# Construction of Frames for Shift-Invariant Spaces

Authors: Stevan Pilipović; Suzana Simić
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We construct a sequence{ϕi(·-j)∣j∈ℤ,i=1,…,r} which constitutes a p-frame for the weighted shift-invariant space Vμp(Φ)={∑i=1r∑j∈ℤci(j)ϕi(·-j)∣{ci(j)}j∈ℤ∈ℓμp,i=1,…,r},
p∈[1,∞], and generates a closed shift-invariant subspace of Lμp(ℝ). The first construction is obtained by choosing functions ϕi, i=1,…,r, with compactly supported Fourier transforms ϕ^i, i=1,…,r. The second construction, with compactly supported ϕi,
i=1,…,r, gives the Riesz basis.


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## Body

## 1. Introduction and Preliminaries

The shift-invariant spacesVμp(Φ), p∈[1,∞], quoted in the abstract, are used in the wavelet analysis, approximation theory, sampling theory, and so forth. They have been extensively studied by many authors [1–18]. The aim of this paper is to construct Vμp(Φ), p∈[1,∞], spaces with specially chosen functions ϕi, i=1,…,r, which generate its p-frame. These results extend and correct the construction obtained in [19]. For the first construction, we take functions ϕi, i=1,…,r, so that the Fourier transforms are compactly supported smooth functions. Also, we derive conditions for the collection {ϕi(·-j)∣j∈ℤ,i=1,…,r} to form a Riesz basis for Vμp(Φ). We note that the properties of the constructed frame guarantee the feasibility of a stable and continuous reconstruction algorithm in Vμp(Φ) [20]. We generalize these results for a shift-invariant subspace of Lμp(ℝd). The second construction is obtained by choosing compactly supported functions ϕi, i=1,…,r. In this way, we obtain the Riesz basis.This paper is organized as follows. In Section2 we quote some basic properties of certain subspaces of the weighted Lp and ℓp spaces. In Section 3 we derive conditions for functions of the form ϕ^i(·)=θ(·+kiπ), ki∈ℤ, i=1,2,…,r, r∈ℕ, to form a Riesz basis for Vμp(Φ). We also show that using functions of the form ϕ^i(·)=θ(·+iπ), i=1,…,r, where θ is compactly supported smooth function whose length of support is less than or equal to 2π, we cannot construct a p-frame for the shift-invariant space Vμp(Φ). In Section 4 we construct a sequence {ϕi(·-j)∣j∈ℤ,i=0,…,r}, where r∈2ℕ or r∈3ℕ, which constitutes a p-frame for the weighted shift-invariant space Vμp(Φ). Our construction shows that the sampling and reconstruction problem in the shift-invariant spaces is robust in the sense of [1]. In Section 5 we construct p-Riesz basis by using compactly supported functions ϕi, i=1,…,r.


## 2. Basic Spaces

Let a functionω be nonnegative, continuous, symmetric, and submultiplicative; that is, ω(x+y)≤ω(x)ω(y), x,y∈ℝd; let a function μ be ω-moderate; that is, μ(x+y)≤Cω(x)μ(y), x,y∈ℝd. Functions μ and ω are called weights. We consider the weighted function spaces Lμp and the weighted sequence spaces ℓμp(ℤd) with ω-moderate weights μ (see [19]). Let p∈[1,∞). Then (with obvious modification for p=∞)
(1)ℒωp={(∫[0,1]d‍(∑j∈ℤd‍|f(x+j)|ω(x+j))pdx)1/pf∣∥f∥ℒωp=(∫[0,1]d(∑j∈ℤd‍|f(x+j)|ω(x+j))pdx)1/p<+∞(∫[0,1]d‍(∑j∈ℤd‍|f(x+j)|ω(x+j))pdx)1/p},Wωp:={(∑j∈ℤd‍supx∈[0,1]d|f(x+j)|pω(j)p)1pf∣∥f∥Wωp=(∑j∈ℤd‍supx∈[0,1]d|f(x+j)|pω(j)p)1/p<+∞{(∑j∈ℤd‍supx∈[0,1]d|f(x+j)|pω(j)p)1pf∣∥f∥Wωp}.In what follows, we use the notationΦ=(ϕ1,…,ϕr)T. Define ∥Φ∥ℋ=∑i=1r∥ϕi∥ℋ, where ℋ=Lωp, ℒωp or Wωp, p∈[1,∞]. With ℱϕ=ϕ^ we denote the Fourier transform of the function ϕ; that is, ϕ^(ξ)=∫ℝd‍ϕ(x)e-iπx·ξdx, ξ∈ℝd.Letc={ci}i∈ℕ∈ℓμp and f,g∈Lωp, p∈[1,∞]. We define, as in [1], the semiconvolution f*′c as (f*′c)(x)=∑j∈ℤdcjf(x-j), x∈ℝd, and 〈f,g〉=∫ℝdf(x)g(x)¯dx.The concept of ap-frame is introduced in [1].It is said that a collection{ϕi(·-j)∣j∈ℤd,i=1,…,r} is a p-frame for Vμp(Φ) if there exists a positive constant C (dependent upon Φ, p, and ω) such that
(2)C-1∥f∥Lμp≤∑i=1r∥{∫ℝdf(x)ϕi(x-j)¯dx}j∈ℤd∥ℓμp≤C∥f∥Lμp,f∈Vμp(Φ).Recall [21] that the shift-invariant spaces are defined by
(3)Vμp(Φ):={f∈Lμp∣f(·)=∑i=1r‍∑j∈ℤd‍cjiϕi(·-j),{cji}j∈ℤd∈ℓμp,i=1,…,r{f∈Lμp∣f(·)=∑i=1r‍∑j∈ℤd‍cjiϕi(·-j),}.Remark 1 (see [22]).
LetΦ∈Wω1 and let μ be ω-moderate. Then Vμp(Φ) is a subspace (not necessarily closed) of Lμp and Wμp for any p∈[1,∞]. Clearly (2) implies that ℓμp and Vμp(Φ) are isomorphic Banach spaces.LetΦ=(ϕ1,…,ϕr)T. Let
(4)[Φ^,Φ^](ξ)=[∑k∈ℤd‍ϕ^i(ξ+2kπ)ϕ^j(ξ+2kπ)¯]1≤i≤r,1≤j≤r,
where we assume that ϕi^(ξ)ϕ^j(ξ)¯ is integrable for any 1≤i, j≤r. Let A=[a(j)]j∈ℤd be an r×∞ matrix and AAT¯=[∑j∈ℤdai(j)ai′(j)¯]1≤i,i′≤r. Then rankA=rankAAT¯.We recall results from [1, 19] which are needed in the sequel.Lemma 2 (see [1]).
The following statements are equivalent.(1)
rank[Φ^(ξ+2jπ)]j∈ℤd is a constant function on ℝd.(2)
rank[Φ^,Φ^](ξ) is a constant function on ℝd.(3)
There exists a positive constantC independent of ξ such that
(5)C-1[Φ^,Φ^](ξ)≤[Φ^,Φ^](ξ)[Φ^,Φ^](ξ)T¯≤C[Φ^,Φ^](ξ),ξ∈[-π,π]d.The next theorem [19] derives necessary and sufficient conditions for an indexed family {ϕi(·-j)∣j∈ℤd,i=1,…,r} to constitute a p-frame for Vμp(Φ), which is equivalent with the closedness of this space in Lμp. 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.Theorem 3 (see [19]).
LetΦ=(ϕ1,…,ϕr)T∈(Wω1)r, p0∈[1,∞], and let μ be ω-moderate. The following statements are equivalent. (i)
Vμp0(Φ) is closed in Lμp0.(ii)
{ϕi(·-j)∣j∈ℤd,i=1,…,r} is a p0-frame for Vμp0(Φ).(iii)
There exists a positive constantC such that
(6)C-1[Φ^,Φ^](ξ)≤[Φ^,Φ^](ξ)[Φ^,Φ^](ξ)T¯≤C[Φ^,Φ^](ξ),ξ∈[-π,π]d.(iv)
There exist positive constantsC1 and C2 (depending on Φ and ω) such that
(7)C1∥f∥Lμp0≤inff=∑i=1rϕi*′ci∑i=1r‍∥{cji}j∈ℤd∥ℓμp0≤C2∥f∥Lμp0,f∈Vμp0(Φ).(v)
There existsΨ=(ψ1,…,ψr)T∈(Wω1)r, such that
(8)f=∑i=1r‍∑j∈ℤd‍〈f,ψi(·-j)〉ϕi(·-j)=∑i=1r‍∑j∈ℤd‍〈f,ϕi(·-j)〉ψi(·-j),f∈Vμp0(Φ).Corollary 4 (see [19]).
LetΦ=(ϕ1,…,ϕr)T∈(Wω1)r, p0∈[1,∞], and let μ be ω-moderate. (i)
If{ϕi(·-j)∣j∈ℤd,i=1,…,r} is a p0-frame for Vμp0(Φ), then the collection {ϕi(·-j)∣j∈ℤd,i=1,…,r} is a p-frame for Vμp(Φ) for any p∈[1,∞].(ii)
IfVμp0(Φ) is closed in Lμp0 and Wμp0, then Vμp(Φ) is closed in Lμp and Wμp for any p∈[1,∞].(iii)
If (7) holds for p0, then it holds for any p∈[1,∞].


## 3. Construction of Frames Using a Band-Limited Function

Considering the length of the support of a functionθ, we have different cases for the rank of matrix [Φ^(ξ+2jπ)]j∈ℤ.First, we consider the next claim.Letθ∈C0∞(ℝ) be a nonnegative function such that θ(x)>0, x∈(-π,π), and suppθ⊆[-π,π]. Moreover, let
(9)ϕ^k(ξ)=θ(ξ+kπ),k∈ℤ,
and Φ=(ϕi,ϕi+1,…,ϕi+r)T, i∈ℤ, r∈ℕ.Then the rank of matrix[Φ^(ξ+2jπ)]j∈ℤ is not a constant function on ℝ and it depends on ξ∈ℝ.As a matter of fact, by the Paley-Wiener theorem,ϕi∈𝒮(ℝ)⊂Wμ1(ℝ), i∈ℤ. For any i∈ℤ, matrix [ϕ^i,ϕ^i](ξ)=∑j∈ℤ|θ(ξ+iπ+2jπ)|2, ξ∈ℝ, has the rank 0 or 1, depending on ξ. Moreover, we have [ϕ^2i,ϕ^2i](π)=0 and [ϕ^2i,ϕ^2i](0)>0. Because of that, the rank of the matrix [Φ^(ξ+2jπ)]j∈ℤ is not a constant function on ℝ and it depends on ξ∈ℝ.Theorem 5.
Letθ∈C0∞(ℝ) be a non-negative function such that θ(x)>0, x∈(-π-ε,π+ε), and suppθ=[-π-ε,π+ε], where 0<ε<1/4. Moreover, let
(10)ϕ^i(ξ)=θ(ξ+kiπ),ki∈ℤ,i=1,2,…,r,
and Φ=(ϕ1,ϕ2,…,ϕr)T.(1)
If|k2-k1|=2 and |ki-kj|≥2 for different i,j≤r, then
(11)rank[Φ^(ξ+2jπ)]j∈ℤ=r,ξ∈ℝ.(2)
If|k2-k1|=2 and, at least for ki1 and ki2, it holds that |ki1-ki2|=1, where 1≤i1, i2≤r, then rank[Φ^(ξ+2jπ)]j∈ℤ is not a constant function on ℝ.Proof.  By the Paley-Wiener theorem, ϕi∈𝒮(ℝ)⊂Wμ1(ℝ), i=1,…,r. All possible cases are described in the following lemmas.Lemma 6.
LetΦ=(ϕk1,ϕk2)T, k2-k1=2, k1,k2∈ℤ. The rank of matrix [Φ^(ξ+2jπ)]j∈ℤ is a constant function on ℝ and equals 2.Proof.
We have the next two cases.
(
1
∘
) If ξ∈(-π-ε-k1π+2ℓπ,-π+ε-k1π+2ℓπ), ℓ∈ℤ, for matrix [Φ^(ξ+2jπ)]j∈ℤ we obtain 2×∞ matrix
(12)[⋯00a1b10⋯⋯0a2b200⋯],
for some ai,bi>0, i=1,2. It is obvious that rank[Φ^(ξ+2jπ)]j∈ℤ=2, ξ∈(-π-ε-k1π+2ℓπ,-π+ε-k1π+2ℓπ), ℓ∈ℤ.
(
2
∘
) For ξ∈[-π+ε-k1π+2ℓπ,π-ε-k1π+2ℓπ], ℓ∈ℤ, there are only two nonzero values a1 and a2 which are in different columns of matrix [Φ^(ξ+2jπ)]j∈ℤ. Since
(13)[Φ^(ξ+2jπ)]j∈ℤ=[⋯00a10⋯⋯0a200⋯]2×∞,
it has the rank 2 for all ξ∈[-π+ε-k1π+2ℓπ,π-ε-k1π+2ℓπ], ℓ∈ℤ.
We conclude that the rank of matrix[Φ^(ξ+2jπ)]j∈ℤ, Φ=(ϕk1,ϕk2)T, k2-k1=2, k1,k2∈ℤ, is a constant function on ℝ and equals 2.Lemma 7.
The rank of matrix[Φ^(ξ+2jπ)]j∈ℤ is not a constant function on ℝ if Φ=(ϕk1,ϕk2)T, k2-k1=1, k1,k2∈ℤ.Proof.
We have four different cases for matrix[Φ^(ξ+2jπ)]j∈ℤ. Suppose, without losing generality, that k1∈2ℤ.
(
1
∘
) If ξ∈(-π-ε+2ℓπ,-π+ε+2ℓπ), ℓ∈ℤ, then
(14)[Φ^(ξ+2jπ)]j∈ℤ=[⋯0a1b10⋯⋯0a200⋯],a1,a2,b1>0,
and rank[Φ^(ξ+2jπ)]j∈ℤ=2, for all ξ∈(-π-ε+2ℓπ,-π+ε+2ℓπ), ℓ∈ℤ.
(
2
∘
) For ξ∈[-π+ε+2ℓπ,-ε+2ℓπ], ℓ∈ℤ, nonzero values a1 and a2 are in the same column of matrix [Φ^(ξ+2jπ)]j∈ℤ. For any choice of a 2 × 2 matrix, we get that the determinant equals 0. So we obtain
(15)rank[⋯0a10⋯⋯0a20⋯]=1,
for all ξ∈[-π+ε+2ℓπ,-ε+2ℓπ], ℓ∈ℤ.
(
3
∘
) If ξ∈(-ε+2ℓπ,ε+2ℓπ), ℓ∈ℤ, then
(16)[Φ^(ξ+2jπ)]j∈ℤ=[⋯00a10⋯⋯0b2a20⋯],
for some a1,a2,b2>0, has the rank 2, for all ξ∈(-ε+2ℓπ,ε+2ℓπ), ℓ∈ℤ.
(
4
∘
) For ξ∈[ε+2ℓπ,π-ε+2ℓπ], ℓ∈ℤ, there are two non-zero values a1 and b2 in different columns of matrix [Φ^(ξ+2jπ)]j∈ℤ and the block with these elements determines the rank 2 for all ξ∈[-ε+2ℓπ,π-ε+2ℓπ], ℓ∈ℤ.
Considering possible cases, we conclude thatrank[Φ^(ξ+2jπ)]j∈ℤ, Φ=(ϕk1,ϕk2)T, k2-k1=1, k1,k2∈ℤ, depends on ξ∈ℝ and equals 1 or 2. This rank is a nonconstant function on ℝ.Proof of Theorem5.
(1) Using Lemmas6 and 7, it is obvious that if |k2-k1|=2 and |ki-kj|≥2 for different i,j≤r, then the position of the first non-zero element in each row of matrix [Φ^(ξ+2jπ)]j∈ℤ is unique for each row. So the rank of matrix [Φ^(ξ+2jπ)]j∈ℤ is a constant function on ℝ and equals r for all ξ∈ℝ.
(2) If|k2-k1|=2 and, at least for ki1 and ki2, it holds that |ki1-ki2|=1, 1≤i1, i2≤r, then, in the row with the index i2 (suppose, without losing generality, that i2∈2ℤ+1), we will have a new column with a non-zero element for ξ∈(-π-ε+2ℓπ,-π+ε+2ℓπ), ℓ∈ℤ, but for ξ∈[ε+2ℓπ,π-ε+2ℓπ], ℓ∈ℤ, the positions of all non-zero elements in that row will appear in the previous columns. It is obvious that the rank of the matrix [Φ^(ξ+2jπ)]j∈ℤ depends on ξ∈ℝ and is not the same for all ξ∈ℝ.As a consequence of Theorems3 and 5(1), we have the following result.Theorem 8.
Let the functionsθ and Φ satisfy all the conditions of Theorem 5(1). Then space Vμp(Φ) is closed in Lμp for any p∈[1,∞], and the family {ϕi(·-j)∣j∈ℤ,1≤i≤r} is a p-Riesz basis for Vμp(Φ) for any p∈[1,∞].The following theorem is a generalisation of Theorem5 and can be proved in the same way, so we omit the proof.Theorem 9.
Letθ∈C0∞(ℝ) be a positive function such that θ(x)>0, x∈(a,b), b>a, and supported by [a,b] where b-a>2π. Moreover, let
(17)ϕi^(ξ)=θ(ξ+kiπ),ki∈ℤ,i=1,2,…,r,r∈ℕ,
and Φ=(ϕ1,ϕ2,…,ϕr)T.(1)
If|k2-k1|=2 and |ki-kj|≥2 for different i,j≤r, then
(18)rank[Φ^(ξ+2jπ)]j∈ℤ=r,ξ∈ℝ.(2)
If|k2-k1|=2 and, at least for ki1 and ki2, it holds that |ki1-ki2|=1, where 1≤i1, i2≤r, then rank[Φ^(ξ+2jπ)]j∈ℤ is not a constant function on ℝ.


## 4. Construction of Frames Using Several Band-Limited Functions

Firstly, we consider two smooth functions with proper compact supports.Lemma 10.
Letθ∈C0∞(ℝ), ψ∈C0∞(ℝ) be positive functions such that
(19)θ(x)>0,x∈(-ε,2π+ε),ψ(x)>0,x∈(ε,2π-ε),suppθ=[-ε,2π+ε],suppψ=[ε,2π-ε],0<ε<14.
Moreover, let ϕ^1(ξ)=θ(ξ), ϕ^2(ξ)=ψ(ξ), ξ∈ℝ, and Φ=(ϕ1,ϕ2)T. Then rank[Φ^(ξ+2jπ)]j∈ℤ=1, ξ∈ℝ.Proof.
Note thatϕi∈𝒮(ℝ)⊂Wμ1(ℝ), i=1,2.
We have the following two cases.
(
1
∘
) If ξ∈(-ε+2ℓπ,ε+2ℓπ), ℓ∈ℤ, then matrix
(20)[Φ^(ξ+2jπ)]j∈ℤ=[⋯0ab0⋯⋯0000⋯],a,b>0,
has a constant rank equal to 1.
(
2
∘
) For ξ∈(ε+2ℓπ,2π-ε+2ℓπ), ℓ∈ℤ, the rank of matrix
(21)[Φ^(ξ+2jπ)]j∈ℤ=[⋯0c0⋯⋯0d0⋯],
where c, d are non-zero values, is equal to 1. An equivalent matrix is obtained for ξ=ε+2ℓπ and ξ=-ε+2ℓπ, so we conclude that rank[Φ^(ξ+2jπ)]j∈ℤ=1, for ξ∈[ε+2ℓπ,2π-ε+2ℓπ], ℓ∈ℤ.
Considering these two cases, the rank of matrix[Φ^(ξ+2jπ)]j∈ℤ is a constant function on ℝ, rank[Φ^(ξ+2jπ)]j∈ℤ=1, ξ∈ℝ.Using functionsθ and ψ from Lemma 10, in the next lemma we construct the p-frame with four functions.Lemma 11.
Let the functionsθ and ψ satisfy all the conditions of Lemma 10. Moreover, let
(22)ϕ^k(ξ)=θ(ξ+2kπ),ϕ^k+2(ξ)=ψ(ξ+2kπ),ϕ^k(ξ)=θ(ξ+2kπ),ϕ^k+2(ξ)=ψk=0,1,
and Φ=(ϕ0,ϕ1,ϕ2,ϕ3)T. Then rank[Φ^(ξ+2jπ)]j∈ℤ=2, ξ∈ℝ.Proof.
The proof is similar to the proof of Lemma10.
(
1
∘
) If ξ∈(-ε+2ℓπ,ε+2ℓπ), ℓ∈ℤ, then matrix
(23)[Φ^(ξ+2jπ)]j∈ℤ=[⋯00a1b10⋯⋯0a2b200⋯⋯00000⋯⋯00000⋯],
where ai,bi>0, i=1,2, has a constant rank equal to 2.
(
2
∘
) For ξ∈[ε+2ℓπ,2π-ε+2ℓπ], ℓ∈ℤ, we have
(24)rank[Φ^(ξ+2jπ)]j∈ℤ=[⋯00c10⋯⋯00d10⋯⋯0c200⋯⋯0d200⋯]=2,
where ci,di>0, i=1,2.
We conclude that the rank of matrix[Φ^(ξ+2jπ)]j∈ℤ is a constant function on ℝ equal to 2.Lemma10 can be easily generalised for an even number of functions ϕi, i=0,…,2r-1, with compactly supported ϕ^i, i=0,…,2r-1. The proof of the next theorem is similar to the previous proofs.Theorem 12.
Let the functionsθ and ψ satisfy all the conditions of Lemma 10. Moreover, let
(25)ϕ^k(ξ)=θ(ξ+2kπ),ϕ^k+r(ξ)=ψ(ξ+2kπ),k=0,…,r-1,r∈ℕ,
and Φ=(ϕ0,ϕ1,…,ϕ2r-1)T.
The following statements hold.(1°)
rank[Φ^(ξ+2jπ)]j∈ℤ=r for all ξ∈ℝ.(2°)
Vμp(Φ) is closed in Lμp for any p∈[1,∞].(3°)
{ϕi(·-j)∣j∈ℤ,0≤i≤2r-1} is a p-frame for Vμp(Φ) for any p∈[1,∞].Now we consider three functions with compact supports.Lemma 13.
Let the functionθ satisfies all the conditions of Lemma 10, and let τ∈C0∞(ℝ) and ω∈C0∞(ℝ) be positive functions such that
(26)τ(x)>0,x∈(ε,π-ε)∪(π+ε,2π-ε),ω(x)>0,x∈(-3π-ε,-π+ε),suppτ=[ε,π-ε]∪[π+ε,2π-ε],suppω=[-3π-ε,-π+ε],0<ε<14.
Moreover, let ϕ^1(ξ)=θ(ξ), ϕ^2(ξ)=τ(ξ), ϕ^3(ξ)=ω(ξ), ξ∈ℝ, and Φ=(ϕ1,ϕ2,ϕ3)T. Then rank[Φ^(ξ+2jπ)]j∈ℤ=2, ξ∈ℝ.Proof.
We have four different forms for matrix[Φ^(ξ+2jπ)]j∈ℤ and in all we show that the rank of matrix is equal to 2.
Now we will show all possible cases. Denote withai, i=1,2,3, and bi, i=1,2, some positive values.
(
1
∘
) Consider
(27)[Φ^(ξ+2jπ)]j∈ℤ=[⋯0a1b100⋯⋯000a20⋯⋯00000⋯],ξ∈(-ε+2ℓπ,ε+2ℓπ).
(
2
∘
) Consider
(28)[Φ^(ξ+2jπ)]j∈ℤ=[⋯0b100⋯⋯00a20⋯⋯0a300⋯],ξ∈[ε+2ℓπ,π-ε+2ℓπ].
(
3
∘
) Consider
(29)[Φ^(ξ+2jπ)]j∈ℤ=[⋯0b1000⋯⋯00a2b20⋯⋯00000⋯],ξ∈(π-ε+2ℓπ,π+ε+2ℓπ).
(
4
∘
) Consider
(30)[Φ^(ξ+2jπ)]j∈ℤ=[⋯0b1000⋯⋯000b20⋯⋯0a3000⋯],vvvvvvξ∈[π+ε+2ℓπ,2π-ε+2ℓπ].Remark 14.
In Lemma13 the support of the function ω must have an empty intersection with the supports of θ and τ. In the opposite case, that is, suppθ∩suppτ∩suppω≠∅, the rank of the matrix [Φ^(ξ+2jπ)]j∈ℤ is a non-constant function on ℝ.Lemma13 can be easily generalised for functions ϕi, i=0,…,3r-1, with compactly supported ϕ^i, i=0,…,3r-1. The proof of the next theorem is similar to the previous proofs.Theorem 15.
Let the functionsθ, τ, and ω satisfy all the conditions of Lemma 13. Moreover, let
(31)ϕ^k(ξ)=θ(ξ+2kπ),ϕ^k+r(ξ)=τ(ξ+2kπ),ϕ^k+2r(ξ)=ω(ξ+2kπ),k=0,…,r-1, r∈ℕ, and Φ=(ϕ0,ϕ1,…,ϕ3r-1)T.
The following statements hold.(1°)
rank[Φ^(ξ+2jπ)]j∈ℤ=2r for all ξ∈ℝ.(2°)
Vμp(Φ) is closed in Lμp for any p∈[1,∞].(3°)
{ϕi(·-j)∣j∈ℤ,0≤i≤3r-1} is a p-frame for Vμp(Φ) for any p∈[1,∞].


## 5. Construction of Frames of Functions with Finite Regularities and Compact Supports

We will recall the well-known construction of theB-spline functions in order to justify the rank properties of the corresponding matrices.LetH(x), x∈ℝ, be the characteristic function of the semiaxis x≥0; that is, H(x)=0 if x<0 and H(x)=1 if x≥0 (Heaviside’s function). We construct a sequence {ϕn}n∈ℕ in the following way. Let ϕ1(x):=(H(x)-H(x-a))/a, a>0, ϕ2:=ϕ1*ϕ1, ϕ3:=ϕ1*ϕ1*ϕ1, …; that is,
(32)ϕn:=ϕ1*ϕ1*⋯*ϕ1︸n-1times,n∈ℕ,
where * denotes the convolution of the functions.We obtain(33)ϕ2(x)=1a2(+(x-2a)H(x-2a))xH(x)-2(x-a)H(x-a)+(x-2a)H(x-2a)),ϕ3(x)=12!a3(H(x-2a)-(x-3a)2H(x-3a))H(x-2a)-(x-3a)2H(x-3a))x2H(x)-3(x-a)2H(x-a)+3(x-2a)2×H(x-2a)-(x-3a)2H(x-3a)),ϕ4(x)=13!a4(H(x-3a)+(x-4a)3H(x-4a))x3H(x)-4(x-a)3H(x-a)+6(x-2a)3H(x-2a)-4(x-3a)3×H(x-3a)+(x-4a)3H(x-4a)).
Continuing in this manner, for all n∈ℕ, we have
(34)ϕn(x)=1an(n-1)!×((-1)nnn(n0)xn-1H(x)-(n1)(x-a)n-1×H(x-a)+(n2)(x-2a)n-1×H(x-2a)-(n3)(x-3a)n-1H(x-3a)+⋯+(-1)n-1(nn-1)(x-(n-1)a)n-1×H(x-(n-1)a)+(-1)n(nn)×(x-na)n-1H(x-na)(-1)nnn).
Calculating the Fourier transform of functions ϕn, n∈ℕ, we get
(35)ϕ^1(ξ)=-iav·p·(1ξ)(eiaξ-1),ϕ^2(ξ)=(-i)2a2v·p·(1ξ2)(eiaξ-1)2,ϕ^3(ξ)=(-i)3a3v·p·(1ξ3)(eiaξ-1)3.
Continuing in this manner, we obtain ϕ^n(ξ)=(-i)n/anv·p·(1/ξn)(eiaξ-1)n, n∈ℕ, where v·p· denotes the principal value.LetΦ=(ϕ1,ϕ2,…,ϕr)T, r∈ℕ. matrix [Φ^(ξ+2jπ)]j∈ℤ has for all ξ∈ℝ the same rank as matrix
(36)R(ξ)=[⋯α-4πβ-4πα-2πβ-2πα0β0α2πβ2πα4πβ4π⋯⋯α-4π2β-4π2α-2π2β-2π2α02β02α2π2β2π2α4π2β4π2⋯⋯α-4π3β-4π3α-2π3β-2π3α03β03α2π3β2π3α4π3β4π3⋯⋯α-4π4β-4π4α-2π4β-2π4α04β04α2π4β2π4α4π4β4π4⋯⋮⋮⋮⋮⋮⋯α-4πrβ-4πrα-2πrβ-2πrα0rβ0rα2πrβ2πrα4πrβ4πr⋯],
where αkm=v·p·(1/(ξ-k))m and βkm=(eia(ξ-k)-1)m. Since rankR(ξ)=r, ξ∈ℝ, we have the next result.Theorem 16.
LetΦ=(ϕk,ϕk+1,…,ϕk+(r-1))T, for k∈ℤ, r∈ℕ. Then Vμp(Φ) is closed in Lμp for any p∈[1,∞] and {ϕk+s(·-j)∣j∈ℤ,0≤s≤r-1} is a p-Riesz basis for Vμp(Φ) for any p∈[1,∞].Remark 17.
Letk be a positive integer. We refer to [23] for the Lp-approximation order k. Shift-invariant spaces generated by a finite number of compactly supported functions in Lp(ℝ), 1≤p≤∞, were studied in [23] by Jia, who gave a characterization of the approximation order providing such shift-invariant spaces. Theorem  3 in [23] shows that the shift-invariant space generated with the family of splines, which we constructed in Section 5, provides Lp-approximation order k.Remark 18.
(1) We refer to [3, 20] for the γ-dense set X={xj∣j∈J}. Let ϕk(x)=ℱ-1(θ(·-kπ))(x), x∈ℝ. Following the notation of [20], we put ψxj=ϕxj, where {xj∣j∈J} is γ-dense set determined by f∈V2(ϕ)=V2(ℱ-1(θ)). Theorems 3.1, 3.2, and  4.1 in [20] give conditions and explicit form of Cp>0 and cp>0 such that inequality cp∥f∥Lμp≤(∑j∈J|〈f,ψxj〉μ(xj)|p)1/p≤Cp∥f∥Lμp holds. This inequality guarantees the feasibility of a stable and continuous reconstruction algorithm in the signal spaces Vμp(Φ) [20].
(2) Since the spectrum of the Gram matrix[Φ^,Φ^](ξ), where Φ is defined in Theorem 16, is bounded and bounded away from zero (see [7]), it follows that the family {Φ(·-j)∣j∈ℤ} forms a p-Riesz basis for Vμp(Φ).
(3) Frames of the above sections may be useful in applications since they satisfy assumptions of Theorems 3.1 and 3.2 in [4]. 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 ϕk1,…,ϕkr.



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