Construction of frames for shift-invariant spaces

We construct a sequence ${\phi_i(\cdot-j)\mid j\in{\ZZ}, i=1,...,r}$ which constitutes a $p$-frame for the weighted shift-invariant space [V^p_{\mu}(\Phi)=\Big{\sum\limits_{i=1}^r\sum\limits_{j\in{\mathbb{Z}}}c_i(j)\phi_i(\cdot-j) \Big| {c_i(j)}_{j\in{\mathbb{Z}}}\in\ell^p_{\mu}, i=1,...,r\Big}, p\in[1,\infty],] and generates a closed shift-invariant subspace of $L^p_\mu(\mathbb{R})$. The first construction is obtained by choosing functions $\phi_i$, $i=1,...,r$, with compactly supported Fourier transforms $\hat{\phi}_i$, $i=1,...,r$. The second construction, with compactly supported $\phi_i,i=1,...,r,$ gives the Riesz basis.


Introduction and preliminaries
The shift-invariant spaces V p µ (Φ), p ∈ [1, ∞], quoted in the abstract, are used in the wavelet analysis, approximation theory, sampling theory, etc. They have been extensively studied in recent years by many authors [2]- [19]). The aim of this paper is to construct V p µ (Φ), p ∈ [1, ∞], spaces with specially chosen functions φ i , i = 1, . . . , r, which generate its p-frame. These results expand and correct the construction obtained in [20]. For the first construction, we take functions φ i , i = 1, . . . , r, so that the Fourier transforms are compactly supported smooth functions. Also, we derive the conditions for the collection {φ i (· − k) | k ∈ Z, i = 1, . . . , r} to form a Riesz basis for V p µ (Φ). We note that the properties of the constructed frame guarantee the feasibility of a stable and continuous reconstruction algorithm in V p µ (Φ) [22]. We generalize these results for a shift-invariant subspace of L p µ (R d ). The second construction is obtained by choosing compactly supported functions φ i , i = 1, . . . , r. In this way, we obtain the Riesz basis.
This paper is organized as follows. In Section 2 we quote some basic properties of certain subspaces of the weighted L p and ℓ p spaces. In Section 3 we derive the conditions for the functions of the form φ i (ξ) = θ(ξ + k i π), k i ∈ Z, i = 1, 2, ..., r, r ∈ N, to form a Riesz basis for V p µ (Φ). We also show that using functions of the form φ i (·) = θ(· + iπ), i = 1, . . . , r, where θ is compactly supported smooth function who's length of support is less than or equal to 2π we can not construct a p-frame for the shift-invariant space V p µ (Φ). In Section 4 we construct a sequence {φ i (· − j) | j ∈ Z d , i = 0, . . . , r}, where r ∈ 2N or r ∈ 3N, which constitutes a p-frame for the weighted shift-invariant space V p µ (Φ). Our construction shows that the sampling and reconstruction problem in the shiftinvariant spaces is robust in the sense of [2]. In Section 5 we construct p-Riesz basis by using compactly supported functions φ i , i = 1, . . . , r.
. Then rank A = rank AA T . We will recall some results from [2] and [20] which are needed in the sequel.
). The following statements are equivalent.
3) There exists a positive constant C independent of ξ such that The next theorem ( [20]) derives necessary and sufficient conditions for an indexed family {φ i (· − j) | j ∈ Z d , i = 1, . . . , r} to constitute a p-frame for V p µ (Φ), which is equivalent with the closedness of this space in L p µ . Thus, it is shown that under appropriate conditions on the frame vectors, there is an equivalence between the concept of p-frames, Banach frames and the closedness of the space they generate.
, and let µ be ω-moderate. The following statements are equivalent.
iv) There exist positive constants C 1 and C 2 (depend on Φ and ω) such that iii) If (2.2) holds for p 0 , then it holds for any p ∈ [1, ∞].

Construction of frames using a compactly supported smooth function
Considering the length of the support of a function θ and defining a function Φ in an appropriate way using θ, we have different cases for the rank of the matrix [ Φ(ξ + 2jπ)] j∈Z described in Theorem 3.1.
First, we consider the next claim: Then the rank of the matrix [ Φ(ξ + 2jπ)] j∈Z is not a constant function on R and it depends on ξ ∈ R.
j∈Z is a constant function on R and equals r.
The rank of the matrix [ Φ(ξ + 2jπ)] j∈Z is a constant function on R and equals 2.
Proof. We have the next two cases.
Proof. In the same way, as in the proof of the Lemma 3.1, we have four different cases for the matrix [ Φ(ξ + 2jπ)] j∈Z . Without losing generality, let we suppose that k 1 ∈ 2Z.
Proof of Theorem 3.1 1) Using Lemma 3.1 and Lemma 3.2, it is obvious that if |k 2 − k 1 | = 2 and |k i − k j | ≥ 2 for different i, j ≤ r, then the position of the first non-zero element in each row of the matrix [ Φ(ξ + 2jπ)] j∈Z is unique for each row. So, the rank of the matrix [ Φ(ξ + 2jπ)] j∈Z is a constant function on R and equals r for all ξ ∈ R.
As a consequence of Theorem 2.1 and Theorem 3.1 1), we have the following result.

Construction of frames using several compactly supported smooth functions
Firstly, we consider two smooth functions with proper compact supports.
Using functions θ and ψ from Lemma 4.1, in the next lemma we construct the p-frame with four appropriate functions.
The rank of the matrix [ Φ(ξ + 2jπ)] j∈Z is a constant function on R and equals 2.
Proof. The proof is similar to the proof of Lemma 4.1.
We conclude that the rank of the matrix [ Φ(ξ+2jπ)] j∈Z is a constant function on R and equals 2.
The following statements hold.
Now we consider three functions with compact supports. Lemma 4.3. Let the function θ satisfies all the conditions of Lemma 4.1, and let τ ∈ C ∞ 0 (R) and ω ∈ C ∞ 0 (R) be positive functions such that Then the rank of the matrix [ Φ(ξ +2jπ)] j∈Z is a constant function on R and equals 2.
Proof. We have four different forms for the matrix [ Φ(ξ + 2jπ)] j∈Z and in all cases the rank of the matrix is 2. Now we will show all possible cases. Denote with a i , i = 1, 2, 3 and b i , i = 1, 2, some positive values.
We obtain Continuing in this manner, for all n ∈ N, we have φ n (x) = 1 a n (n − 1)!
Calculating the Fourier transform of functions φ n , n ∈ N, we get Continuing in this manner, we obtain φ n (ξ) = (−i) n a n v.p. 1 ξ n (e iaξ − 1) n , n ∈ N, where v.p. denotes the principal value.
(1) We refer to [4] and [22] for the γ-dense set X = {x j | j ∈ J}. Let φ k (x) = F −1 (θ(· − kπ))(x), x ∈ R. Following the notation of [22], we put ψ xj = φ xj where {x j | j ∈ J} is γ-dense set determined by f ∈ V 2 (φ) = V 2 (F −1 (θ)). Theorems 3.1, 3.2 and 4.1 in [22] give the conditions and explicit form of C p > 0 and c p > 0 such that the inequality ≤ C p f L p µ holds. This inequality guarantee the feasibility of a stable and continuous reconstruction algorithm in the signal spaces V p µ (Φ) ( [22]). (2) Since the spectrum of the Gram matrix [ Φ, Φ](ξ), where Φ is defined in Theorem 5.1, is bounded and bounded away from zero (see [8]), it follows that the family {Φ(· − j) | j ∈ Z} forms a p-Riesz basis for V p µ (Φ). (3) Frames of the above sections may be useful in applications since they satisfy assumptions of Theorem 3.1 and Theorem 3.2 in [5]. They show that error analysis for sampling and reconstruction can be tolerated, or that the sampling and reconstruction problem in shift-invariant space is robust with respect to appropriate set of functions φ k1 , . . . , φ kr .