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

*Correspondence: jczhangqingyue@mail.nankai.edu.cn 1College of Science, Tianjin University of Technology, Tianjin, China Abstract 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). MSC: 42C15; 42C40; 41A58

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 L 2 (R d ) [9][10][11]. And Jia, Micchelli, Aldroubi, Sun and Tang discussed the closedness of shift invariant subspaces in L p (R d ) [1,12,13]. In this paper, we consider the closedness of shift invariant subspaces in L p,q (R d+1 ).
In order to provide our main result which extends the result in [1], we introduce some definitions and notations.
The definition of L p,q (R d+1 ) is as follows. We define mixed sequence spaces p,q (Z d+1 ) as follows: Given a function f , define 1] .
The norms are defined above and with usual modification in the case of p or q = ∞. L p,q is a generalization of L p (the definition of L p see [14,Sect. 1]). Clearly, For a given sequence c and a function φ, consists of all distributions whose Fourier coefficients belong to p,q . When p = q = 1, WC 1,1 becomes the Wiener class WC.
The following proposition shows that the shift invariant subspaces in L p,q (1 < p, q < ∞) are well defined.
(i) V p,q ( ) is closed in L p,q .
(ii) There exist some positive constants C 1 and C 2 satisfying (iii) There exist constants B 1 , B 2 > 0 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 ∈ (L 2 ) (r) . Then the following are equivalent: (ii) There exist some positive constants C 1 and C 2 such that Then there exists a finite index set , η λ ∈ [-π, π] d+1 , 0 < δ λ < 1/4, nonsingular 2π -periodic r × r matrix P λ (ξ ) with all entries in the Wiener class and K λ ⊂ Z d+1 with cardinality(K λ ) = k 0 for all λ ∈ , having the following properties: where B(x 0 , δ) denotes the open ball in R d+1 with center x 0 and radius δ; where 1,λ and 2,λ are functions from R d+1 to C k 0 and C r-k 0 , respectively, satisfying The following lemma can be proved similarly to [7,Theorem 3.4]. And we leave the details to the interested reader.

Lemma 2.4
Let c ∈ 1 . Then one has: Proof (i) By Young's inequality and the triangle inequality, one has

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: (iv) ⇒ (iii): Conversely, if f = r j=1 c j * sd θ j , then, by Proposition 1.3 and the triangle inequality Taking the infimum for (3.1), one gets Let B 1 = 1/ max 1≤j≤r θ j L p,q and B 2 = r j=1 ψ j L ∞,∞ . Then one has For convenience, let T : ( p,q ) (r) → V p,q ( ) be a mapping which is defined by and let f inf = inf f = r j=1 c j * sd θ j r j=1 c j p,q . Then, obviously, · inf is a norm. Assume f n ⊂ Ran(T) (n ≥ 1) is a Cauchy sequence. Here Ran(T) denotes the range of T. Without loss of generality, let f nf n-1 inf < 2 -n . Using the definition of · inf , there is C n ∈ ( p,q ) (r) (n ≥ 2) such that TC n = f nf n-1 and C n ( p,q ) (r) < 2 -n for any n ≥ 2. By the completeness of ( p,q ) (r) and ∞ n=2 C n ( p,q ) (r) < ∞, one has Z = ∞ n=2 C n ∈ ( p,q ) (r) and f 1 + TZ ∈ Ran(T). Note that TC inf ≤ C ( p,q ) (r) for any C ∈ ( p,q ) (r) . One has when n → ∞. Therefore, Ran(T) is closed. Since V p,q ( ) = Ran(T), one sees that V p,q ( ) is closed.

Concluding remarks
In this paper, we study the closedness of shift invariant subspaces in L p,q (R d+1 ). We first define the shift invariant subspaces generated by the shifts of finite functions in L p,q (R d+1 ). Then we give some necessary and sufficient conditions for the shift invariant subspaces in L p,q (R d+1 ) to be closed. However, in this paper, we only consider the closedness of shift invariant subspace of L p,q (R d+1 ). Studying the L p,q -frames in a shift invariant subspace of mixed Lebesgue L p,q (R d ) is the goal of future work.

Funding
This work was supported partially by the National Natural Science Foundation of China under Grants Nos. 11371200, 11326094 and 11401435. This work was also partially supported by the Program for Visiting Scholars at the Chern Institute of Mathematics.