y + l)|q ⎦
d l∈Zd y∈[0,1]
< ∞.
Let W0 (Lp,q ) (1 ≤ p, q < ∞) be the space of all continuous functions in W (Lp,q ). For 1 ≤ p < ∞, a function f belongs to Wiener amalgam spaces W (Lp ) = W (Lp )(Rd+1 ), if it satisfies ess supx∈[0,1]d+1 |f (x + k)|p < ∞. f pW (Lp ) := k∈Zd+1
For p = ∞, a measurable function f belongs to W (L∞ ) = W (L∞ )(Rd+1 ) if it satisfies f W (L∞ ) := sup ess supx∈[0,1]d+1 |f (x + k)| < ∞. k∈Zd+1
It is easy to see that W (Lp ) ⊂ W (Lp,p ). Let W0 (Lp ) (1 ≤ p ≤ ∞) be the space of all continuous functions in W (Lp ). For any c ∈ p , d ∈ 1 (1 ≤ p ≤ +∞), define the discrete convolution c(k)d(l − k). (c ∗ d)(l) = k∈Zd+1
For any c ∈ p (1 ≤ p ≤ +∞), f ∈ W (L1 ), define the semi-discrete convolution c(k)f (x − k). (c ∗sd f )(x) = k∈Zd+1
The following is the discrete version of Young’s inequality. Proposition 2.3 [10, Theorem 6.18] If c ∈ p , d ∈ 1 (1 ≤ p ≤ ∞), then c ∗ dp ≤ cp d1 . Given a function φ ∈ W (L1 ), the following lemma shows that φ∗sd maps p to Lp and the semi-discrete version of Young’s inequality is provided. Lemma 2.4 Let φ ∈ W (L1 ) and c ∈ p (1 ≤ p ≤ ∞). Then c ∗sd φLp ≤ cp φW (L1 ) .
3
(2.1)
Proof. For p = ∞, (2.1) holds obviously. In the case 1 ≤ p < ∞, by Fubini’s theorem, p p c(k)φ(x − k) dx c ∗sd φLp = Rd+1 k∈Zd+1 p c(k)φ(x + l − k) dx = [0,1]d+1 l∈Zd+1 k∈Zd+1 p c(k)φ(x + l − k) dx. = [0,1]d+1 l∈Zd+1 k∈Zd+1 Fix x for the moment, let dx be the sequence {dx (k) = φ(x + k) : k ∈ Zd+1 }. Then c(k)φ(x + l − k) = c(k)dx (l − k) = (c ∗ dx )(l). k∈Zd+1
k∈Zd+1
Proposition 2.3 leads to p c(k)φ(x + l − k) = c ∗ dx pp l∈Zd+1 k∈Zd+1 ⎞p ⎛ ≤ cpp dx p1 = cpp ⎝ |φ(x + k)|⎠ k∈Zd+1
≤
cpp φpW (L1 ) .
Therefore, one has c ∗sd φpLp ≤
[0,1]d+1
cpp φpW (L1 ) dx = cpp φpW (L1 ) .
One completes the proof. The following lemma provides a discrete version of Young’s inequality in discrete mixed spaces. Lemma 2.5 If c ∈ p,q , d ∈ 1 (1 ≤ p, q < ∞), then c ∗ dp,q ≤ cp,q d1 . Proof. Fix k1 ∈ Z for the moment, denote ck1 = {ck1 (l) = c(k1 , l) : l ∈ Zd } and dk1 = {dk1 (l) = d(k1 , l) : l ∈ Zd }, then c ∗ dpp,q
q ⎤p/q ⎣ = c(k1 , k2 )d(n1 − k1 , n2 − k2 ) ⎦ n1 ∈Z n2 ∈Zd k1 ∈Z k2 ∈Zd
⎡
4
p = (ck1 ∗ dn1 −k1 )(·) q n1 ∈Z k1 ∈Z ⎛ ⎞p ⎝ ≤ (ck1 ∗ dn1 −k1 )(·)q ⎠ . n1 ∈Z
k1 ∈Z
The discrete version of Young’s inequality (Proposition 2.3) leads to ⎛ ⎞p ⎝ ck1 q dn1 −k1 1 ⎠ . c ∗ dpp,q ≤ n1 ∈Z
k1 ∈Z
Let a = {a(n) = cn q : n ∈ Z} and b = {b(n) = dn 1 : n ∈ Z}. Using Proposition 2.3 again, one has ⎛ ⎞p ⎛ ⎞p ⎝ ck1 q dn1 −k1 1 ⎠ = a ∗ bpp ≤ app bp1 = ck1 pq ⎝ dk1 1 ⎠ . n1 ∈Z
k1 ∈Z
k1 ∈Z
Therefore, c ∗ dpp,q
≤
k1 ∈Z
= =
⎛ ck1 pq ⎝ ⎛ ⎝
k1 ∈Z
⎞p
dk1 1 ⎠
k1 ∈Z
⎞p/q ⎛
|c(k1 , k2 )|q ⎠
k1 ∈Z k2 ∈Zd cpp,q dp1 .
⎝
⎞p |d(k1 , k2 )|⎠
k1 ∈Z k2 ∈Zd
The following is H¨ older’s inequality for mixed Lebesgue spaces. Proposition 2.6 [21, Theorem 1.1.3] Let 1 ≤ p, q ≤ ∞ , with Then f gL1,1 ≤ f Lp,q f Lp ,q .
3
1 p
+ p1 = 1 and
1 q
+ q1 = 1.
Shift-invariant subspaces in Lp,q
For φ ∈ W (L1,1 ), shift-invariant subspaces in mixed Legesgue spaces Lp,q are defined as follows ⎫ ⎧ ⎬ ⎨ Vp,q (φ) = c(k1 , k2 )φ(· − k1 , · − k2 ) : c(k1 , k2 ) : k1 ∈ Z, k2 ∈ Zd ∈ p,q . ⎭ ⎩ d k1 ∈Z k2 ∈Z
It is obvious that the double sum pointwisely converges almost everywhere. In fact, c(k1 , k2 ) ∈ p,q tells that c(k1 , k2 ) ∈ ∞ . This with φ ∈ W (L1,1 ) leads to |c(k1 , k2 )φ(x − k1 , y − k2 )| ≤ c∞ |φ(x − k1 , y − k2 )| k1 ∈Z k2 ∈Zd
k1 ∈Z k2 ∈Zd
≤ c∞ φW (L1,1 ) (a.e.). 5
The following theorem shows that shift-invariant subspaces are well-defined in Lp,q . 1,1 Theorem 3.1 for any c ∈ p,q , the function ! Let 1 ≤ p, q < ∞ and φ ∈ W (L ). Then ! p,q f = k1 ∈Z k2 ∈Zd c(k1 , k2 )φ(· − k1 , · − k2 ) belongs to L . Furthermore,
f Lp,q ≤ cp,q φW (L1,1 ) . Proof. For fixed x ∈ R and k1 ∈ Z, let ck1 (k2 ) = {c(k1 , k2 ) : k2 ∈ Zd }, φx−k1 (y) = φ(x − k1 , y). Then c(k1 , k2 )φ(x − k1 , y − k2 ) = ck1 (k2 )φx−k1 (y − k2 ) = (ck1 ∗sd φx−k1 )(y). k2 ∈Zd
k2 ∈Zd
This and Lemma 2.4 lead to ⎡ ⎣ f pLp,q =
q ⎤ p q ⎦ c(k1 , k2 )φ(x − k1 , y − k2 ) dy dx R Rd k ∈Z k2 ∈Zd 1 p dx (c ∗ φ )(·) = k1 sd x−k1 R k ∈Z q 1 L ⎡ ⎤p ⎣ ≤ (ck1 ∗sd φx−k1 )(·)Lq ⎦ dx R
≤
⎡ ⎣
R
⎡ ⎣
= R
k1 ∈Z
⎤p ck1 q φx−k1 W (L1 ) ⎦ dx
k1 ∈Z
⎤p c(k1 , ·)q φ(x − k1 , ·)W (L1 ) ⎦ dx.
(3.1)
k1 ∈Z
Let d = {d(k1 ) = c(k1 , ·)q : k1 ∈ Z} and h(x) = φ(x, ·)W (L1 ) . Then by (3.1) and Lemma 2.4 again, one has p q c(k , ·) φ(x − k , ·) 1 ) dx 1 1 W (L R k ∈Z 1 p p d(k )h(x − k ) = 1 1 dx = d ∗sd hLp R k ∈Z 1 ≤ dpp hpW (L1 ) = cpp,q φpW (L1,1 ) . Hence, f Lp,q ≤ cp,q φW (L1,1 ) . In order to obtain the main theorem, we need the following results. Proposition 3.2 [11] Let φ ∈ W L1,1 . Then φ satisfies 2 " φ (ξ + 2πk) > 0, ξ ∈ Rd+1 , k∈Zd+1
6
if and only if there exists a function d(k1 , k2 )φ(· − k1 , · − k2 ) g= k1 ∈Z k2 ∈Zd
such that φ (· − α) , g = δ0,α , where d = {d(k1 , k2 ) : k1 ∈ Z, k2 ∈ Zd } ∈ 1 . Lemma 3.3 The function g in Proposition 3.2 belongs to W (L1,1 ). Proof. It is easy to see gW (L1,1 ) = sup sup |g (x + n, y + l)| n∈Z x∈[0,1] l∈Zd y∈[0,1]
d
sup sup d(k1 , k2 )φ(x + n − k1 , y + l − k2 ) = d n∈Z x∈[0,1] l∈Zd y∈[0,1] k1 ∈Z k2 ∈Zd ≤ sup sup |d(k1 , k2 )||φ(x + n − k1 , y + l − k2 )|
d
n∈Z x∈[0,1] l∈Zd y∈[0,1] k1 ∈Z k2 ∈Zd
≤
k1 ∈Z k2
|d(k1 , k2 )|
∈Zd
sup
n∈Z x∈[0,1]
l∈Zd
sup |φ(x + n − k1 , y + l − k2 )|
y∈[0,1]d
= d1 φW (L1,1 ) < ∞. Now, we show the main result in this section which gives the norm equivalences. As usual, for quantities X and Y , X ≈ Y and X Y mean that there exist constants c1 , c2 , c such that c1 X ≤ Y ≤ c2 X and X ≤ cY , respectively. Theorem 3.4 Let 1 < p, q < ∞. Assume that φ ∈ W (L1 ) satisfies 2 " φ (ξ + 2πk) > 0,
ξ ∈ Rd+1 .
k∈Zd+1
! ! Then for any c ∈ p,q , the function f = k1 ∈Z k2 ∈Zd c(k1 , k2 )φ(· − k1 , · − k2 ) belongs to Lp,q , and cp,q ≈ f Lp,q ≈ f W (Lp,q ) . Proof. We first consider cp,q ≈ f Lp,q . Theorem 3.1 leads to f Lp,q cp,q , thus one only needs to prove cp,q f Lp,q . By Proposition 3.2, there exists a function g such that φ (· − α) , g = δ0,α . Therefore,
c(k1 , k2 ) =
R
Rd
f (x, y)g (x − k1 , y − k2 )dxdy. 7
# $ Let b = b(k1 , k2 ) : k1 ∈ Z, k2 ∈ Zd ∈ p ,q with
1 p
+
1 p
= 1 and
1 q
+
1 q
= 1. Then
c(k1 , k2 )b(k1 , k2 ) | c, b | = k1 ∈Z,k2 ∈Zd = b(k1 , k2 ) f (x, y)g (x − k1 , y − k2 )dxdy R Rd k1 ∈Z,k2 ∈Zd = f (x, y) b(k1 , k2 )g (x − k1 , y − k2 )dxdy . R Rd k1 ∈Z,k2 ∈Zd Using Proposition 2.6, Lemma 3.3 and Theorem 3.1, one has | c, b | ≤ f Lp,q b(k1 , k2 )g (x − k1 , y − k2 ) k1 ∈Z,k2 ∈Zd ≤ f Lp,q bp ,q gW (L1,1 ) .
,q
Lp
Thus cp,q ≤ f Lp,q gW (L1,1 ) ,
(3.2)
that is, cp,q f Lp,q . Therefore, one obtains cp,q ≈ f Lp,q . Now, we prove f Lp,q ≈ f W (Lp,q ) . It is easy to see that f pLp,q
p
= R
=
Rd
1
⎛ ⎝
n∈Z 0
≤
q
|f (x, y)| dy q
⎞p/q
l∈Zd
⎡
sup ⎣
n∈Z x∈[0,1]
dx
[0,1]d
|f (x + n, y + l)|q dy ⎠
dx
⎤p/q sup |f (x + n, y + l)|q ⎦
l∈Zd y∈[0,1]
d
= f pW (Lp,q ) .
(3.3)
On the other hand, p p c(k1 , k2 )φ(· − k1 , · − k2 ) f W (Lp,q ) = k1 ∈Z k2 ∈Zd W (Lp,q ) q ⎤p/q ⎡ ⎣ = sup sup c(k1 , k2 )φ(x + n − k1 , y + l − k2 ) ⎦ d n∈Z x∈[0,1] l∈Zd y∈[0,1] k1 ∈Z k2 ∈Zd
8
≤
⎡
sup ⎣
n∈Z x∈[0,1]
⎡
⎛ sup ⎝
d l∈Zd y∈[0,1]
⎛
n∈Z
k1 ∈Z k2
|c(k1 , k2 )||φ(x + n − k1 , y + l − k2 )|⎠ ⎦
k1 ∈Z k2 ∈Zd
⎣ ⎝ ≤ |c(k1 , k2 )| l∈Zd
⎞q ⎤p/q
∈Zd
⎞q ⎤p/q sup
(x,y)∈[0,1]×[0,1]d
|φ(x + n − k1 , y + l − k2 )|⎠ ⎦
.
Denote d(n, l) = sup(x,y)∈[0,1]×[0,1]d |φ(x + n, y + l)|, then by Lemma 2.5, one obtains ⎞q ⎤p/q ⎡ ⎛ ⎣ ⎝ |c(k1 , k2 )|d(n − k1 , l − k2 )⎠ ⎦ = |c| ∗ dpp,q f pW (Lp,q ) = l∈Zd k1 ∈Z k2 ∈Zd p |c|p,q dp1 = cpp,q φpW (L1 ) n∈Z
≤
≤ f pLp,q gpW (L1,1 ) φpW (L1 ) ,
(3.4)
where the last inequality follows by (3.2) and |c| = {|c(k1 , k2 )| : k1 ∈ Z, k2 ∈ Zd }. Combining (3.3) and (3.4), one has f Lp,q ≈ f W (Lp,q ) . Therefore, one completes the proof.
4
Nonuniform sampling in shift-invariant subspaces In this section, we mainly discuss nonuniform sampling in shift-invariant subspaces.
Definition 4.1 A set X = {(xk , yj ) : k, j ∈ J} ⊂ Rd+1 is said to be strongly separated if inf (k,j)=(k ,j ) |(xk , yj ) − (xk , yj )| = δ1 > 0 and inf k=k |xk − xk | = δ2 > 0, where % |(xk , yj ) − (xk , yj )| = (xk − xk )2 + |yj − yj |2 , and J is a countable index set. The following theorem provides a sufficient condition under which the problem of nonuniform sampling in shift-invariant subspaces is well-defined. It tells us the stability of nonuniform sampling in shift-invariant subspaces. Theorem 4.2 If X = {(xk , yj ) : k, j ∈ J, xk ∈ R, yj ∈ Rd } is strongly separated and f ∈ W0 (Lp,q ). Then ⎡ ⎛ ⎞p/q ⎤1/p d/q 1/p 1 1 ⎥ ⎢ ⎝ q⎠ |f (xk , yj )| +1 +1 f W (Lp,q ) . ⎦ ≤ ⎣ δ1 δ2 k∈J
j∈J
Proof. For fixed xk , |f (xk , yj )|q =
|f (xk , yj )|q
l∈Zd yj ∈[l,l+1]d
j∈J
≤ ≤
sup
l∈Zd yj ∈[l,l+1]d
y∈[l,l+1]d
+1
sup
1
l∈Zd
=
δ1
1 +1 δ1 9
|f (xk , y)|q
d
d
y∈[l,l+1]d
sup
l∈Zd y∈[l,l+1]
d
|f (xk , y)|q |f (xk , y)|q .
Therefore, ⎛ ⎞p/q ⎡ ⎤p/q pd/q 1 ⎝ ⎣ |f (xk , yj )|q ⎠ ≤ +1 sup |f (xk , y)|q ⎦ δ1 d d y∈[l,l+1] k∈J
j∈J
k∈J
= ≤ = =
1 +1 δ1 1 +1 δ1 1 +1 δ1 1 +1 δ1
l∈Z
pd/q
⎡
⎣
n∈Z xk ∈[n,n+1]
pd/q n∈Z
pd/q pd/q
1 +1 δ2
l∈Zd
sup y∈[l,l+1]d
⎡
|f (xk , y)|q ⎦
⎣
sup x∈[n,n+1]
⎤p/q
⎤p/q sup
d l∈Zd y∈[l,l+1]
|f (x, y)|q ⎦
⎡ ⎤p/q 1 +1 sup ⎣ sup |f (x + n, y + l)|q ⎦ δ2 d x∈[0,1] d y∈[0,1]
n∈Z
l∈Z
1 + 1 f pW (Lp,q ) . δ2
Then one has ⎡ ⎛ ⎞p/q ⎤1/p d/q 1/p 1 1 ⎥ ⎢ ⎝ q⎠ |f (xk , yj )| +1 +1 f W (Lp,q ) . ⎦ ≤ ⎣ δ1 δ2 k∈J
j∈J
Let f be a continuous function. We define the oscillation (or modulus of continuity) of f by oscδ (f )(x) = sup|y|≤δ |f (x + y) − f (x)|. Lemma 4.3 Let φ ∈ W (L1 ), then oscδ (φ) ∈ W (L1 ). Proof. Without loss of generality, assume that δ ≤ 1. Then for j ∈ Zd+1 , sup x∈[0,1]d+1
|oscδ (φ)(x + j)| = ≤
sup |φ(x + j + y) − φ(x + j)|
sup
x∈[0,1]d+1 |y|≤δ
sup |φ(x + y + j)| +
sup
x∈[0,1]d+1 |y|≤δ
≤ 2
sup x∈[−1,2]d+1
sup
k∈[−1,2]d+1 ∩Zd+1
Summing over j ∈ Zd+1 , one obtains sup |oscδ (φ)(x + j)| ≤ 2 j∈Zd+1
x∈[0,1]d+1
x∈[0,1]d+1
|φ(x + j + k)|.
j∈Zd+1
sup
k∈[−1,2]d+1 ∩Zd+1
k∈[−1,2]d+1 ∩Zd+1
j∈Zd+1
= 2
≤ 22d+3 φW (L1 ) < +∞. 10
|φ(x + j)|
|φ(x + j)|
≤ 2
sup x∈[0,1]d+1
x∈[0,1]d+1
sup x∈[0,1]d+1
|φ(x + j + k)| |φ(x + j + k)|
Therefore, oscδ (φ) ∈ W (L1 ).
Lemma 4.4 Let φ ∈ W0 (L1 ). Then limδ→0 oscδ (φ)W (L1 ) = 0. Proof. By Lemma 4.3, one has oscδ (φ) ∈ W (L1 ). Given > 0, there exists an integer L0 > 0 such that (4.1) sup |oscδ (φ)(x + j)| < . 2 x∈[0,1]d+1 |j|≥L0
Moreover, since φ is continuous, there exists δ0 > 0 such that sup x∈[0,1]d+1
sup |φ(x + j + y) − φ(x + j)| ≤
|y|≤δ
for all |j| < L0 and all δ < δ0 . Thus sup |oscδ (φ)(x + j)| = |j|
d+1
sup
|j|
≤
|j|
≤
sup |φ(x + j + y) − φ(x + j)|
|y|≤δ
2(2L0 )d+1
. 2
Combining (4.1) and (4.2), one obtains oscδ (φ)W (L1 ) ≤ sup j∈Zd+1
d+1
2(2L0 )d+1
x∈[0,1]d+1
(4.2)
|oscδ (φ)(x + j)| < ,
∀ 0 < δ ≤ δ0
Thus limδ→0 oscδ (φ)W (L1 ) = 0.
Lemma 4.5 Let φ ∈ W0 (L1 ). If f ∈ Vp,q (φ), then the oscillation (or modulus of continuity) oscδ (f ) belongs to Lp,q . Moreover for all > 0, there exists δ0 > 0 such that oscδ (f )Lp,q ≤ f Lp,q uniformly for all f ∈ Vp,q (φ) and 0 < δ < δ0 . ! Proof. If f = k∈Zd+1 c(k)φ(· − k) ∈ V p,q (φ), then oscδ (f )(x) = ≤ ≤
sup |f (x + y) − f (x)|
|y|≤δ
sup |y|≤δ
k∈Zd+1
=
|c(k)||φ(x + y − k) − φ(x − k)|
k∈Zd+1
|c(k)| sup |φ(x − k + y) − φ(x − k)| |y|≤δ
|c(k)|oscδ (φ)(x − k)
k∈Zd+1
= [|c| ∗sd (oscδ )(φ)](x),
11
where |c| = {|c(k)| : k ∈ Zd+1 }. By Theorem 3.1, Theorem 3.4 and Lemma 4.3, there exists M > 0 such that oscδ (f )Lp,q
≤ |c| ∗sd (oscδ )(φ)Lp,q ≤ |c|p,q oscδ (φ)W (L1,1 ) = cp,q oscδ (φ)W (L1,1 ) ≤ M f Lp,q oscδ (φ)W (L1,1 ) ≤ M f Lp,q oscδ (φ)W (L1 ) .
(4.3)
Using Lemma 4.4, for any > 0, there exists δ0 > 0 such that , ∀ 0 < δ ≤ δ0 . oscδ (φ)W (L1 ) < M This with (4.3) leads to oscδ (f )Lp,q ≤ f Lp,q .
Definition 4.6 A set X = {(xj , yk ) : k, j ∈ J, xk ∈ R, yj ∈ Rd } is γ0 -dense in Rd+1 if Rd+1 = ∪j,k Bγ (xj , yk )
for every γ > γ0 ,
where Bγ (xj , yk ) is the open ball with center (xj , yk ) and radius γ, and J is a countable index set. A bounded uniform partition of unity {βj,k }j,k∈J associated to a strongly separated sampling set X = {(xj , yk ) : j, k ∈ J} is a set of functions such that (i). 0 ≤ βj,k ≤ 1, ∀ j, k ∈ J, (ii). suppβj,k ⊂ Bγ (xj , yk ), ! ! (iii). j∈J k∈J βj,k = 1. If f ∈ W0 (Lp,q ), we write QX f =
f (xj , yk )βj,k
j∈J k∈J
for the quasi-interpolant of the sequence cj,k = f (xj , yk ). In order to obtain the reconstruction algorithm, we need the following lemma. Lemma 4.7 Let P be any bounded projection from Lp,q onto Vp,q (φ) with φ ∈ W0 (L1 ). Then there exists γ0 = γ0 (p, q, P ) such that the operator I − P QX is a contraction on Vp,q (φ) for every strongly-separated γ-dense set X with γ ≤ γ0 . Proof. For f ∈ Vp,q (φ), one has f − P QX f Lp,q
= P f − P QX f Lp,q ≤ P op f − QX f Lp,q ≤ P op oscγ f Lp,q ≤ P op f Lp,q .
Here, the last inequality follows by Lemma 4.5. One can choose γ0 such that for any γ < γ0 , P op < 1. Hence, it leads to a contraction. Now we give a fast iterative algorithm to reconstruct f ∈ Vp,q (φ) from its samples {f (xj , yk ) : j, k ∈ J}. 12
Theorem 4.8 Let φ ∈ W0 (L1 ). Assume that P is a bounded projection from Lp,q onto Vp,q (φ). Then there exists a density γ > 0 (γ = γ(p, q, P )) such that any f ∈ Vp,q (φ) can be recovered from its samples {f (xj , yk ) : (xj , yk ) ∈ X} on any γ-dense set X = {(xj , yk ) : j, k ∈ J} by the following iterative algorithm: ( f 1 = P QX f (4.4) fn+1 = P QX (f − fn ) + fn . The iterates fn converges to f uniformly and in Lp,q norms. Furthermore, the convergence is geometric, that is, f − fn Lp,q ≤ M αn for some α = α(γ) < 1 and M < ∞. Proof. Let en = f − fn be the error after n iterations. By (4.4), en+1 = f − fn+1 = f − fn − P QX (f − fn ) = (I − P QX )en . By Lemma 4.7, one may choose γ so small that I − P QX op = α < 1. Then one obtains en+1 Lp,q ≤ αen Lp,q ≤ αn e0 Lp,q . Thus en Lp,q → 0, when n → ∞. The proof is completed.
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