Construction of scaling partitions of unity

Partitions of unity in ${\mathbf R}^d$ formed by (matrix) scales of a fixed function appear in many parts of harmonic analysis, e.g., wavelet analysis and the analysis of Triebel-Lizorkin spaces. We give a simple characterization of the functions and matrices yielding such a partition of unity. For invertible expanding matrices, the characterization leads to easy ways of constructing appropriate functions with attractive properties like high regularity and small support. We also discuss a class of integral transforms that map functions having the partition of unity property to functions with the same property. The one-dimensional version of the transform allows a direct definition of a class of nonuniform splines with properties that are parallel to those of the classical B-splines. The results are illustrated with the construction of dual pairs of wavelet frames.


Introduction
A function g : R d → C is said to have the (scaling) partition of unity property with respect to a real invertible (1.1) 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 L 2 (R d ) [5] and the more general case of dual wavelet frame pairs [9,10]. In this paper we will give a surprisingly simple characterization of the scaling partition of unity property. In the case where A is an expanding matrix, i.e., a real matrix A such that all its eigenvalues have 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. 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 L 2 (R d ) and their associated dual frames.

Characterization of the partition of unity property
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 d × d matrix A. Then the following hold: (2.1) (ii) On the other hand, take any function ϕ : R d → C such that (2.1) holds. Then, fixing (iii) Take again any function ϕ : R d → C such that (2.1) holds. Then the partition of unity condition (1.1) holds if and only if the two limits lim N →±∞ ϕ(A N γ) exist and Proof. For the proof of (i), assume that the infinite series ∞ j=−∞ g(A j γ) is convergent for all γ ∈ R d \ {0}. Then Taking now ϕ(γ) := ∞ j=0 g(A j γ), γ ∈ R d \ {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 .

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 satisfies the partition of unity condition j∈Z g(a j γ) = 1, γ ∈ R \ {0}. Clearly, g ∈ C 1 (R). Note that for any choice of a norm || · || on R d , the function g can be lifted to a radial function g : R d → R, by defining g(γ) := g(||γ||), γ ∈ R d ; the function g satisfies the partition of unity condition with respect to the d × d diagonal matrix A = aI.
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: (iii) There exists a constant C > 0 such that 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(A j γ) ≤ 1 for every j ∈ Z and all γ ∈ R d \ {0}; thus, j∈Z |g(A j γ)| 2 ≤ j∈Z g(A j γ) = 1.
The condition ||γ|| ≤ ||Aγ||, γ ∈ R d , 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.

An integral transform preserving partitions of unity
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.

The integral transform on R d
Fix a measurable function g : R d → C and consider formally the integral operator K g that maps a function f : where || · || is an arbitrary norm on R d . The set of functions f for which the transform is well-defined clearly depends on the choice of the function g. Typically, we assume that g is supported on an annulus has support in an annulus a(R 1 , R 2 ) and g is a bounded function with support in an annulus a(R 3 , R 4 ), then h is well-defined and supported on the annulus a(R 3 R 1 , R 4 R 2 ).
The following proposition describes a case where the integral transform is well-defined for all f ∈ L 1 (R) and generates a family of partitions of unity.
and there exists a constant C > 0 for which In particular, if f ∈ L 1 (R d ) is chosen such that R d f (t) dt = 1, the function h has the partition of unity property with respect to the matrix A. If the function g is nonnegative, then the transform K g maps nonnegative functions f to nonnegative functions h = K g f.
Proof. The assumptions imply that g is bounded, so it is clear that the integral in thus, by Lebesgue's dominated convergence theorem, The rest of the proof is clear.
A similar but more general result can be obtained by replacing the expression g( γ ||t|| ) in (3.1) by a function g(t, γ) that yields a partition of unity in the second variable. We leave the exact formulation to the interested reader.

An example of the integral transform on R and a class of splines
In this subsection, we will study the one-dimensional version of the integral transform in (3.1). We will fix a constant c ∈ (0, 1), and consider the set Furthermore, we will fix g := χ S . Then the integral transform K g in (3.1), which we denote simply as K here, takes the form Note that for any fixed γ ∈ R, In particular, the integral in (3.4) is well-defined for all γ ∈ R whenever f ∈ L 1 loc (R). We leave the short proof of the following result to the reader.
(ii) If f ∈ C k (R) for some k ∈ N ∪ {0} and f is supported away from the origin, then h ∈ C k+1 (R).
Proof. As (i) clearly follows from Proposition 3.1, we only have to prove (ii). Letting For γ > 0, the function h is obviously differentiable, and thus under the stated assumptions h is (k + 1) times continuously differentiable for γ > 0. Similarly, h is (k + 1) times continuously differentiable for γ < 0; and since the function h vanishes on a neighborhood of zero, h is even infinitely differentiable at γ = 0.
Observe that Proposition 3.3(i) implies that h ∈ C(R) and j∈Z h(c j γ) = 2 for γ ∈ R\{0}. We could of course obtain this construction via Proposition 2.2 as well.
We will now use the integral transform K to give a direct definition of a class of splines with attractive properties.
Let us collect some of the key properties of the spline functions h n : Proposition 3.7 The functions h n , n ∈ N, have the following properties: (i) h n is a spline, with knots at the points ±c n , ±c n−1 , . . . , ±1.
(ii) h n is even.
(iii) For n ≥ 2, h n ∈ C n−2 (R). (vi) 1 Q n−1 h n satisfies the partition of unity condition (vii) There exists a constant C > 0 such that (viii) For n ≥ 2, the functions h n satisfy the recursion formula Proof. Most of the results are immediate consequences of results that are already proved. Indeed, (i) follows from (viii), which will be proved below; (ii) follows from the definition and Lemma 3.2; and (iii) and (vi) are obtained from Proposition 3.3 and Example 3.6. In addition, (iv) is proved by a straightforward induction, (v) is a consequence of (iv) plus a direct calculation of Q 1 ; and (vii) follows from the partition of unity exactly as in the proof of Proposition 2.4(iii).
We will now prove the only item that remains, namely (viii). Since h n is even for all n ∈ N, we will assume that γ ≥ 0. To get started, direct calculations based on the expressions in Example 3.6 show that the recursion formula holds for n = 2 and n = 3. Thus, we will now consider n ≥ 4. Define the function H n by H n (γ) := γ 0 h n (t) dt, γ ≥ 0. (3.7) We will perform an inductive proof of the recursion formula for h n , assuming that it holds for h k for all k = 2, . . . , n − 1. Now, using Lemma 3.2 and the induction hypothesis, Then a direct calculation using integration by parts yields that Now, it follows from (3.7) and Lemma 3.2 that Also, Hence, based on (3.8), after solving for h n (γ), we obtain (3.6).
The splines in Definition 3.5 are indeed well-known: as noted from the recursion formula (3.6), they are the symmetrized version of the nonuniform B-splines with knots at c n , c n−1 , . . . , 1, see [1,2]. Here, we have provided another perspective in obtaining them. Their properties also serve as a concrete illustration of the general properties we derived in Propositions 3.1 and 3.3. Other related papers on polynomial splines with geometric knots include [6,8,11].
As a further comment on the one-dimensional transform K in (3.3), we observe that it can be lifted to a transform acting on functions on R d : Example 3.8 In this example, we describe a way of lifting the transform K to generate radial functions on R d .
(i) We can easily lift the integral transform to an operator that yields a radial function h : R d → R as output. Indeed, taking an arbitrary norm || · || on R d , define the integral transform K, acting on functions f ∈ L 1 loc (R), by Clearly, in terms of the transform K in (3.3), we have h(γ) = Kf (||γ||) = h(||γ||). Furthermore, if f ∈ L 1 (R), then (ii) As a special case of (i) and based on the nonuniform B-splines h n in (3.5), we can define a family of radial functions h n on R d by Each of these radial functions is supported on an annulus, and they can be easily calculated using the recursion formula in Proposition 3.7. Also, h n satisfies the partition of unity condition 4 Wavelet frames in L 2 (R d ) and dual frames In this section, we apply the results on the scaling partition of unity to construct dual pairs of matrix-based wavelet frames in L 2 (R d ). Since wavelet frames is a well-studied area by itself (see, e.g., [14,13,12]), 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 D A : L 2 (R d ) → L 2 (R d ) by (D A f )(x) := | det A| 1/2 f (Ax); and, for ν ∈ R d , the translation operator T ν : . Fixing a function ψ ∈ L 2 (R d ), a d × d matrix A and a translation parameter b > 0, the associated wavelet system is given by {D A j T bk ψ} j∈Z,k∈Z d . Denoting the canonical norm on L 2 (R d ) by || · || 2 , the wavelet system {D A j T bk ψ} j∈Z,k∈Z d is said to form a frame for L 2 (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 {D A j T bk ψ} j∈Z,k∈Z d and 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 [10], while versions of (ii) can be found, e.g., in [4,7]; see also [3]. We define the Fourier transform on L 1 (R d ) by F f (γ) = f (γ) := Lemma 4.1 Let A denote an invertible d × d matrix with real entries, and let b > 0. Then the following hold: (ii) Assume that the matrix A is expanding and suppose that for some ψ, ψ ∈ L 2 (R d ), Then {D c j T kb ψ} j,k∈Z and {D c j T kb ψ} j,k∈Z are dual wavelet frames for L 2 (R).
, the frame property of {D c j T kb ψ} 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. If γ ∈ supp ψ, then ψ(c j γ) can only be nonzero for j = −n + 1, −n + 2, . . . , n − 1; thus (4.4) implies that ψ(γ) = b Q n−1 for γ ∈ supp ψ. It follows that ψ(γ) = b Q n−1 ψ(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 {D c j T kb ψ} j,k∈Z , {D c j T kb ψ} j,k∈Z are indeed dual frames.
Note that a different dual frame {D c j T kb ψ} j,k∈Z associated with {D c j T kb ψ} j,k∈Z could have been obtained via the results in [9]. Also, by combining the result with the lifting transform in Example 3.8, it is easy to construct radial dual wavelet frames {D c j I T kb ψ} j∈Z,k∈Z d and {D c j I T kb ψ} j∈Z,k∈Z d for L 2 (R d ), where ψ, ψ ∈ L 2 (R d ); 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., [14,13,12]. However, in this case the frame generators themselves are splines, while in our construction the splines occur in the Fourier domain.
(iii) If r(γ) = 1 for γ ∈ [0, R 1 ] for some R 1 > 0, choose an index set J as in (ii) and for b > 0, define the function ψ : Then for sufficiently small values of b, {D A j T bk ψ} j∈Z,k∈Z d and {D A j T bk ψ} j∈Z,k∈Z d are dual frames for L 2 (R d ).
Proof. The matrix A T is expanding, so Proposition 2.4(iii) implies that there exists a constant C > 0 such that The result in (i) now follows from Remark 4.2.
Consequently, both ψ and ψ are supported on the closed ball B(0, R max j∈J λ −1 j ). The rest of the proof of (iii) is similar to the proof of Theorem 4.3, where Lemma 4.1 and Remark 4.2 are applied. Specifically, we see that whenever b ≤ (2R max j∈J λ −1 j ) −1 , {D A j T bk ψ} j∈Z,k∈Z d and {D A j T bk ψ} j∈Z,k∈Z d are Bessel sequences. Also, the partition of unity condition (4.6) together with (ii) shows that and hence, {D A j T bk ψ} j∈Z,k∈Z d and {D A j T bk ψ} j∈Z,k∈Z d are dual frames for L 2 (R d ).