The closedness of shift invariant subspaces in \(L^{p,q} (\mathbb{R}^{d+1} )\)

(1)

R E S E A R C H

Open Access

The closedness of shift invariant

subspaces in

L

p,q

(

R

d+1

)

Qingyue Zhang

1*

*_{Correspondence:}

[email protected]

1_{College of Science, Tianjin}

University of Technology, Tianjin, China

Abstract

In this paper, we consider the closedness of shift invariant subspaces inLp,q₍_Rd+1_{). We} first define the shift invariant subspaces generated by the shifts of finite functions in

Lp,q₍_Rd+1_{). Then we give some necessary and suﬃcient conditions for the shift} invariant subspaces inLp,q₍_Rd+1_{) to be closed. Our results improve some known} results in (Aldroubi et al. in J. Fourier Anal. Appl.7:1–21,2001).

MSC: 42C15; 42C40; 41A58

Keywords: Mixed Lebesgue spacesLp,q₍_Rd+1_{); Closedness of shift invariant} subspaces; Shift invariant subspaces

1 Introduction and main result

Lp,q₍_Rd+1_{) (1 <}_p_,_q_{< +}_∞_{) are called mixed Lebesgue spaces which generalize Lebesgue}

spaces [2–6]. They are very important for the study of sampling and equation problems, since we can consider functions to be independent quantities with diﬀerent properties [5–8]. Recently, Torres, Ward, Li, Liu and Zhang studied the sampling theorem on the shift invariant subspaces inLp,q(Rd+1) [6–8]. In this environment, we study the closedness of shift invariant subspaces inLp,q₍_Rd+1_).

The closedness is an expected property for shift invariant subspaces, which is widely considered in the study of shift invariant subspaces. de Boor, DeVore, Ron, Bownik and Shen studied the closedness of shift invariant subspaces inL2₍_Rd_{) [}₉_–₁₁_{]. And Jia,}

Mic-chelli, Aldroubi, Sun and Tang discussed the closedness of shift invariant subspaces in

Lp₍_Rd_{) [}₁_,₁₂_,₁₃_{]. In this paper, we consider the closedness of shift invariant subspaces in} Lp,q(Rd+1).

In order to provide our main result which extends the result in [1], we introduce some deﬁnitions and notations.

The deﬁnition ofLp,q₍_Rd+1_{) is as follows.}

Deﬁnition 1.1 For 1 <p,q< +∞.Lp,q₌_Lp,q₍_Rd+1_{) is made up of all functions}_f _satisfying

fLp,q=

R

Rd

f(x,y)qdy

p

q

dx

1

p

< +∞.

(2)

We deﬁne mixed sequence spacesp,q₍_Zd+1_{) as follows:}

p,q=p,qZd+1 =

c:cp,q=

n∈Z

l∈Zd

_c(n,l)q

p

q1p

< +∞

.

Given a functionf, deﬁne

fLp,q:=

k1∈Z

[0,1]d

k2∈Zd

f(·+k1,x2+k2)

q dx2

1/q

Lp_[0,1]

.

For 1≤p,q≤ ∞, letLp,q =Lp,q(Rd+1) be the linear space of all functionsf for which

fLp,q <∞. The norms are deﬁned above and with usual modiﬁcation in the case of

porq=∞.Lp,q _{is a generalization of}_Lp _{(the deﬁnition of}_Lp _{see [}₁₄_{, Sect. 1]). Clearly,}

for 1≤p,q≤ ∞, one hasL∞,∞_⊂_L∞_and_L∞,∞_⊂_Lp,q_⊂_L1,1_.

Letfˆ(ω) denote the Fourier transform off ∈L1₍_Rd+1_):

ˆ

f(ω) =

Rd+1f(x)e –iωx_dx_.

For a given sequencecand a functionφ,c∗sdφ=k∈Zd+1c(k)φ(·–k) is called

semi-convolution ofcandφ.

Assume thatBis a Banach space. (B)(r)_denotes_r_copies_B_×_B_{× · · · ×}_B_of_B_{. If}_C₌

(c1,c2, . . . ,cr)T∈(B)(r), then one deﬁnes the norm ofCbyC₍_B₎(r)=

r

j=1cjB. WCp,q

(1≤p,q≤ ∞) consists of all distributions whose Fourier coeﬃcients belong to

p,q. Whenp=q= 1,WC1,1becomes the Wiener classWC.

Suppose that = (θ1,θ2, . . . ,θr)T and = (ψ1,ψ2, . . . ,ψs)T are two vector functions

which satisfyθj(ω)ψj(ω) (1≤j≤r, 1≤j≤s) are integrable. One deﬁnes

[,](ω) =

k∈Zd+1

θj(ω+ 2kπ)ψj(ω+ 2kπ)

1≤j≤r,1≤j≤s

.

Remark 1.2 By [14, Theorem 3.1 and Theorem 3.2], [,](ω)∈WC for any , ∈

L∞,∞ _⊂_L∞ _⊂_L2_{. Therefore, for any}_∈_L∞,∞_{, using the continuity of [}_,_](_ω_{) and}

rank[,](ω) =rank((ω+ 2kπ))k∈Zd+1, one obtains, for any n≥0, the set _n={ω: rank((ω+ 2kπ))_k_∈_Zd+1>n}is open.

The following proposition shows that the shift invariant subspaces inLp,q_{(1 <}_p_,_q_<_∞₎

are well deﬁned.

Proposition 1.3([8, Lemma 2.2]) Letθ∈Lp,q,where1 <p,q<∞.Then,for any c∈p,q,

c∗sdθLp,q≤ cp,q_θLp,q.

Deﬁnition 1.4 For= (θ1,θ2, . . . ,θr)T∈(L∞,∞)(r), the multiply generated shift invariant

subspace in the mixed Lebesgue spacesLp,q_{is deﬁned by}

Vp,q() =

_r

j=1

k∈Zd+1

cj(k)θj(·–k) :cj=

cj(k) :k∈Zd+1∈p,q, 1≤j≤r

(3)

The following is our main result.

Theorem 1.5 Assume= (θ1,θ2, . . . ,θr)T∈(L∞,∞)(r)and1 <p,q<∞.Then the following four conditions are equivalent.

(i) Vp,q()is closed inLp,q.

(ii) There exist some positive constantsC1andC2satisfying

C1[,](ω)≤[,](ω)[,](ω)T≤_C2_[_,_](_ω_), ∀_ω∈_[–_π_,_π_]d+1_.

(iii) There exist constantsB1,B2> 0satisfying

B1fLp,q≤ inf

f=r_j₌₁cj∗sdφj

r

j=1

cjp,q≤B2fLp,q, ∀f∈Vp,q().

(iv) There is= (ψ1,ψ2, . . . ,ψr)T∈(L∞,∞)(r)satisfying

f =

r

j=1

k∈Zd+1

f,ψj(·–k)

θj(·–k)

=

r

j=1

k∈Zd+1

f,θj(·–k)

ψj(·–k), ∀f ∈Vp,q().

The paper is organized as follows. In the next section, we give some three useful lemmas and two propositions. In Sect.3, we give the proof of Theorem1.5. Finally, concluding remarks are presented in Sect.4.

2 Some useful lemmas and propositions

In this section, we give three useful lemmas and two propositions which are needed in the proof of Theorem1.5.

Proposition 2.1([1, Lemma 1]) Let∈(L2₎(r)_._{Then the following are equivalent}_:

(i) rank((ω+ 2kπ))_k_∈_Zd+1is a constant for anyω∈Rd+1.

(ii) There exist some positive constantsC1andC2such that

C1[,](ω)≤[,](ω)[,](ω)T_≤_C2_[_,_](_ω_), _∀_ω_∈_[–_π_,_π_]d+1_.

Proposition 2.2([1, Lemma 2]) Let∈(L2₎(r)_satisfy_rank₍₍_ξ_{+ 2}_k_π₎₎

k∈Zd+1=k₀≥1for allξ∈Rd+1_._{Then there exists a ﬁnite index set}_,_η

λ∈[–π,π]d+1, 0 <δλ< 1/4,nonsingular

2π-periodic r×r matrix Pλ(ξ)with all entries in the Wiener class and Kλ⊂Zd+1 with

cardinality(Kλ) =k0for allλ∈,having the following properties:

(i)

[–π,π]d+1⊂

λ∈

B(δλ,δλ/2),

(4)

(ii)

Pλ(ξ)(ξ) =

1,λ(ξ)

2,λ(ξ)

, ξ∈Rd+1andλ∈,

where1,λand2,λare functions fromRd+1toCk0andCr–k0,respectively,satisfying

rank1,λ(ξ+ 2πk) _k_∈_K

λ=k0, ∀ξ∈B(δλ,δλ/2)

and

2,λ(ξ) = 0, ∀ξ∈B(δλ, 8δλ/5) + 2πZd+1.

Furthermore,there exist2π-periodicC∞functions hλ(ξ),λ∈,onRd+1such that

λ∈

hλ(ξ) = 1, ∀ξ∈Rd+1

and

supphλ(ξ)⊂B(δλ,δλ/2) + 2πZd+1.

The following lemma can be proved similarly to [7, Theorem 3.4]. And we leave the details to the interested reader.

Lemma 2.3 Assume that f ∈Lp,q(1 <p,q<∞)and g∈L∞,∞.Then

_R_Rdf(x1,x2)g(x1–k1,x2–k2)dx1dx2:k1∈Z,k2∈Z

d

p,q

≤ fLp,qgL∞,∞.

Lemma 2.4 Let c∈1_._{Then one has}_:

(i) Ifθ∈Lp,q₍_{1 <}_p_,_q_<_∞_),_then

c∗sdθLp,q≤ c1θLp,q.

(ii) Ifθ∈L∞,∞,then

c∗sdθL∞,∞≤ c1θL∞,∞.

Proof (i) By Young’s inequality and the triangle inequality, one has

c∗sdθLp,q =

n∈Z

[0,1]d

l∈Zd

c∗sdθ(·+n,y+l)

q dy

1/q

Lp_[0,1]

=

n∈Z

[0,1]d

l∈Zd

n∈Z

l∈Zd

cn,lθ

·+n–n,y+l–l

q dy

1/q

Lp_[0,1]

≤

n∈Z

[0,1]d

n∈Z

l∈Zd

cn_,lθ·+n–n,y+l–l

q dy

1/q

(5)

≤

n∈Z

[0,1]d

n∈Z

l∈Zd

|cn_,l| l∈Zd

θ·+n–n,y+l

q

dy

1/q

Lp_[0,1]

≤

n∈Z

l∈Zd

|cn,l|

[0,1]d

l∈Zd

θ·+n–n,y+l

q dy

1/q

Lp_[0,1]

≤

n∈Z

l∈Zd

|cn,l|

n∈Z

[0,1]d

l∈Zd

θ·+n–n,y+l

q dy

1/q

Lp_[0,1]

≤

n∈Z

l∈Zd

|cn_,l| n∈Z

[0,1]d

l∈Zd

θ·+n–n,y+l

q dy

1/q

Lp_[0,1]

≤

n∈Z

l∈Zd

|cn_,l| n∈Z

[0,1]d

l∈Zd

θ·+n,y+l

q dy

1/q

Lp_[0,1]

=c1θLp,q.

The desired result (i) in Lemma2.4is obtained.

(ii) The desired result (ii) in Lemma2.4can be found in [8, Lemma 2.4].

Lemma 2.5 Assume thatθ ∈Lp,q _{(1 <}_p_,_q_<_∞₎_and

k∈Zd+1θ(·–k) = 0.Then for any function h onRd+1_satisfying

h(x)≤D1 +|x| –d–2 and h(x) –h(y)≤D|x–y|1 +min|x|,|y| –d–2, (2.1)

one has

lim n→∞2

–n(d+1)

k∈Zd+1

h2–nk θ(·–k)

Lp,q

= 0.

Here D in(2.1)is a positive constant.

Proof Sinceθ∈Lp,q_{, for any}_ε_{> 0, there is}_N0_≥_{2 satisfying}

|l|≥N0

[0,1]d

k∈Zd

θ(·+l,x+k)

q dx

1/q

Lp[0,1]

< (2.2)

and

l∈Z

[0,1]d

k∈Ed_N 0

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

<, (2.3)

whereEd_N0={(k1, . . . ,kd) : there exists some 1≤i0≤dsuch that|ki0|>N0}. Set

θ1(x1, . . . ,xd+1) =θ(x1, . . . ,xd+1)χON₀(x1, . . . ,xd+1)

+

(k1,...,kd+1)∈EdN+1₀

(6)

whereON0=_|_k_i_|≤_N0,1_≤_i_≤_d+1[(k1, . . . ,kd+1) + [0, 1]d+1_{] and}_χ

Sis the characteristic function

ofS.

Thus_k_∈_Zd+1θ1(·–k) =_k_∈_Zd+1θ(·–k) = 0 andθ1–θLp,q< 5. In fact

θ1–θLp,q

=

l∈Z

[0,1]d

k∈Zd

(θ1–θ)(·+l,x+k)

q dx

1/q

Lp_[0,1]

≤

[0,1]d

k∈Zd

(θ1–θ)(·,x+k)

q dx

1/q

Lp[0,1]

+

l=0

[0,1]d

k∈Zd

(θ1–θ)(·+l,x+k)

q dx

1/q

Lp_[0,1]

=I1+I2.

First of all, one treatsI1: by (2.2) and (2.3), one has

I1≤

[0,1]d

₍θ1–θ)(·,x) q dx 1/q + [0,1]d

k=0

(θ1–θ)(·,x+k)

q dx

1/q

Lp_[0,1]

≤

[0,1]d

(θ1–θ)(·,x) q

dx

1/q

Lp_[0,1]

+

[0,1]d

k=0

(θ1–θ)(·,x+k)

q dx

1/q

Lp_[0,1]

≤

[0,1]d

(k1,...,kd+1)∈EdN+10

θ(·+k1, . . . ,xd+1+kd+1)

q dx

1/q

Lp_[0,1]

+

[0,1]d

k∈Ed_N 0

θ(·,x+k)

q dx

1/q

Lp_[0,1]

≤

[0,1]d

|l|>N0,k∈Zd

+

l∈Z,k∈Ed_N 0

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

+

l∈Z

[0,1]d

k∈Ed_N 0

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

≤

[0,1]d

|l|>N0,k∈Zd

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

+

[0,1]d

l∈Z,k∈Ed_N 0

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

(7)

≤ |l|>N0

[0,1]d

k∈Zd

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

+

l∈Z

[0,1]d

k∈Ed_N 0

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

+

<++= 3. Next, one treatsI2:

I2≤

|l|>N0

[0,1]d

k∈Zd

₍θ1–θ)(·+l,x+k)

q dx

1/q

Lp_[0,1]

+

|l|≤N0,l=0

[0,1]d

k∈Zd

(θ1–θ)(·+l,x+k)

q dx

1/q

Lp_[0,1]

=

|l|>N0

[0,1]d

k∈Zd

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

+

|l|≤N0,l=0

[0,1]d

k∈Ed_N 0

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

≤

|l|>N0

[0,1]d

k∈Zd

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

+

l∈Z

[0,1]d

k∈Ed N₀

θ(·+l,x+k)

q dx

1/q

Lp_[0,1]

<+= 2.

Therefore, one hasθ1–θLp,q< 5.

Using Lemma2.4and (2.1), there exists some positive constantCsuch that

2–n(d+1)

k∈Zd+1

h2–nk φ(·–k) –φ1(·–k)

Lp,q

≤2–n(d+1)

k∈Zd+1

h2–nk φ1–φLp,q≤C.

Thus

2–n(d+1)

k∈Zd+1

h2–nk θ1(·–k)

Lp,q

= 2–n(d+1)

j1∈Z

[0,1]d

j2∈Zd

k1∈Z,k2∈Zd

h2–nk1, 2–nk2

×θ1(·+j1–k1,x2+j2–k2)

qdx2

1/q

Lp_[0,1]

= 2–n(d+1)

j1∈Z

[0,1]d

j2∈Zd

k1∈Z,k2∈Zd

h2–nk1, 2–nk2 –h

2–nj1, 2–nj2

(8)

×θ1(·+j1–k1,x2+j2–k2)

qdx2

1/q

Lp_[0,1]

≤2–n(d+2)C1(N0)

j1∈Z

[0,1]d

j2∈Zd

k1∈Z,k2∈Zd

1 + 2–n(k1,k2) –(d+2)

×θ1(·+j1–k1,x2+j2–k2)

q dx2

1/q

Lp_[0,1]

= 2–n(d+2)C1(N0)

j1∈Z

[0,1]d

k1∈Z

k2∈Zd

1 + 2–n(k1,k2) –(d+2)

×

j2∈Zd

θ1(·+j1–k1,x2+j2)

q dx2

1/q

Lp[0,1]

≤2–n(d+2)C1(N0)

k1∈Z,k2∈Zd

1 + 2–n(k1,k2) –(d+2)

×

j1∈Z

[0,1]d

j2∈Zd

θ1(·+j1–k1,x2+j2)

q dx2

1/q

Lp_[0,1]

≤2–nC2(N0)

j1∈Z

[0,1]d

j2∈Zd

θ1(·+j1,x2+j2)

q dx2

1/q

Lp_[0,1]

= 2–n_C2₍_N0₎_θ

1Lp,q≤2–nC2(N0)θLp,q+ 5 .

HereCi(N0) (i= 1, 2) are positive constants depending only onN0andd. This completes

the proof.

3 Proof of Theorem1.5

In this section, we give the proof of Theorem1.5. The main steps of the proof are as follows: (iv)⇒(iii)⇒(i)⇒(ii)⇒(iv).

(iv)⇒(iii):

Letf =r_j=1_k_∈_Zd+1f,ψj(·–k)θj(·–k). Then, by Lemma2.3, one has

inf f=r_j₌₁cj∗sdφj

r

j=1

cjp,q≤

r

j=1

f,ψj(·–k1,·–k2)

:k1∈Z,k2∈Zdp,q

≤

r

j=1

fLp,qψjL∞,∞=fLp,q r

j=1

ψjL∞,∞.

Conversely, iff =r_j=1cj∗sdθj, then, by Proposition1.3and the triangle inequality

fLp,q =

r

j=1 cj∗sdθj

Lp,q

≤

r

j=1

cj∗sdθjLp,q

≤

r

j=1

cjp,qθ_jLp,q≤max

1≤j≤rθjL

p,q

r

j=1

(9)

Taking the inﬁmum for (3.1), one gets

fLp,q≤max

1≤j≤rθjL

p,q inf

f=r_j₌₁cj∗sdθj

r

j=1

cjp,q.

LetB1= 1/max1≤j≤rθjLp,q andB2=r

j=1ψjL∞,∞. Then one has

B1fLp,q≤ inf

f=r_j₌₁cj∗sdφj

r

j=1

cjp,q≤B2fLp,q, ∀f∈Vp,q().

(iii)⇒(i):

For convenience, letT: (p,q)(r)_→_V

p,q() be a mapping which is deﬁned by

TC=

r

j=1

cj∗sdθj, C= (c1,c2, . . . ,cr)T∈

p,q (r),

and letfinf=inff=r_j₌₁cj∗sdθj

r

j=1cjp,q. Then, obviously, · infis a norm. Assumefn⊂ Ran(T) (n≥1) is a Cauchy sequence. HereRan(T) denotes the range ofT. Without loss of generality, letfn–fn–1inf< 2–n. Using the deﬁnition of · inf, there isCn∈(p,q)(r)

(n≥2) such thatTCn=fn–fn–1andCn₍p,q₎(r)< 2–nfor anyn≥2. By the completeness

of (p,q)(r)_and∞

n=2Cn(p,q₎(r)<∞, one hasZ=∞_n=2C_n∈(p,q)(r)andf1+TZ∈Ran(T).

Note thatTCinf≤ C(p,q)(r)for anyC∈(p,q)(r). One has

fn–f1–TZinf=

T

_∞

k=n+1 Ck

inf

≤ ∞

k=n+1 Ck

(p,q₎(r)

≤ ∞

k=n+1

Ck₍p,q₎(r)→0,

whenn→ ∞. Therefore,Ran(T) is closed. SinceVp,q() =Ran(T), one sees thatVp,q() is closed.

(i)⇒(ii):

Similarly to [1, Proof of (i)⇒(iii)], one can prove (i)⇒(ii) by usingL∞,∞_⊂_L∞_{, and}

substitutingLp,q_,_L∞,∞_{, Proposition}_2.1_{and Lemma}_2.5_for_Lp_,_L∞_{, Lemma 1 and Lemma 3}

in [1], respectively. (ii)⇒(iv):

Assume thathλ(ω),Pλ(ω) and1,λ(ω) are as in Proposition2.2. Deﬁne

Dλ(ω) =Pλ(ω)T

[1,λ,1,λ](ω)–1 0

0 I

Pλ(ω)Hλ(ω). (3.2)

HereHλ(ω) is a function with period 2πwhich satisﬁessuppHλ⊂B(ηλ,δλ) + 2πZd+1and

Hλ(ω) = 1 onsupphλ. ThusDλ∈(WC)(r×r). Let= (ψ1,ψ2, . . . ,ψr)Tbe deﬁned by

(ω) =

λ∈

hλ(ω)Dλ(ξ)(ω). (3.3)

Then, by Lemma 2.4, one has ∈L∞,∞. For any f ∈Vp,q(), using the deﬁnition of

(10)

A(ω)T₍_ω_{). Putting}

g=

r

j=1

k∈Zd+1

f,ψj(·–k)

θj(·–k).

By the periodicity ofhλ(ω) andDλ(ω), (3.2), (3.3) and Proposition2.2, one has ˆ

g(ω) =A(ω)T[,](ω)(ω) =

λ∈

A(ω)TPλ(ω)–1

×

[1,λ,1,λ](ω) 0

0 0

[1,λ,1,λ](ω)–1 0

0 I

1,λ(ω)

0

hλ(ω)

=

λ∈

A(ω)TPλ(ω)–1

1,λ(ω)

0

hλ(ω)

=

λ∈

A(ω)TPλ(ω)–1Pλ(ω)(ω)hλ(ω)

=

λ∈

A(ω)T(ω)hλ(ω)

=A(ω)T(ω) =ˆf(ω).

Thusfˆ(ω) =gˆ(ω). Thereforef=g, namely

f =

r

j=1

k∈Zd

f,ψj(·–k)

θj(·–k).

Similar arguments show that

f =

r

j=1

k∈Zd

f,θj(·–k)

ψj(·–k).

4 Concluding remarks

In this paper, we study the closedness of shift invariant subspaces inLp,q₍_Rd+1_{). We ﬁrst}

deﬁne the shift invariant subspaces generated by the shifts of ﬁnite functions inLp,q₍_Rd+1_).

Then we give some necessary and suﬃcient conditions for the shift invariant subspaces in

Lp,q₍_Rd+1_{) to be closed.}

However, in this paper, we only consider the closedness of shift invariant subspace of Lp,q₍_Rd+1_{). Studying the}_Lp,q_{-frames in a shift invariant subspace of mixed Lebesgue} Lp,q₍_Rd_{) is the goal of future work.}

Funding

(11)

Competing interests

The author declares that he has no competing interests.

Authors’ contributions

QZ provided the questions and gave the proof for the main result. He read and approved the manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 27 February 2018 Accepted: 23 May 2018

References

1. Aldroubi, A., Sun, Q., Tang, W.:p-frames and shift invariant subspaces ofLp_{. J. Fourier Anal. Appl.}₇_{(1), 1–21 (2001)} 2. Benedek, A., Panzone, R.: The spaceLp_{with mixed norm. Duke Math. J.}₂₈_{(3), 301–324 (1961)}

3. Benedek, A., Calderón, A.P., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Natl. Acad. Sci. USA48(3), 356–365 (1962)

4. Francia, J.L., Ruiz, F.J., Torrea, J.L.: Calderón–Zygmund theory for operator-valued kernels. Adv. Math.62(1), 7–48 (1986) 5. Fernandez, D.L.: Vector-valued singular integral operators onLp_{-spaces with mixed norms and applications. Pac. J.}

Math.129(2), 257–275 (1987)

6. Torres, R., Ward, E.: Leibniz’s rule, sampling and wavelets on mixed Lebesgue spaces. J. Fourier Anal. Appl.21(5), 1053–1076 (2015)

7. Li, R., Liu, B., Liu, R., Zhang, Q.: Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue spaces Lp,q₍_Rd+1_{). J. Math. Anal. Appl.}₄₅₃_{(2), 928–941 (2017)}

8. Li, R., Liu, B., Liu, R., Zhang, Q.: TheLp,q_{-stability of the shifts of ﬁnitely many functions in mixed Lebesgue space} Lp,q₍_Rd+1_{). Acta Math. Sin. (2018).}_{https://doi.org/10.1007/s10114-018-7333-1}

9. de Boor, C., DeVore, R.A., Ron, A.: The structure of ﬁnitely generated shift-invariant spaces inL2(Rd). J. Funct. Anal. 119(1), 37–78 (1994)

10. Bownik, M.: The structure of shift-invariant subspaces ofL2(Rn). J. Funct. Anal.177(2), 282–309 (2000)

11. Ron, A., Shen, Z.: Frames and stable bases for shift-invariant subspaces ofL2(R). Can. J. Math.47(5), 1051–1094 (1995)

12. Jia, R.Q., Micchelli, C.A.: On linear independence for integer translates of a ﬁnite number of functions. Proc. Edinb. Math. Soc.36(1), 69–85 (1992)

13. Jia, R.Q.: Stability of the shifts of a ﬁnite number of functions. J. Approx. Theory95(2), 194–202 (1998)