Quincunx fundamental refinable functions in arbitrary dimensions

In this paper, we generalize the family of Deslauriers–Dubuc’s interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique quincunx interpolatory masks exists and such a family of masks is of real value and has the full-axis symmetry property. In dimension d = 2, we give the explicit form of such unique quincunx interpolatory masks, which implies the nonnegativity property of such a family of masks.


Introduction and Motivation
We say that a d × d integer matrix M is a dilation matrix if lim n→∞ M −n = 0, that is, all the eigenvalues of M are greater than 1 in modulus.An M-refinable function (or distribution) φ satisfies the refinement equation: where a : Z d → C is called a refinement mask (or low-pass filter) for φ.In the sequel, we assume that the refinement mask a is finitely supported and normalized; i.e., a ∈ l 0 (Z d ) and ∑ k∈Z d a(k) = 1, where by l(Z d ) we denote the linear space of all sequences v : Z d → C of complex numbers on Z d and by l 0 (Z d ) we denote the linear space of all sequences v = {v(k)} k∈Z d ∈ l(Z d ) such that the cardinality of {k ∈ Z d : v(k) = 0} is finite.It is often convenient to use the (formal) Fourier seris v of a sequence v = {v(k)} k∈Z d ∈ l 0 (Z d ), which is defined to be: where k . ., k d ) and ξ = (ξ 1 , . . ., ξ d ) in R d .In terms of Fourier series, the refinement Equation (1) can be also stated in the frequency domain as: where for a function f ∈ L 1 (R d ), its Fourier transform f is defined to be f (ξ) := R d f (x)e −ix•ξ dx, which can be naturally extended to functions in L 2 (R d ) or tempered distributions.
Let a ∈ l 0 (Z d ) be a mask.For a nonnegative integer κ ∈ N 0 := N ∪ {0}, we say that a satisfies the sum rules of order κ + 1 with respect to a dilation matrix M (or a lattice where Π d κ denotes the set of all polynomials in R d of (total) degree at most κ.Note that (3) depends only on the lattice MZ d := {Mk : k ∈ Z d }; that is, if two lattices MZ d and NZ d generated by two dilation matrices M and N are the same MZ d = NZ d , then a satisfies the sum rules of order κ + 1 with respect to M if and only if a satisfies the sum rules of order κ + 1 with respect to N.
To find the solution for (1) or (2), one starts with an initial function φ 0 and employs iteratively the subdivision scheme Q n a,M φ 0 , n = 1, 2, • • • , where, We say that the subdivision scheme Q n a,M φ 0 converges in L p (R d ) if there exists a function φ ∈ L p (R d ) such that: lim n→∞ Q n a,M φ 0 − φ p = 0.In such a case, φ satisfies (1).It was shown in [1] that the subdivision scheme Q n a,M φ 0 converges in L p (R d ) if and only if, where ∇ j v := v − v(• − e j ) is the difference operator with e j being the jth coordinate unit vector in R d , δ is the Kronecker delta satisfying δ(0) = 1 and δ(k) = 0 for all k ∈ Z d \{0}, and S a,M : l(Z d ) → l(Z d ) is the subdivision operator defined to be: In this paper, we are interested in fundamental refinable functions associated with quincunx interpolatory masks and quincunx dilation matrices in arbitrary dimensions.A function φ is said to be fundamental if it is continuous and φ(k) = δ(k), k ∈ Z d .A dilation matrix M is said to be a quincunx dilation matrix if | det M| = 2.For example, in dimension one (d = 1), M = 2 is the dyadic dilation factor; in dimension d = 2, such dilation matrices include: and in dimension d = 3, an example of quincunx dilation matrices is: It is easy to show that the quincunx lattice Q d := MZ d generated by a quincunx dilation matrix M is: which is independent of the quincunx dilation matrix M. In dimension one, the quincunx lattice is simply the lattice of even integers; while in a higher dimension, it is also called the checkerboard lattice.
One can show that if φ is a fundamental refinable function associated with a refinement mask a and a quincunx dilation matrix M as in (1), then it is necessary that a is a quincunx interpolatory mask: The subdivision scheme associated with a quincunx interpolatory mask and a quincunx dilation matrix is called a quincunx interpolatory subdivision scheme.If the quincunx interpolatory subdivision scheme Q n a,M φ 0 converges to φ ∈ L p (R d ), then φ is called a quincunx fundamental refinable function.Interpolatory subdivision schemes play a crucial role in computer-aided geometric design (CAGD) [2], sampling theory, and wavelet/framelet analysis [3][4][5][6][7][8][9][10].
In dimension d = 1, Deslauriers and Dubuc [11] proposed a family of interpolatory subdivision schemes associated with a family {a 2n−1 : n ∈ N} of quincunx interpolatory masks (with respect to the dyadic dilation factor M = 2).Such a family is unique in the following sense: (1) a 2n−1 is a quincunx interpolatory mask: (3) a 2n−1 satisfies the sum rules of order 2n with respect to the dyadic dilation M = 2.
In fact, the interpolatory mask a 2n−1 can be explicitly written as: It is easily seen that such a family enjoys many agreeable properties: symmetry, real-value, non-negativity ( a 2n−1 (ξ) 0), minimal support, and so on.Moreover, the family of Deslauriers and Dubuc's interpolatory masks a 2n−1 , n ∈ N, is closely related to the family of Daubechies' orthonormal masks a db n in the sense that | a db n (ξ)| 2 = a 2n−1 (ξ) and can be obtained by utilizing the Riesz factorization technique [12].
On the one hand, in dimension d = 2, a very simple extension of the dyadic dilation is to consider the dilation matrix M = 2I 2 = diag(2, 2).For such a dilation matrix, Dynet al. [13] constructed the so-called butterfly interpolatory subdivision scheme; Deslauriers et al. [14] obtained several continuous fundamental refinable functions; Mongeau and Deslauriers [15] obtained several C 1 fundamental refinable functions; using convolutions of box splines, Riemenschneider and Shen [16] constructed a family of interpolatory subdivision schemes with symmetry; and Han and Jia [17] constructed a family of optimal interpolatory subdivision schemes with many desirable properties.
On the other hand, in higher dimensions (d 2), a more natural way of extending the dyadic dilation factor with respect to the lattice of even integers is the quincunx dilation matrix.Han and Jia in ( [18], Theorem 3.3) constructed a family of quincunx interpolatory subdivision schemes associated with a family, {a (m,n) : m, n ∈ N 0 , m + n odd}, of 2-dimensional unique quincunx interpolatory masks, which can be viewed as the generalization of the family of Deslauriers and Dubuc's interpolatory masks in dimension one to dimension two in the following sense: (1) a (m,n) is a quincunx interpolatory mask; i.e., (6) holds with d = 2. ( (3) a (m,n) satisfies the sum rules of order m + n + 1 with respect to the quincunx lattice Q 2 defined as in (5) for d = 2.
The uniqueness of such a family a (m,n) implies that a (m,n) is minimally supported among all the quincunx interpolatory masks which satisfies the sum rules of order m + n + 1.Moreover, the mask a (m,n) is real-valued and full-axis symmetric (see (15)).
A natural question is whether the above result is still true in higher dimensions d 3. It should be pointed out that the results in [18] are already highly nontrivial.Many of the results are tailored for dimension two.It was not clear whether techniques in that paper can be carried over directly to higher dimensions.In this paper, we develop new techniques that extend many results in [18] to arbitrary dimensions and give a complete picture of the unique family of quincunx interpolatory masks in arbitrary dimension d ∈ N.That is, we prove the following main theorem.Theorem 1.There exists a family, of d-dimensional masks that are unique in the following sense: (1) a m is a quincunx interpolatory mask; i.e., (6) holds. ( (3) a m satisfies the sum rules of order m 1 + • • • + m d + 1 with respect to the quincunx lattice Q d defined as in (5).
In addition, a m is real-valued and full-axis symmetric.
Note that a (d − 1)-dimensional mask a : Z d−1 → C can be regarded as a d-dimensional mask by identifying Z d−1 as a subset Z d−1 × {0} in Z d .The above result not only provides a natural generalization of the Deslauriers and Dubuc's family of interpolatory masks to arbitrary dimensions, but also shows the close connection between low-dimensional and high-dimensional quincunx interpolatory masks: any such quincunx interpolatory mask a m in R d can be regarded as a quincunx interpolatory mask a (m,0,...,0) in R d+k for any k ∈ N 0 and (m, 0, . . ., 0) ∈ Z d+k .
The remainder of this paper is devoted to proving the above theorem and is organized as follows.In Section 2, we introduce some necessary lemmas and definitions in order to simply give the proof of our main result.Some properties of the unique quincunx interpolatory masks in arbitrary dimensions are discussed.In Section 3, we present the explicit form of the unique quincunx interpolatory masks in dimension d = 2, thereby showing the nonnegativity property of such a family in dimension two.Some remarks are given in the last section.

Quincunx Interpolatory Masks in Arbitrary Dimensions
In this section, we prove the main theorem, which relies essentially on multivariate polynomial interpolation.Before proceeding further, we introduce some notation and definitions. For Elements in N d 0 are ordered lexicographically so that ν = (ν 1 , . . ., and ν < µ for some 1 d.We say that ν µ if ν j µ j for j = 1, . . ., d.We denote, The polynomial space Π d n can be written as A simple example of an X 2 n satisfying the node nonfiguration in R 2 is: For node configuration in R d , we define it recursively as follows.Let, and each set of points, One can easily show that the property of node configuration in R d is invariant under linear transforms.That is, if Let X 1 n := {x 0 , . . ., x n } be a set of n + 1 distinct points in R. Since hyperplanes in dimension one are just points, the above definition of node configuration in R d for X d n can be also defined recursively based on X 1 n .For convention, we assign X 0 n := ∅ the empty set.For two index subsets X and Λ in R d , we denote by: (β µ ) β∈X ;µ∈Λ a matrix A of size (#X ) × (#Λ).The vector (β µ ) β∈X ;µ is then the column vector of A indexed by µ ∈ Λ while the vector (β µ ) β;µ∈Λ is the row vector of A indexed by β ∈ X .
We have the following result (c.f.[19], Theorem 4) regarding the uniqueness of multivariate polynomial interpolation associated with such a set X d n .
Lemma 1. Suppose that X d n ⊆ R d satisfies the node configuration in R d and p ∈ Π d n is a polynomial.If p vanishes on X d n , then p vanishes everywhere.Consequently, the square matrix, is non-singular.
Proof.We prove the result by induction on the pair (d, n) ∈ N × N 0 .
(1) For d = 1, the set X 1 n contains n + 1 distinct points in R and the matrix in ( 7) is simply the Vandermonde matrix.The result obviously holds.
(2) Now suppose the statement holds for the pair (d , n) for any dimension d d − 1 and n ∈ N 0 .
(3) To prove the statement holds for the pair (d, n) for any n ∈ N 0 , we proceed by induction on n.
(3.1)The statement is obviously true for n = 0 (X d 0 is a singleton).(3.2) Suppose the statement holds for n − 1 with n 1. (3.3) Let p ∈ Π n be any polynomial that vanishes on X d n .Then by the node configuration property of X d n , there exists an orthogonal transform A : R d → R d so that for all |µ| = n, Ax µ = (t 0 , t ν ); for some t 0 ∈ R, t ν ∈ R d−1 .That is, the orthogonal transform turns the hyperplane H n containing {x µ ∈ X d n : |µ| = n} perpendicular to the first coordinate axis.Hence, is a set of points lies in a hyperplane perpendicular to the first coordinate.Note that the set has the same cardinality as {x µ ∈ X d n : |µ| = n}.Since the node configuration property is invariant under orthogonal transforms, we see that V d−1 n satisfies the node configuration in R d−1 .Define q n (y) = p(A −1 y) = p(x).Then, the polynomial q n (y 1 , y 2 , . . ., y d ) y 1 =t 0 is a polynomial of d − 1 variables, has a degree at most n, and vanishes on the set V d−1 n .By the induction hypothesis in item (3.2), q n (t 0 , y 2 , . . ., y d ) vanishes everywhere.Consequently, we must have: for some d-variate polynomial q n−1 (y 1 , y 2 , . . ., y d ) of the total degree at most n − 1.Again, by our induction hypothesis in item (3.2), q n−1 vanishes everywhere.Consequently, q n and thus p vanishes everywhere.
The statement for (d, n) for any n ∈ N 0 has been proven.Therefore, by induction, the statement holds for any integer pair (d, n) ∈ N × N 0 .
Next, we show that the matrix (β µ ) β∈X d n ;µ∈Λ d n is non-singular.Suppose this is not the case.Then However, it implies that the non-trivial polynomial p(x) = ∑ µ∈Λ d n c µ x µ vanishes on X d n and hence p vanishes everywhere.This a contradiction.Therefore, (β µ ) β∈X d n ;µ∈Λ d n must be non-singular.
Note that X 0 n = ∅, the statement in Lemma 1 also holds for any (d, n) ∈ N 0 × N 0 .Next, we show that for points on a rectangular grid in R d , they can be extended to be a set that satisfies the node configuration in R d .
with |m| being odd.Define: Then, G 0 m and G 1 m can be both extended to two sets Y 0 m and Y 1 m , respectively, such that they both satisfy the node configuration in Obviously, we have G m ⊆ G m .Define |m| + 1 hyperplanes: for j = 0, . . ., |m|−1 2 , as well as sets of nodes It is easy to show that: which extends G 1 m .We next show that Y 1 m satisfies the node configuration in R d .In fact, from the above definitions, X j lies on the hyperplane H j and does not intersect with any other hyperplane H k for k = 0, . . ., |m| and k = j.Moreover, for each X j , one can simply use some linear transforms A j : X j → O d j so that A j X j = O d j .In fact, define for j = 0, . . ., |m|−1 2 , the linear transforms A 2j , A 2j+1 : R d → R d as, Then, we have A j H j = {x ∈ R d : |x| = j} and A j X j = O d j for j = 0, . . ., |m|.Note that when regarded as a set in R d−1 , the index set O d j satisfies the node configuration in R d−1 .Since each transform A j is an affine transform, it preserves the node configuration of the set X j in R d−1 .Consequently, by the definition of node configuration in R d , we conclude that Y 1 m satisfies the node configuration in R d .A similar approach can be applied to G 0 m in order to obtain Y 0 m ⊇ G 0 m .The corresponding hyperplanes in this cases are: for j = 0, . . ., |m|−1 2 , and Y 0 m := ∪ |m| j=0 X j with X j := G m ∩ H j for G m defined as in (10).Hence, the columns of these two matrices are linearly dependent.The following lemma provides a more precise description of the linear dependence of the columns of these two matrices.
We denote the full-axis symmetry group E by: We say that a sequence {a(β One can easily show that G τ m defined in ( 9) is full-axis symmetric for τ = 0, 1; i.e., for each τ = 0, 1, By considering the symmetry property of the set G τ m , we can further reduce the dependence of columns and rows of matrices in (12). and, Then the matrices, (β µ ) β∈G 0,+ m ;µ∈Λ 0 m and (β µ ) are square matrices and are both non-singular.
By Lemmas 3 and 4, we can choose an index subset Λ ⊆ N d 0 so that, and the square matrix, (β µ ) β∈G 1 m ;µ∈Λ is non-singular, where Λ 0 m , Λ m are defined as in ( 17), (11), respectively.Then we can uniquely solve the following system of linear equations for {c β : β ∈ G 1 m }: Note that {c β : β ∈ G 1 m } must be full-axis symmetric: Hence, we obtain for each Comparing ( 20) and ( 21), we see that {c β : β ∈ G 1 m } must be full-axis symmetric.Define, Then obviously a m is a quincunx interpolatory mask supported on G m .Note that a m (E ε β) = a m (β) = 1  2 δ(β) for all β ∈ G 0 m .Together with the symmetry property of {a m (β) : β ∈ G 1 m }, we conclude that a m is full-axis symmetric: The real-value property of a m is obvious.Next, we show that a m satisfies the sum rules of order |m| + 1 with respect to the quincunx lattice Q d , which is equivalent to: Note that by our definition of {a β : β ∈ G 0 m }, we have, Moreover, by our construction, we already have, We prove it by considering, as the union of two index sets.For µ = (µ 1 , . . ., µ d ) ∈ Λ m \Λ, there must exist µ j odd in view of Λ 0 m ⊆ Λ.By the full-axis symmetry property of a m and similar to (19), we must have, , we have, which must be 0 since p µ (x) is a linear combination of x ν with ν ∈ Λ m .Hence (22) holds.
Consequently, a m is a quincunx interpolatory mask supported on G m and satisfying the sum rules of order |m| + 1 with respect to the quincunx lattice Q d .
We finally show the uniqueness of a m .Suppose there is another quincunx interpolatory mask b : Z d → C supported on G m and satisfying the sum rules of order |m| + 1 with respect to the quincunx lattice Q d .Then, we have, By the non-singularity of (β µ ) β∈G 1 m ;µ∈Λ , we must have a m (β) = b(β) for all β ∈ G 1 m .Consequently, a m = b.We are done.
In view of the proof of Theorem 1, we can have a more generalized result as follows.

Explicit Form of the Bivariate Quincunx Interpolatory Masks
In dimension d = 1, the unique quincunx interpolatory mask a m with m = 2n + 1 is the Deslaruier-Dubuc interpolatory mask, which has the explicit form ( [20,21]): In dimension d = 2, Han and Jia [1] show that the quincunx interpolatory masks a m in Theorem 1 for m = (2n, 1) and m = (2n − 1, 2) are given by: In this paper, we give the explicit form of the full family a m of Theorem 1 in dimension d = 2.
Before we proceed to the proof of Theorem 3, let us introduce some notation and lemmas to simplify our proof.Define, and, Then, by letting x = cos ξ 1 and y = cos ξ 2 , the masks a m in ( 24) and (25) can be written as: First, consider the sum rule definition in (3).In the case of the quincunx dilation matrix in dimension d = 2, a mask a satisfies the sum rules of order κ + 1 with respect to a quincunx dilation matrix M (| det M| = 2), equivalent to: The following lemma shows that our masks a m in (28) indeed satisfy the sum rules of order at least 2n + 1 with respect to the quincunx dilation matrix.
Lemma 5. Let a m be defined as in (28).Define: Then, we have, It follows that, that is, a m satisfies the sum rules of order 2n + 1.
Proof.Note that Q m (x, y) is the linear combination of terms of the form: with α = 2k 0 for m 2 = 2k 0 and α = 2k 0 + 1 for m 2 = 2k 0 + 1.Using the Leibniz rule, one can easily check that, Hence (30) holds.That is, at x = y = −1, Q m (x, y) is a polynomial of the form: Thus, at ξ 1 = π, ξ 2 = π, we have, It follows that, We are done.
Lemma 6.Let a m be defined as in (28).Then the function: Proof.It is easy to show that for m = (2n + 1 − 2k 0 , 2k 0 ), we have: and for m = (2n + 1 − (2k 0 + 1), 2k 0 + 1), we have: with = cos ξ 1 , y = cos ξ 2 .Hence, we only need to show that the right-hand sides of (33) and (34) are independent of the variable y.We next prove (33).The proof for (34) is analogous.One can show that, 1 −1 where the Γ function is defined by Γ(x) = ∞ 0 t x−1 e −t dt and we have, Recalling the definition of g n,k 0 (t, y) in ( 27), the right-hand side of (33) is the linear combination of terms of the form: Moreover, Note that by the independency of G n,k 0 and H n,k 0 with respect to y, we have (by substituting y = 1), Using integration by parts, it is easy to show that, α n,k 0 := (1 − t 2 ) n−1−2k 0 (1 − t) 2k 0 +1 dt satisfy, Consequently, by our induction hypothesis, we have, Therefore, by induction, we conclude that (35) holds for any integer pair (n, k 0 ) with n > 2k 0 0 and hence item (1) holds.

Lemma 4 .
Let G m , G 0 m , and G 1 m be defined as in(8) and (9) for odd |m|.Define,

m
must be non-singular.We are done.Recall inTheorem 1 that a m : Z d → C with m = (m 1 , . . ., m d ) ∈ N d 0 is supposed to be a mask defined on a lattice G m := [−m 1 , m 1 ]×, [−m d , m d ] ∩ Z d and satisfies the sum rules of order |m| + 1 with respect to the quincunx lattice Q d := {β ∈ Z d : |β| even}.Now we are ready to prove our main result in Theorem 1.