A function g : ℝ^d → ℂ is said to have the (scaling) partition of unity property with respect to a real invertible d × d matrix A if

Partitions of unity of this form appear in several parts of analysis, e.g., wavelet analysis and the theory for Triebel-Lizorkin spaces, and the question of how to construct them has attracted some attention. In particular, this issue comes up in connection with the analysis of tight wavelet frames in L²(ℝ^d) [1] and the more general case of dual wavelet frame pairs [2, 3].

In this paper we will give a surprisingly simple characterization of the scaling partition of unity property. In the special case where A is an expanding matrix, i.e., a real matrix with all its eigenvalues having absolute value strictly greater than one, the characterization leads to easy ways of constructing appropriate functions g with attractive properties like high regularity and small support. Under certain conditions, nonnegativity of the function g can be guaranteed. We also discuss a class of integral transforms that can be used to generate functions with the partition of unity property. The one-dimensional version of the transform leads in a natural fashion to a definition of a recursively given family of nonuniform splines. These splines have some similarities with the classical B-splines: their regularity and support grow with the order, and they satisfy the de Boor recursion formula [4, 5]. However, there are also differences: all the splines have support within [−1, 1], and they satisfy a scaling partition of unity condition instead of the translation partition of unity condition. Finally, the key results are applied to the construction of dual pairs of wavelet frames.

The paper is organized as follows. In section 2, we characterize the scaling partition of unity condition and provide explicit and easily verifiable sufficient conditions in the case where A is an expanding matrix. Section 3 deals with the above mentioned one-dimensional integral transform and its lifting to higher dimensions. Finally, section 4 applies the results to obtain easy constructions of wavelet frames in L²(ℝ^d) and their associated dual frames.

We first establish a characterization of the scaling partition of unity property. Despite its simplicity we have not been able to find it stated in the literature.

Theorem 2.1. Consider a function g : ℝ^d → ℂ and any real invertible d × d matrix A. Then the following hold:

Assume that the infinite series ∑j=-∞∞g(Ajγ) is convergent for all γ ∈ ℝ^d \ {0}. Then there is a function φ : ℝ^d → ℂ such that (2.1)g(γ)=φ(γ)−φ(Aγ),∀γ∈ℝd∖{0}.

Proof. For the proof of (i), assume that the infinite series ∑j=-∞∞g(Ajγ) is convergent for all γ ∈ ℝ^d \ {0}. Then

Taking now φ(γ) : =∑j=0∞g(Ajγ),γ∈ℝd\{0}, yields the result. For the proof of (ii), by direct calculation and for any M, N ∈ ℕ,

Then (ii) follows immediately; and (iii) is a consequence of (ii).         □

Note that the function φ satisfying (2.1) for a given function g is not unique. In the sequel φ will denote any such function, not necessarily the one constructed in the proof of Theorem 2.1.

Via Theorem 2.1, we can now show that any expanding matrix A leads to the partition of unity property for a large class of functions g. The following result and its proof hold whenever ||·|| denotes an arbitrary norm on ℝ^d.

Proposition 2.2 Let A be any expanding d × d matrix, and consider any function φ : ℝ^d → ℂ which is continuous at γ = 0 and satisfies the conditions that φ(0) = 1 and lim||γ||→∞φ(γ)=0. Then the function g(γ) : =φ(γ)-φ(Aγ) satisfies the partition of unity condition (1.1).

Proof. By Lemma 5.2 in Hernandez et al. [6], a matrix A is expanding if and only if there exist constants C ∈ (0, 1] and α > 1 such that

for all γ ∈ ℝ^d and N ∈ ℕ ∪ {0}. Thus, the assumption lim||γ||→∞φ(γ)=0 immediately implies that limN→∞φ(ANγ)=0 for all γ ∈ ℝ^d \ {0}. Replacing γ by A-Nγ in the inequality (2.2) shows that ||A-Nγ||≤C-1α-N||γ|| for all γ ∈ ℝ^d and N ∈ ℕ ∪ {0}; thus the assumptions imply that limN→-∞φ(ANγ)=1. The result now follows from Theorem 2.1.         □

Example 2.3 We first give an example of a partition of unity based on a diagonal matrix, and then a construction that works for arbitrary expanding matrices.

(i) Consider an even, continuous and nonnegative function k : ℝ → ℝ such that ∫0∞k(t)dt=1. Then the function

satisfies the conditions in Proposition 2.2. Thus, for any a > 1, the function

satisfies the partition of unity condition ∑j∈ℤg(ajγ)=1,γ∈ℝ\{0}. Clearly, g ∈ C¹(ℝ). Note that for any choice of a norm ||·|| on ℝ^d, the function g can be lifted to a radial function g~ : ℝd→ℝ, by defining g~(γ) : =g(||γ||),γ∈ℝd; the function g~ satisfies the partition of unity condition with respect to the d × d diagonal matrix A=aI.

(ii) Let ||·|| be the Euclidean norm on ℝ^d. The function φ(γ) : = e^−||γ||2, γ ∈ ℝ^d, satisfies the conditions in Proposition 2.2. Thus, for any expanding d × d matrix A, the function

satisfies the partition of unity condition (1.1). Clearly, g ∈ C^∞(ℝ^d).         □

Proposition 2.2 makes it easy to construct partitions of unity for arbitrary expanding matrices A. Furthermore, several properties of the generating function g can be controlled directly in terms of the function φ, e.g., regularity and support. We now prove that nonnegativity of g can also be guaranteed by choosing φ to be a radial function with respect to a given norm ||·|| on ℝ^d:

Proposition 2.4. Let ||·|| be an arbitrary norm on ℝ^d and consider an expanding d × d matrix A such that ||γ||≤||Aγ|| for all γ ∈ ℝ^d. Let r :[0, ∞) → ℝ denote a continuous decreasing function such that r(0) = 1 and r(s) → 0 as s → ∞. Letting φ(γ) : = r(||γ||), γ ∈ ℝ^d, the function g(γ)=φ(γ)-φ(Aγ) has the following properties:

g ≥ 0.

Proof. Since the function r is decreasing, (i) follows immediately from the assumption that ||γ||≤||Aγ|| for all γ ∈ ℝ^d. The partition of unity (ii) follows from Proposition 2.2, so we only need to prove (iii). In order to do so, the nonnegativity of g and (ii) imply that 0≤g(Ajγ)≤1 for every j ∈ ℤ and all γ ∈ ℝ^d \ {0}; thus, ∑j∈ℤ|g(Ajγ)|2≤∑j∈ℤg(Ajγ)=1.

In order to prove the lower bound in (2.3), let η ∈ ℝ^d \ {0}. Then by (ii), there exists j_η ∈ ℤ such that ϵη :=g(Ajηη)>0. Thus, we can choose an open set I_η containing η such that g(Ajηγ)≥ϵη/2 for all γ ∈ I_η. Letting B(0, 1) denote the closed unit ball in ℝ^d with respect to the norm ||·||, the open sets I_η, η ∈ B(0, 1), form a cover of B(0, 1); thus, we can select a finite subcover, i.e., B(0, 1) ⊂ I_η1 ∪ I_η2 ∪ ⋯ ∪ I_ηn for some η₁, …, η_n ∈ B(0, 1). It follows that for any γ ∈ ℝ^d with ||γ|| ≤ 1, γ must lie in I_ηℓ for some ℓ ∈ {1, …, n}; thus,

This proves the lower bound in (2.3) for γ belonging to the closed unit ball in ℝ^d. Taking now an arbitrary γ ∈ ℝ^d \ {0}, the argument in the proof of Proposition 2.2 shows that there exists N ∈ ℕ such that ||A-Nγ||≤1; thus, by a change of variable,

This completes the proof.         □

The condition ||γ||≤||Aγ||,γ∈ℝd, is clearly necessary for the nonnegativity of g(γ)=φ(γ)-φ(Aγ) whenever φ is a function of the type considered in Proposition 2.4. Note that the condition does not follow from A being expanding, as we shall see in the example below.

Example 2.5 Take ||·|| to be the Euclidean norm on ℝ² and let A=(023/40). The eigenvalues are ±3/2, so A is indeed expanding. However A(10)=(03/4), so the condition ||γ||≤||Aγ|| is clearly violated.         □

In this section, we consider certain integral transforms that map a function g having the scaling partition of unity property to another function with the same property. We first discuss the transform on ℝ^d and then specialize to the one-dimensional case, where explicit calculations are much easier. It turns out that the one-dimensional case leads to a definition of a class of splines in a natural way.

In this section, we apply the results on the scaling partition of unity to construct dual pairs of matrix-based wavelet frames in L²(ℝ^d). Since wavelet frames is a well-studied area by itself (see, e.g., [10–12]), we will not make any attempt to motivate them or highlight their applications but just state the definitions and results that are strictly necessary for our discussion.

Given an invertible d × d matrix A with real entries, we define the scaling operator DA:L2(ℝd)→L2(ℝd) by (DAf)(x) := |detA|1/2f(Ax); and, for ν ∈ ℝ^d, the translation operator Tν:L2(ℝd)→L2(ℝd) by T_νf(x) := f(x − ν). Fixing a function ψ ∈ L²(ℝ^d), a d × d matrix A and a translation parameter b > 0, the associated wavelet system is given by {DAjTbkψ}j∈ℤ,k∈ℤd. Denoting the canonical norm on L²(ℝ^d) by || · ||₂, the wavelet system {DAjTbkψ}j∈ℤ,k∈ℤd is said to form a frame for L²(ℝ^d) if there exist constants A, B > 0 such that

if at least the upper condition in (4.1) is satisfied, it is called a Bessel sequence. Two Bessel sequences {DAjTbkψ}j∈ℤ,k∈ℤd and {DAjTbkψ~}j∈ℤ,k∈ℤd, where ψ,ψ~∈L2(ℝd), are said to form dual frames if

We will need the following result, which gives sufficient conditions for wavelet systems to form Bessel sequences, frames, and dual frames. It exists in several variants in the literature: (i) was first stated explicitly in Lemvig [3], while versions of (ii) can be found, e.g., in [6, 13]; see also [14]. We define the Fourier transform on L¹(ℝ^d) by Ff(γ)=f^(γ) : =∫ℝdf(x)e-2πix·γ dx,γ∈ℝd, with the usual extension to L²(ℝ^d).

Lemma 4.1. Let A denote an invertible d × d matrix with real entries, and let b > 0. Then the following hold:

If ψ ∈ L²(ℝ^d) and B : =1bdess supγ∈ℝd∑j∈ℤ∑k∈ℤd|ψ^((AT)jγ)ψ^((AT)jγ−k/b)           | <∞,then{DAjTbkψ}j∈ℤ,k∈ℤd is a Bessel sequence. If furthermore A : =1bd ess infγ∈ℝd(∑j∈ℤ|ψ^((AT)jγ)|2            −∑j∈ℤ∑k≠0|ψ^((AT)jγ)ψ^((AT)jγ−k/b)|)>0, then {DAjTbkψ}j∈ℤ,k∈ℤd is a frame for L²(ℝ^d) with bounds A, B.

Remark 4.2. It follows from Lemma 4.1 that if ψ^ is supported on the closed ball B(0, R) of radius R in ℝ^d and b ≤ (2R)⁻¹, then {DAjTbkψ}j∈ℤ,k∈ℤd is a Bessel sequence when ess supγ∈ℝd∑j∈ℤ|ψ^((AT)jγ)|2<∞; and it is a frame when

If both ψ^ and ψ~^ are supported on B(0, R), then (4.2) is satisfied for m ∈ ℤ^d\{0} when b ≤ (2R)⁻¹; in this case the condition (4.2) consists of the single equation

Up to the factor b^d, this essentially means that the function ψ^¯ψ~^ satisfies the scaling partition of unity property with respect to the matrix AT.

Proposition 3.7 and Lemma 4.1 lead to the following frame result on L²(ℝ) for the splines h_n in (3.4):

Theorem 4.3 Given any n ∈ ℕ and c ∈ (0, 1), consider the spline h_n in (3.4). Fix b ∈ (0, cⁿ⁻¹/2] and define the functions ψ,ψ~∈L2(ℝ) by ψ^ := h_n and

Then {DcjTkbψ}j,k∈ℤ and {DcjTkbψ~}j,k∈ℤ are dual wavelet frames for L²(ℝ).

Proof. Since supp ψ^⊂[-1,1]⊆[-c-n+1,c-n+1], the frame property of {DcjTkbψ}j,k∈ℤ follows directly from Proposition 3.7(vii) and Remark 4.2. Now, by the partition of unity condition in Proposition 3.7(vi), we have

The expression on the right-hand side of (4.3) clearly defines a bounded function, with compact support [−c⁻ⁿ⁺¹, −cⁿ⁻¹] ∪ [cⁿ⁻¹, c⁻ⁿ⁺¹] which is bounded away from the origin. Thus the function ψ~ is well-defined and {DcjTkbψ~}j,k∈ℤ is a Bessel sequence by Lemma 4.1(i) and Remark 4.2.

If γ∈supp ψ^, then ψ^(cjγ) can only be nonzero for j = −n + 1, −n + 2, …, n− 1; thus (4.4) implies that ψ~^(γ)=bQn-1 for γ∈supp ψ^. It follows that ψ^(γ)¯ψ~^(γ)=bQn-1ψ^(γ)¯ for all γ ∈ ℝ; using again (4.4) now shows that

Hence we conclude from Lemma 4.1(ii) and Remark 4.2 that {DcjTkbψ}j,k∈ℤ, {DcjTkbψ~}j,k∈ℤ are indeed dual frames.         □

Note that a different dual frame {DcjTkbψ~}j,k∈ℤ associated with {DcjTkbψ}j,k∈ℤ could have been obtained via the results in Lemvig [2]. Also, by combining the result with the lifting transform in Example 3.8, it is easy to construct radial dual wavelet frames {DcjITkbψ}j∈ℤ,k∈ℤd and {DcjITkbψ~}j∈ℤ,k∈ℤd for L²(ℝ^d), where ψ,ψ~∈L2(ℝd); we leave the details to the reader.

We also note that the unitary extension principle and its many variants is a classical tool to construct wavelet frames based on splines, see, e.g., [10−12]. However, in this case the frame generators themselves are splines, while in our construction the splines occur in the Fourier domain. Figure 1 shows the graphs of the splines ψ^ and ψ~^ in Theorem 4.3 when c = 1/2, n = 3 and b = cⁿ⁻¹/2.

We will now establish a result about the construction of wavelet frames in L²(ℝ^d), based on Proposition 2.4:

Theorem 4.4 Let || · || denote any norm on ℝ^d, and consider an expanding d × d matrix A such that ||γ||≤||ATγ|| for all γ ∈ ℝ^d. Let r:[0, ∞) → ℝ be a continuous decreasing function supported on [0, R] for some R > 0 such that r(0) = 1. Consider the function ψ : ℝ^d → ℝ defined via ψ^(γ) : =r(||γ||)-r(||ATγ||),γ∈ℝd. Then the following hold:

Whenever b ≤ (2R)⁻¹, {DAjTbkψ}j∈ℤ,k∈ℤd is a wavelet frame for L²(ℝ^d).

Proof. The matrix AT is expanding, so Proposition 2.4(iii) implies that there exists a constant C > 0 such that C≤∑j∈ℤ|ψ^((AT)jγ)|2≤1 for all γ ∈ ℝ^d \ {0}. The result in (i) now follows from Remark 4.2.

In order to prove (ii), we will first show that the function ψ^ is supported on the annulus a(R1||AT||-1,R). (This annulus is well-defined as ||γ||≤||ATγ||≤||AT||||γ|| implies that ||AT||≥1.) If ||γ|| ≥ R, we also have that ||ATγ||≥R, so indeed ψ^(γ)=0. Now, assume that ||γ||≤R1||AT||-1. Then ||γ|| ≤ R₁ and

thus ψ^(γ)=r(||γ||)-r(||ATγ||)=1-1=0 as claimed.

Now, applying Proposition 2.4(ii) to the expanding matrix AT, we have

Since AT is expanding, there exist constants C′ ∈ (0, 1] and α > 1 such that ||(AT)jγ||≥C′αj||γ|| for all γ ∈ ℝ^d and j ∈ ℕ ∪ {0}. Thus, for j ∈ ℕ ∪ {0} and any γ∈ supp ψ^,

It follows that for γ∈suppψ^, we have ψ^((AT)jγ)=0 whenever C′αjR1||AT||-1≥R, i.e., for j sufficiently large. On the other hand, since ||(AT)-jγ||≤(C′)-1α-j||γ|| for all γ ∈ ℝ^d and j ∈ ℕ ∪ {0}, it also follows that for γ∈suppψ^,

thus ψ^((AT)-jγ)=0 whenever (C′)-1α-jR≤R1||AT||-1, i.e., for j sufficiently large. This completes the proof of (ii).

Finally, to establish (iii), we first note that for every j ∈ ℤ, there exist constants λ_j, μ_j > 0 such that

Indeed, for j = 0, we simply take λ₀ = μ₀ = 1. If j ∈ ℕ, then C′αj||γ||≤||(AT)jγ||≤||(AT)j||||γ|| for all γ ∈ ℝ^d. On the other hand, ||(AT)-jγ||≤(C′)-1α-j||γ|| and ||γ||=||(AT)j(AT)-jγ||≤||(AT)j||||(AT)-jγ|| for all γ ∈ ℝ^d. Next, using the fact that suppψ^⊂a(R1||AT||-1,R), it follows from (4.7) that supp ψ^((AT)j·)⊂a(R1||AT||-1μj-1,Rλj-1) for every j ∈ ℤ. Thus the definition of ψ~^ in (4.5) shows that ψ~^ is a bounded function, with