Acta Mathematica Sinica, English Series Published online: March 20, 2018 https://doi.org/10.1007/s10114-018-7333-1 http://www.ActaMath.com
Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2018
The Lp,q -stability of the Shifts of Finitely Many Functions in Mixed Lebesgue Spaces Lp,q (Rd+1 ) Rui LI
Bei LIU
College of Science, Tianjin University of Technology, Tianjin 300384, P. R. China E-mail :
[email protected] [email protected]
Rui LIU1) School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China E-mail :
[email protected]
Qing Yue ZHANG College of Science, Tianjin University of Technology, Tianjin 300384, P. R. China E-mail :
[email protected] Abstract The stability is an expected property for functions, which is widely considered in the study of approximation theory and wavelet analysis. In this paper, we consider the Lp,q -stability of the shifts of finitely many functions in mixed Lebesgue spaces Lp,q (Rd+1 ). We first show that the shifts + 2πk)|2 > 0. Then φ(· − k) (k ∈ Zd+1 ) are Lp,q -stable if and only if for any ξ ∈ Rd+1 , k∈Zd+1 |φ(ξ we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces Lp,q (Rd+1 ) to be Lp,q -stable which improves some known results. Keywords
Mixed Lebesgue spaces, Lp,q -stability, semi-convolution
MR(2010) Subject Classification
1
46B15, 42C15, 42C40, 41A58
Introduction and Motivation
In this article, we study the Lp,q -stability of the shifts of finitely many functions in mixed Lebesgue spaces Lp,q (Rd+1 ). Mixed Lebesgue spaces are a generalization of Lebesgue spaces, which consider the integrability of each variable separately [1, 2, 5, 6, 16, 19]. The definition of mixed Lebesgue spaces Lp,q (Rd+1 ) is as follows. Definition 1.1 Let 1 < p, q < +∞. Then Lp,q (Rd+1 ) consists of all measurable functions f on Rd+1 such that pq p1 q f Lp,q = |f (x1 , x2 )| dx2 dx1 < +∞. R
Rd
Received July 13, 2017, accepted October 18, 2017 Supported by the National Natural Science Foundation of China (Grant Nos. 11371200, 11401435, 11601383 and 11671214) and Hundred Young Academia Leaders Program of Nankai University 1) Corresponding author
Li R. et al.
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The corresponding sequence spaces of mixed Lebesgue spaces Lp,q (Rd+1 ) are pq p1
p,q d+1 q p,q (Z ) = c : c = |c(k1 , k2 )| < +∞ . k1 ∈Z
k2 ∈Zd
It is easy to see that · Lp,q is a norm, then for f, g ∈ Lp,q , f + gLp,q ≤ f Lp,q + gLp,q .
(1.1)
Mixed Lebesgue spaces were first described in detail by Benedek and Panzone in [1], they were also explored by Rubio de Francia et al. in [6, 16]. Mixed Lebesgue spaces play an important role in the study of sampling and equation problems, since we can consider functions to be independent quantities with different properties [10, 16]. It is generally known that the stability is an expected property for functions in sampling problems [12–15, 17, 18, 20, 21]. The stability of the function system can ensure that signals can be recovered when the disturbance occurs. The Lp,q -stability of the shifts of finitely many functions in mixed Lebesgue spaces Lp,q (Rd+1 ) is provided. Definition 1.2 Let φ1 , . . . , φn ∈ Lp,q (Rd+1 ) (1 < p, q < ∞). The shifts φj (· − k) (1 ≤ j ≤ n, k ∈ Zd+1 ) are said to be Lp,q -stable if there are two positive constants C1 and C2 such that
n n
n cj p,q ≤ cj ∗sd φj ≤ C cj p,q C1 2 j=1
for all cj ∈
p,q
(Z
d+1
Lp,q
j=1
j=1
), 1 ≤ j ≤ n.
Here, the semi-convolution c ∗sd φ is the sum
c ∗sd φ = c(k)φ(· − k) k∈Zd+1
for given a sequence c and a function φ. In particular, if p = q, then we call that the shifts φj (· − k) (1 ≤ j ≤ n, k ∈ Zd+1 ) are Lp -stable. In [8], Jia and Micchelli gave a characterization for L2 -stability, and they used the H¨ older inequality extend the result to the case 1 < p ≤ ∞. Then they studied the characterization of Lp -stability (1 ≤ p ≤ ∞) of the shifts of a finite number of compactly supported functions in [7, 8]. Furthermore, Jia [9] proved that the shifts of φ1 , . . . , φn are Lp -stable if and only if for any ξ ∈ Zd , the sequences {φk (ξ + 2βπ)}β∈Zd (k = 1, . . . , n) are linearly independent. This is a generalization of the previous results of Jia and Micchelli in [7, 8]. 1.1 Main Results In order to provide our main results in Lp,q (Rd+1 ) which extend the results in [9], we introduce some notations. Here and after, let fˆ(ξ) denote the Fourier transform of f ∈ L1 (Rd+1 ): ˆ f (x)e−iξx dx. f (ξ) = Rd+1
For any 1 ≤ p < ∞, f ∈ L ([0, 1] p
d+1
) means that f Lp ([0,1]d+1 ) =
[0,1]d+1
|f (x)| dx p
p1
< ∞;
When p = ∞, f ∈ L∞ ([0, 1]d+1 ) means that f L∞ ([0,1]d+1 ) = ess supx∈[0,1]d+1 |f (x)| < ∞.
The Lp,q -stability in Mixed Spaces
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Given a function f , define f Lp,q := k1 ∈Z
[0,1]d
k2 ∈Zd
1q q |f (· + k1 , x2 + k2 )| dx2
.
Lp ([0,1]d )
For 1 ≤ p, q ≤ ∞, let Lp,q = Lp,q (Rd+1 ) be the linear space of all functions f for which f Lp,q < ∞. The norms are defined above and with usual modification in the case of q = ∞. Considering triangle inequality, for f ∈ Lp,q and g ∈ Lp,q , f + gLp,q ≤ f Lp,q + gLp,q .
(1.2)
Clearly, for 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, L∞,∞ ⊂ Lp,q ⊂ L1,1 = L1 .
(1.3)
Now we show the first main result which gives a necessary and sufficient condition for the shifts φ(· − k) (k ∈ Zd+1 ) to be Lp,q -stable. Theorem 1.3 Let φ ∈ Lp,q (Rd+1 ) (1 < p, q < ∞). Then the shifts φ(· − k) (k ∈ Zd+1 ) are Lp,q -stable if and only if for any ξ ∈ Rd+1 ,
+ 2πk)|2 > 0. |φ(ξ (1.4) k∈Zd+1
For the Lp,q -stability of the shifts of finitely many functions in mixed Lebesgue spaces L (Rd+1 ), we have the following main result. p,q
Theorem 1.4 Let φ1 , . . . , φn ∈ Lp,q (Rd+1 ) (1 < p, q < ∞). Then the shifts φj (· − k) (1 ≤ j ≤ n, k ∈ Zd+1 ) are Lp,q -stable if and only if for any ξ ∈ Rd+1 , the sequences {φj (ξ + 2πk)}k∈Zd+1 (1 ≤ j ≤ n) are linearly independent. The paper is organized as follows. In the next subsection, we give some basic notations and definitions. Some useful lemmas and propositions are presented in Section 2. Finally, in Section 3, we give proofs of Theorems 1.3 and 1.4. 1.2 Basic Notations and Definitions In order to investigate the Lp,q -stability of functions in mixed Lebesgue spaces Lp,q (Rd+1 ), we introduce some basic notations and definitions. For any c ∈ 2 (Zd+1 )(or c ∈ 1 (Zd+1 )), define the discrete Fourier transform on torus T by
cˆ(ξ) = c(k)e−2πikξ . k∈Zd+1
Here, T denotes the additive group of the reals modulo 1 (that is Rd+1 /Zd+1 ). For g ∈ L1 (T), define
g(x)e−2πikx dx.
gˆ(k) = T
Let B = {ˆ a(ξ) : a is a sequence, and a ∈ 1 (Zd+1 )}. Lemma 1.5 If f ∈ B and f (ξ) = 0 for every ξ ∈ T, then by Wiener’s lemma (see [11, p. 266]) 1 f is also in B.
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For any f, g ∈ L2 (Rd+1 ), define their convolution (f ∗ g)(x) = f (x − y)g(y)dy = Rd+1
Rd+1
f (y)g(x − y)dy.
For any 1 < p < ∞, f ∈ Lp (Rd+1 ) means that
p p1 f Lp = |f (x + k)| dx < ∞. [0,1]d+1
For φ ∈ L
∞,∞
, define S1 (φ) =
k∈Zd+1
(c ∗sd φ)(x) =
c(k)φ(x − k) : c ∈ 1 (Zd+1 ) .
k∈Zd+1
2
Some Useful Lemmas and Propositions
In this section, we prove some useful lemmas which are needed in the proofs of Theorems 1.3 and 1.4. Proposition 2.1 ([7, Theorem 2.1])
Let φ ∈ Lp and c ∈ p (Zd+1 ) (1 ≤ p ≤ ∞). Then
c ∗sd φLp ≤ cp φLp . The following lemma provides a mixed space’s version of Proposition 2.1. Lemma 2.2
Let φ ∈ Lp,q (Rd+1 ), where 1 < p, q < ∞. Then for any c ∈ p,q (Zd+1 ), c ∗sd φLp,q ≤ cp,q φLp,q .
Proof
Let c = {c(k1 , k2 ) : k1 ∈ Z, k2 ∈ Zd } ∈ p,q . Then
q pq
p
c(k1 , k2 )φ(x1 − k1 , x2 − k2 ) dx2 dx1 . c ∗sd φLp,q =
R
Rd
k1 ∈Z k2 ∈Zd
For fixed k1 and x1 , let ck1 = {ck1 (k2 ) = c(k1 , k2 ) : k2 ∈ Zd } and φx1 −k1 (x2 ) = φ(x1 − k1 , x2 ). Then
c(k1 , k2 )φ(x1 − k1 , x2 − k2 ) = ck1 (k2 )φx1 −k1 (x2 − k2 ) = (ck1 ∗sd φx1 −k1 )(x2 ). k2 ∈Zd
k2 ∈Zd
Using Proposition 2.1, one has c ∗sd φpLp,q = ≤
p
dx1 (c ∗ φ )(·) k sd x −k 1 1 1 q R
L
k1 ∈Z
R
k1 ∈Z
R
k1 ∈Z
R
k1 ∈Z
p
(ck1 ∗sd φx1 −k1 )(·)Lq
dx1
p
q q ≤ ck1 φx1 −k1 L
dx1
p
= c(k1 , ·)q φ(x1 − k1 , ·)Lq
dx1 .
(2.1)
Denote d = {d(k1 ) = c(k1 , ·)q : k1 ∈ Z} and h(x) = φ(x, ·)Lq . Then by Proposition 2.1,
p
c(k1 , ·)q φ(x1 − k1 , ·)Lq dx1 = |(d ∗sd h)(x1 )|p dx1
R
k1 ∈Z
R
The Lp,q -stability in Mixed Spaces
5
≤ dpp hpLp = cpp,q φpLp,q . This with (2.1) leads to c ∗sd φLp,q ≤ φLp,q cp,q .
2
The following proposition gives two necessary and sufficient conditions of L -stability for the shifts of the function φ ∈ L∞,∞ . Proposition 2.3 ([7, Theorem 3.3]) Let φ ∈ L∞,∞ be L2 -stable, if and only if one of the following conditions holds: (1) for every ξ ∈ Rd+1 ,
+ 2πk)|2 > 0; |φ(ξ k∈Zd+1
(2) there exists a function g ∈ S1 (φ) such that φ(· − α), g = δ0,α ,
for all α ∈ Zd+1 .
Lemma 2.4 The g in Proposition 2.3 belongs to L∞,∞ . That is, for c = {c(k), k ∈ Zd+1 } ∈ 1 and φ ∈ L∞,∞ ,
c(k)φ(· − k) ∈ L∞,∞ . g= k∈Zd+1
Proof
It is easy to see gL∞,∞ = ess sup
ess sup
x1 ∈[0,1] n ∈Z x2 ∈[0,1]d n2 ∈Zd 1
≤ ess sup
|g(x1 + n1 , x2 + n2 )|
ess sup
x∈[0,1] n ∈Z x2 ∈[0,1]d n2 ∈Zd k1 ∈Z k2 ∈Zd 1
≤
|c(k1 , k2 )| ess sup
|c(k1 , k2 )||φ(x1 + n1 − k1 , x2 + n2 − k2 )|
ess sup
x∈[0,1] n ∈Z x2 ∈[0,1]d n2 ∈Zd 1
k1 ∈Z k2 ∈Zd
|φ(x1 + n1 − k1 , x2 + n2 − k2 )|
= c1 φL∞,∞ .
Proposition 2.5 ([19, Theorem 1.1.3]) Then
Let 1 < p, q < ∞, with
1 p
+
1 p
= 1 and
1 q
+
1 q
= 1.
f gL1 ≤ f Lp,q gLp ,q . Based on H¨older’s inequality and Fubini’s theorem, we have the following lemma. Lemma 2.6 then
Let 1 < p, q < ∞, with
1 p
+
1 p
= 1 and
1 q
+
1 q
= 1. If f ∈ Lp,q and g ∈ Lp ,q ,
f ∗ gL∞,∞ ≤ f Lp,q gLp ,q . Proof has
Let x = (x1 , x2 ), where x1 ∈ R and x2 ∈ Rd . From the definition of convolution, one (f ∗ g)(x + k) = (f ∗ g)(x1 + k1 , x2 + k2 ) = f (y1 , y2 )g(x1 + k1 − y1 , x2 + k2 − y2 )dy2 dy1 . R
Rd
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It is easy to see |(f ∗ g)(x1 + k1 , x2 + k2 )| ≤ |f (y1 , y2 + n2 )||g(x1 + k1 − y1 , x2 + k2 − y2 − n2 )|dy2 dy1 . R
[0,1]d
n2 ∈Zd
Repeatedly using Fubini’s theorem, one obtains
|(f ∗ g)(x1 + k1 , x2 + k2 )| k2 ∈Zd
= R
[0,1]d
n2 ∈Zd
= R
[0,1]d n ∈Zd 2
|f (y1 , y2 + n2 )|
|g(x1 + k1 − y1 , x2 + k2 − y2 − n2 )|dy2 dy1
k2 ∈Zd
|f (y1 , y2 + n2 )|
|g(x1 + k1 − y1 , x2 + k2 − y2 )|dy2 dy1 .
k2 ∈Zd
H¨older’s inequality leads to
|f (y1 , y2 + n2 )| |g(x1 + k1 − y1 , x2 + k2 − y2 )|dy2 [0,1]d
n ∈Zd
2
≤ |f (y1 , · + n2 )|
k2 ∈Zd
Lq ([0,1]d )
n2 ∈Zd
|g(x1 + k1 − y1 , · + k2 )|
Lq ([0,1]d )
k2 ∈Zd
= f (y1 , ·)Lq g(x1 + k1 − y1 , ·)Lq , where
1 q
+
1 q
= 1. Then
ess sup x2 ∈[0,1]d
|(f ∗ g)(x1 + k1 , x2 + k2 )| ≤
k2 ∈Zd
R
f (y1 , ·)Lq g(x1 + k1 − y1 , ·)Lq dy1 .
Similarly, Fubini’s theorem and H¨ older’s inequality tell that, for any x ∈ [0, 1]d+1 ,
ess sup |(f ∗ g)(x1 + k1 , x2 + k2 )| k∈Z x2 ∈[0,1] k2 ∈Zd d
≤
k1 ∈Z n1 ∈Z
=
[0,1] n ∈Z 1
≤
[0,1]
[0,1]
f (y1 + n1 , ·)Lq g(x1 + k1 − y1 − n1 , ·)Lq dy1
f (y1 + n1 , ·)Lq
g(x1 + k1 − y1 , ·)Lq dy1
k1 ∈Z
p p1
q f (y + n , ·) dy 1 1 L 1
[0,1]
n1 ∈Z
p p1
g(y + k , ·) dy 1 1 1 Lq
k1 ∈Z
= f Lp,q f Lp ,q . Therefore, one gets the conclusion f ∗ gL∞,∞ = ess sup
ess sup
x1 ∈[0,1] k ∈Z x2 ∈[0,1]d k2 ∈Zd 1
|(f ∗ g)(x1 + k1 , x2 + k2 )|
≤ f Lp,q gLp ,q . Proposition 2.7 ([4, Proposition 8.9])
Suppose that f ∈ Lp , g ∈ L1 (1 ≤ p ≤ ∞), then
f ∗ gLp ≤ f Lp gL1 .
The Lp,q -stability in Mixed Spaces
7
The following lemma provides a mixed spaces version of Young’s inequality. Lemma 2.8
If f ∈ Lp,q (Zd+1 ), g ∈ L1 (Rd+1 ) (1 < p, q < ∞), then we have f ∗ gLp,q ≤ f Lp,q gL1 .
Let fx1 = f (x1 , ·) and f ∗ gpLp,q =
gx1 = f (x1 , ·), then
q pq
f (y1 , y2 )g(x1 − y1 , x2 − y2 )dy2 dy1
dx2 dx1
R Rd R Rd
q pq
=
fy1 ∗ gx1 −y1 (x2 )dy1 dx2 dx1 R Rd R p = fy1 ∗ gx1 −y1 (·)dy1 dx1 .
Proof
R
R
Lq
Minkowski’s inequality and Proposition 2.7 lead to p p fy1 ∗ gx1 −y1 (·)Lq dy1 dx1 f ∗ gLp,q ≤ R R p ≤ fy1 Lq gx1 −y1 L1 dy1 dx1 . R
R
Let h(x1 ) = fx1 Lq and b(x1 ) = gx1 L1 , then by Proposition 2.7 again p fy1 Lq gx1 −y1 L1 dy1 dx1 = |(h ∗ b)(x1 )|p dx1 = h ∗ bpLp ≤ hpLp bpL1 , R
R
R
where
hpLp =
and
R
fx1 pLq dx1 = p
bpL1 =
R
R
gx1 L1 dx1
Rd
|f (x1 , x2 )|q dx2
pq
dx1 = f pLp,q p
= R
Rd
|g(x1 , x2 )|dx2 dx1
= gpL1 .
Finally, one has f ∗ gpLp,q ≤ f pLp,q gpL1 . Proposition 2.9 ([3, Corollary 1.11]) If f ∈ L1 (T) and all of the Fourier coefficients of f are equal to zero, then f is identically zero. Lemma 2.10 If f ∈ L1 (Rd+1 ) and f(2πn) = 0 for all n ∈ Zd+1 , then k∈Zd+1 f (· − k) = 0. Proof Let g(x) = k∈Zd+1 f (x − k) ∈ L1 (T). Its Fourier coefficients are
f (x − k)e−2πinx dx = f (x)e−2πinx dx = f(2πn) = 0. c(n) = [0,1]d+1 k∈Zd+1
From Proposition 2.9, we have g(x) = 3
Rd+1
k∈Zd+1
f (x − k) = 0.
Proof of Main Results
In this section, we give the proofs of Theorems 1.3 and 1.4.
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3.1 Proof of Theorem 1.3 (⇐) From Lemma 2.2, it is easy to see c ∗sd φLp,q ≤ M2 cp,q , where M2 = φLp,q . Now we prove that there exists a positive constant M1 such that M1 cp,q ≤ c ∗sd φLp,q . First, we assume that φ ∈ L∞,∞ and f = c ∗sd φ. By Proposition 2.3, there exists a function g ∈ S1 (φ) such that φ(· − α), g = δ0,α . Therefore, (1.3) tells that f (x)g(x − k)dx = Rd+1
Rd+1 l∈Zd+1
=
c(l)φ(x − l)g(x − k)dx
c(l)
l∈Zd+1
Rd+1
φ(y − l + k)g(y)dy
= c(k).
Let b = {b(k) : k ∈ Zd+1 } ∈ p ,q , where p1 + p1 = 1 and 1q + q1 = 1. It is easy to see
c(k)b(k)
| c, b | =
k∈Zd+1
=
b(k) f (x)g(x − k)dx
Rd+1 k∈Zd+1
= f (x) b(k)g(x − k)dx
. Rd+1
k∈Zd+1
Applying Proposition 2.5, one has
b(k)g(x − k) | c, b | ≤ f Lp,q k∈Zd+1
Lp ,q
≤ f Lp,q bp ,q gLp ,q . The last inequality follows by Lemmas 2.2, 2.4 and (1.3). Thus cp,q ≤ f Lp,q gLp ,q .
(3.1)
Let M1 = 1/gLp ,q . Then one obtains M1 cp,q ≤ f Lp,q , that is M1 cp,q ≤ c ∗sd φLp,q . To deal with the case φ ∈ Lp,q (Rd+1 ) (1 < p, q < ∞), one smooths φ by convolving it with 2 the function ϕ. Here ϕ(x) := e−π|x| and |x| is the Euclidean norm of x. Let ρ = φ ∗ ϕ. Then by Lemma 2.6, ρL∞,∞ ≤ φLp,q ϕLp ,q , where p1 + p1 = 1 and 1q + q1 = 1. Moreover, since ϕ(ξ) = e−|ξ| /(4π) never vanishes, ρ also satisfies (1.4). These with what has been proved, ϕL1 = 1 and Lemma 2.8, one has 2
M1 cp,q ≤ c ∗sd ρLp,q = c ∗sd φ ∗ ϕLp,q ≤ c ∗sd φLp,q ϕL1 = c ∗sd φLp,q .
The Lp,q -stability in Mixed Spaces
9
Thus, we obtain for any φ ∈ Lp,q (Rd+1 ), M1 cp,q ≤ c ∗sd φLp,q . Hence the shifts φ(· − k) (k ∈ Zd+1 ) are Lp,q -stable. 0 + 2πk)|2 = 0, that is for all k ∈ Zd+1 , (⇒) Suppose that for some ξ0 ∈ Rd+1 , k∈Zd+1 |φ(ξ 0 + 2πk) = 0. Now we prove that the shifts φ(· − k) (k ∈ Zd+1 ) are not Lp,q -stable. Without φ(ξ loss of generality, assume that φ(2πk) = 0 for all k ∈ Zd+1 (by considering the function e−iξ0 x φ(x) if necessary). By (1.3) and Lemma 2.10, it follows that
φ(· − k) = 0. (3.2) k∈Zd+1
For each integer n > 0, let en be the sequence on Zd+1 given by ⎧ ⎨ 1, if |ki | ≤ n for every i = 1, 2, . . . , d + 1, en (k) = ⎩ 0, otherwise with k = (k1 , k2 , . . . , kd+1 ) ∈ Zd+1 , then en ∈ p,q . To prove that the shifts φ(· − k) are not Lp,q -stable, it suffices to show that en ∗sd φLp,q → 0 as n → ∞. en p,q
(3.3)
To this end, one first truncates φ as follows. For each integer N > 0, let φN be the function on Rd+1 given by ⎧ ⎨ φ(x), if |xi | ≤ N for every i = 1, 2, . . . , d + 1, φN (x) = ⎩ 0, otherwise, and ψN is the function on Rd+1 given by ⎧
⎪ (φ − φN )(x − k), ⎨ ψN (x) = k∈Zd+1 ⎪ ⎩ 0,
if x ∈ [0, 1)d+1 , otherwise.
The construction of ψN implies that
p,q (φ − φN )(· − k1 , · − k2 ) ψN L = k1 ∈Z k ∈Zd
Lq [0,1]
2
≤ |(φ − φN )(· − k1 , · − k2 )|
k1 ∈Z
k2 ∈Zd
d
Lq [0,1]
Lp [0,1]
d
Lp [0,1]
= φ − φN Lp,q .
(3.4)
Now we define ϕN := φN + ψN . (1.2) and (3.4) lead to φ − ϕN Lp,q = φ − φN − ψN Lp,q ≤ φ − φN Lp,q + ψN Lp,q ≤ 2φ − φN Lp,q . Lemma 2.2 shows that en ∗sd (φ − ϕN )Lp,q ≤ φ − ϕN Lp,q en p,q ≤ 2φ − φN Lp,q en p,q .
(3.5)
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Therefore by (1.1), one has en ∗sd φLp,q en ∗sd ϕN Lp,q ≤ + 2φ − φN Lp,q . en p,q en p,q √ Take N to be the integer part of n. By the dominated convergence theorem, when N → +∞ (or n → +∞), φ − φN Lp,q → 0. Thus it remains to estimate en ∗sd ϕN Lp,q . en p,q For this purpose, one observes that ϕN is compactly supported, that is for some i, |xi | > N.
ϕN (x) = 0
(3.6)
Second, using (3.2) and the construction of ϕN ,
ϕN (· − k) = (φN + ψN )(· − k) = φ(· − k) = 0. k∈Zd+1
k∈Zd+1
(3.7)
k∈Zd+1
By (3.6), (3.7) and n > N , we know that as long as for some i, |xi | > n + N, then
(en ∗sd ϕN )(·) = en (k)ϕN (· − k) = 0.
(3.8)
k∈Zd+1
At the same time, (en ∗sd ϕN )(·) =
en (k)ϕN (· − k) = 0,
(3.9)
k∈Zd+1
when |xi | < n − N for every i = 1, 2, . . . , d + 1. Put x = (x1 , x2 , . . . , xd+1 ) and k = (k1 , k2 , . . . , kd+1 ). It follows from (3.8) and (3.9) that en ∗sd ϕN pLp,q pq q = |(en ∗sd ϕN )(x1 , x2 , . . . , xd+1 )| dx2 · · · dxd+1 dx1 Rd R ≤ n−N ≤|x1 |≤n+N
···
|xi | ≤ n+N,
|(en ∗sd ϕN )(x1 , x2 , . . . , xd+1 )|q dx2 · · · dxd+1
pq dx1
2 ≤ i ≤ d+1
d+1
+
···
l=2
|x1 |
|xi | ≤ n+N, i = l andi = 2,. . ., d+1
n−N <|xl |
|(en ∗sd ϕN )(x1 , x2 , . . . , xd+1 )| dx2 · · · dxd+1 ··· ≤ q
k1 ∈Z
···
pq q |ϕN (x1 − k1 , . . . , xd+1 − kd+1 )| dx2 · · · dxd+1 dx1
kd+1 ∈Z
+ |x1 |
dx1
|xi |≤n+N, i=2,...,d+1
n−N ≤|x1 |≤n+N
pq
d+1 l=2
···
|xi | ≤ n+N, i = l andi = 2,. . . ,d + 1
n−N <|xl |
The Lp,q -stability in Mixed Spaces
Since
k1 ∈Z
···
pq q |ϕN (x1 − k1 , . . . , xd+1 − kd+1 )| dx2 · · · dxd+1 dx1 .
···
k1 ∈Z
11
kd+1 ∈Z
kd+1 ∈Z
|ϕN (x1 − k1 , . . . , xd+1 − kd+1 )| is a periodic function, then
en ∗sd ϕN pLp,q 2d (n + N )d ≤ 4N [0,1]
···
k1 ∈Z
[0,1]
kd+1 ∈Z
d2d−1 (n + N )d−1
[0,1]
···
k1 ∈Z
≤ [2
[0,1]
pq q |ϕN (x1 − k1 , . . . , xd+1 − kd+1 )| dx2 · · · dxd+1 dx1
+ 2(n + N )
dp q+1
···
[0,1]
···
4N
[0,1]
[0,1]
pq q |ϕN (x1 − k1 , . . . , xd+1 − kd+1 )| dx2 · · · dxd+1 dx1
kd+1 ∈Z
dp q
N (n + N )
+2
dp q+1
p q
p q
d N (n + N )
(d−1)p q+1
] [0,1]
···
k1 ∈Z
[0,1]
···
[0,1]
pq q |ϕN (x1 − k1 , . . . , xd+1 − kd+1 )| dx2 · · · dxd+1 dx1 .
kd+1 ∈Z
Since n > N, then there exists constant C1 such that p
en ∗sd ϕN pLp,q ≤ C1 (N ndp/q + n(N nd−1 ) q )ϕN pLp,q . Moreover, en pp,q
=
k1 ∈Z
=
···
k2 ∈Z
|k1 |≤n
|en (k1 , . . . , kd+1 )|
q
kd+1 ∈Z
···
|k2 |≤n
pq
pq
1
|kd+1 |≤n
≥ C2 n1+dp/q , where C2 is constant. Thus one has en ∗sd ϕN pLp,q C1 ≤ en plp,q C2
N + n
N n
pq ϕN pLp,q .
(3.10)
By (3.5), (3.11) ϕN Lp,q ≤ φLp,q + 2φ − φN Lp,q . √ Take N to be the integer part of n in consideration, then one concludes from (3.10) and (3.11) that en ∗sd ϕN pLp,q →0 en plp,q This verifies (3.3), therefore “(⇒)” has been proved.
as n → ∞.
Li R. et al.
12
3.2 Proof of Theorem 1.4 (⇒) One proves this by contradiction. If for some ξ0 ∈ Rd+1 , the sequences {φj (ξ0 + 2πk)}k∈Zd+1 (1 ≤ j ≤ n) are linearly dependent, then there exist constants rj (j = 1, . . . , n), not all zero, such that n
rj φj (ξ0 + 2πk) = 0
for all k ∈ Zd+1 .
j=1
n
Let φ := j=1 rj φj . Then by Theorem 1.3, φ is not Lp,q -stable, hence the shifts φj (· − k) (1 ≤ j ≤ n, k ∈ Zd+1 ) are not Lp,q -stable. This proves “(⇒)”. (⇐) Given a1 , . . . , an ∈ p,q , let f = nj=1 aj ∗sd φj . Then by (1.1) and Lemma 2.2,
n n n
n p,q aj p,q ≤ C a f Lp,q = aj ∗sd φj ≤ ∗ φ ≤ φ aj p,q , j sd j j L Lp,q
j=1
Lp,q
j=1
j=1
j=1
where C = max1≤j≤n {φ1 Lp,q , . . . , φn Lp,q }. Next, we prove the lower bound. Assume that φj ∈ L∞,∞ (1 ≤ j ≤ n), then φj ∈ L2 . Therefore,
2 j (ξ)|2 dξ |φj (ξ + 2πk)| dξ = |φ [0,2π]d+1
Rd+1
k∈Zd+1
= (2π)
d+1 2
Rd+1
= (2π)
d+1 2
|φj (x)|2 dx
[0,1]d+1 k∈Zd+1
≤ (2π)
d+1 2
|φj (x + k)|2 dx
φj L2 < ∞.
Then in this case, one has {φj (ξ + 2πk)}k∈Zd+1 ∈ 2 (Zd+1 ) (1 ≤ j ≤ n) . Since the sequences {φj (ξ + 2πk)}k∈Zd+1 (1 ≤ j ≤ n) are linearly independent, its Gram matrix ([φj , φk ](ξ))1≤j,k≤n is nonsingular for ξ ∈ T, where
[φj , φk ](ξ) = φj (ξ + 2πl)φk (ξ + 2πl). l∈Zd+1
By Lemma 1.5, there exist bj,k ∈ 1 (Zd+1 ) (j, k = 1, . . . , n) such that matrix (bj,k (ξ))1≤j,k≤n is the inverse of ([φj , φk ](ξ))1≤j,k≤n . Let gj :=
n
bj,k ∗sd φk .
k=1
Lemma 2.4 leads to gj ∈ L∞,∞ (Rd+1 ). Therefore, for 1 ≤ j, k ≤ n, inverse matrix leads to [gj , φk ](ξ) =
n
bj,m (ξ)[φm , φk ](ξ) = δj,k .
m=1
Hence gj , φk (· − α) = (2π)− = (2π)
d+1 2
− d+1 2
Rd+1
k (ξ)eiαξ dξ gj (ξ)φ
[0,2π]d+1 l∈Zd+1
k (ξ + 2πl)eiαξ dξ gj (ξ + 2πl)φ
The Lp,q -stability in Mixed Spaces
13
= (2π)− and for all 1 ≤ j ≤ n, k ∈ Zd+1 , f, gj (· − k) = =
d+1 2
δj,k δ0,α
n l=1
n
al ∗sd φl , gj (· − k)
al (m)φl (x − m), gj (· − k)
l=1 m∈Zd+1
= (2π)−
d+1 2
aj (k).
Similar to the proof of (3.1), one has aj p,q ≤ f Lp,q gj Lp ,q . Let Mj = 1/gj Lp ,q . Then we obtain Mj aj p,q ≤ f Lp,q . Put M = min1≤j≤n {M1 , . . . , Mn }, then M
n
aj p,q ≤ f Lp,q .
j=1
Similar to the the argument of (⇐) in Theorem 1.3, one extends the result for φj ∈ L∞,∞ to the case φj ∈ Lp,q (Rd+1 ) (1 ≤ j ≤ n) by convolving φj with the function ϕ, where ϕ(x) := 2 cp,q ≤ n c ∗sd φj Lp,q . e−π|x| and |x| is the Euclidean norm of x. Thus, one obtains M j=1 Hence the shifts φ(· − k) (k ∈ Zd+1 ) are Lp,q -stable. Acknowledgements We thank the referees very much for elaborate and valuable suggestions which helped to improve this paper. References [1] Benedek, A., Panzone, R.: The space Lp with mixed norm. Duke Math. J., 28, 301–324 (1961) [2] Benedek, A., Calder´ on, A. P., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Nat. Acad. Sci. USA, 48, 356–365 (1962) [3] Duoandikoetxea, J.: Fourier Analysis, American Mathematical Society, Rhode Island, 2000 [4] Folland, G. B.: Real Analysis, John Wiley & Sons, New York, 1999 [5] Fernandez, D. L.: Vector-valued singular integral operators on Lp -spaces with mixed norms and applications. Pac. J. Math., 129(2), 257–275 (1987) [6] Francia, J. L., Ruiz, F. J., Torrea, J. L.: Calder´ on–Zygmund theory for operator-valued kernels. Adv. Math., 62(1), 7–48 (1986) [7] Jia, R. Q., Micchelli, C. A.: Using the refinement equations for the construction of pre-wavelets II: Powers of two. in Curves and Surfaces (P. J. Laurent, A. Le M´ehaut´e, and L. L. Schumaker, eds.), Academic Press, New York, 1991, 209–246 [8] Jia, R. Q., Micchelli, C. A.: On linear independence for integer translates of a finite number of functions. Proc. Edinburgh Math. Soc., 36, 69–85 (1992) [9] Jia, R. Q.: Stability of the shifts of a finite number of functions. J. Approx. Theory, 95(2), 194–202 (1998) [10] Rui, L., Bei, L., Rui, L., et al.: Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue spaces Lp,q (Rd+1 ). J. Math. Anal. Appl., 453, 928–941 (2017) [11] Rudin, W.: Functional Analysis, McGraw Hill Book Company, New York, 1973 [12] Shannon, C.: Communication in the presence of noise. Proc. I.R.E., 37(1), 10–21 (1949) [13] Shannon, C.: Communication in the presence of noise. Proc. IEEE, 72, 1192–1202 (1984) [14] Sun, Q.: Nonuniform average sampling and reconstruction of signals with finite rate of innovation. SIAM J. Math. Anal., 38(5), 1389–1422 (2006) [15] Sun, W., Zhou, X.: Sampling theorem for multiwavelet subspaces. Chinese Sci. Bull., 44(14), 1283–1286 (1999) [16] Torres, R., Ward. E.: Leibniz’s Rule, Sampling and wavelets on mixed Lebesgue spaces. J. Fourier Anal. Appl., 21(5), 1053–1076 (2015)
14
Li R. et al.
[17] Vaidyanathan, P. P., Vrcelj, B.: Biorthogonal partners and applications. IEEE Trans. Signal Process., 49, 1013–1027 (2001) [18] 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) [19] Ward, E.: New estimates in harmonic analysis for mixed Lebesgue spaces, Ph.D. Thesis, University of Kansas, 2010 [20] Zhang, Q., Sun, W.: Invariance of shift invariant spaces. Sci. China Ser. A, 55(7), 1395–1401 (2012) [21] Zhou, X., Sun, W.: On the sampling theorem for wavelet subspaces. J. Fourier Anal. Appl., 5(4), 347–354 (1999)
