The closedness of shift invariant subspaces in Lp,q(Rd+1).

In this paper, we consider the closedness of shift invariant subspaces in Lp,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 in Lp,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).

Introduction and main result

() 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 in [6-8]. In this environment, we study the closedness of shift invariant subspaces in .

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 in [9-11]. And Jia, Micchelli, Aldroubi, Sun and Tang discussed the closedness of shift invariant subspaces in [1, 12, 13]. In this paper, we consider the closedness of shift invariant subspaces in .

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

The definition of is as follows.

Definition 1.1

For . is made up of all functions f satisfying

We define mixed sequence spaces as follows:

Given a function f, define For , let be the linear space of all functions f for which . The norms are defined above and with usual modification in the case of . is a generalization of (the definition of see [14, Sect. 1]). Clearly, for , one has and .

Let denote the Fourier transform of :

For a given sequence c and a function ϕ, is called semi-convolution of c and ϕ.

Assume that is a Banach space. denotes r copies of . If , then one defines the norm of C by .

() consists of all distributions whose Fourier coefficients belong to . When , becomes the Wiener class .

Suppose that and are two vector functions which satisfy (, ) are integrable. One defines

Remark 1.2

By [14, Theorem 3.1 and Theorem 3.2], for any . Therefore, for any , using the continuity of and , one obtains, for any , the set is open.

The following proposition shows that the shift invariant subspaces in () are well defined.

Proposition 1.3

([8, Lemma 2.2])

Let , where . Then, for any ,

Definition 1.4

For , the multiply generated shift invariant subspace in the mixed Lebesgue spaces is defined by

The following is our main result.

Theorem 1.5

Assume and . Then the following four conditions are equivalent.

is closed in .

There exist some positive constants and satisfying

There exist constants satisfying

There is satisfying

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 Theorem 1.5. Finally, concluding remarks are presented in Sect. 4.

Some useful lemmas and propositions

In this section, we give three useful lemmas and two propositions which are needed in the proof of Theorem 1.5.

Proposition 2.1

([1, Lemma 1])

Let . Then the following are equivalent:

is a constant for any .

There exist some positive constants and such that

Proposition 2.2

([1, Lemma 2])

Let satisfy for all . Then there exists a finite index set Λ, , , nonsingular 2π-periodic matrix with all entries in the Wiener class and with for all , having the following properties: Furthermore, there exist 2π-periodic functions , , on such that and

where denotes the open ball in with center and radius δ;

where and are functions from to and , respectively, satisfying and

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 () and . Then

Lemma 2.4

Let . Then one has:

If (), then

If , then

Proof

(i) By Young’s inequality and the triangle inequality, one has The desired result (i) in Lemma 2.4 is obtained.

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

Lemma 2.5

Assume that () and . Then for any function h on satisfying one has Here D in (2.1) is a positive constant.

Since , for any , there is satisfying and where .

Set where and is the characteristic function of S.

Thus and . In fact

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

Next, one treats : Therefore, one has .

Using Lemma 2.4 and (2.1), there exists some positive constant C such that Thus Here () are positive constants depending only on and d. This completes the proof. □

Proof of Theorem 1.5

In this section, we give the proof of Theorem 1.5. The main steps of the proof are as follows: .

:

Let . Then, by Lemma 2.3, one has

Conversely, if , then, by Proposition 1.3 and the triangle inequality Taking the infimum for (3.1), one gets Let and . Then one has

:

For convenience, let be a mapping which is defined by and let . Then, obviously, is a norm. Assume () is a Cauchy sequence. Here denotes the range of T. Without loss of generality, let . Using the definition of , there is () such that and for any . By the completeness of and , one has and . Note that for any . One has when . Therefore, is closed. Since , one sees that is closed.

:

Similarly to [1, Proof of ], one can prove by using , and substituting , , Proposition 2.1 and Lemma 2.5 for , , Lemma 1 and Lemma 3 in [1], respectively.

:

Assume that , and are as in Proposition 2.2. Define Here is a function with period 2π which satisfies and on . Thus . Let be defined by Then, by Lemma 2.4, one has . For any , using the definition of , there exists a distribution with period 2π which satisfies . Putting By the periodicity of and , (3.2), (3.3) and Proposition 2.2, one has Thus . Therefore , namely Similar arguments show that

Concluding remarks

In this paper, we study the closedness of shift invariant subspaces in . We first define the shift invariant subspaces generated by the shifts of finite functions in . Then we give some necessary and sufficient conditions for the shift invariant subspaces in to be closed.

However, in this paper, we only consider the closedness of shift invariant subspace of . Studying the -frames in a shift invariant subspace of mixed Lebesgue is the goal of future work.