Dual Univariate Interpolatory Subdivision of Every Arity: Algebraic Characterization and Construction

1 Introduction

Subdivision schemes are iterative methods for the construction of curves and surfaces exploited in various applications, ranging from computer-aided geometric design, computer graphics and animation (see, e.g., MR2415757 ; Warren:2001:SMG:580358 ) to the construction of wavelets and frames (see, e.g., MR1968118 ; MR1971300 ). Here we focus on univariate stationary subdivision MR1079033 . Convergent subdivision schemes are characterized by a compactly supported function $φ \in C^{0} (R)$ , called basic limit function, which satisfies a refinement equation of the type

φ (x) = \sum k \in Z a_{k} φ (m x - k + τ), x \in R,

(1)

where $m \in N ∖ {1}$ is the arity, $a = {a_{k} \in R}_{k \in Z}$ , is a compactly supported sequence of real values called mask and $τ \in [0, 1) \cap Q$ is a shift parameter. If a compactly supported function $φ$ satisfies (1), we can always consider it to have $s u p p (φ) = [- s, s]$ , $s > 0$ , changing the indexing and the shift parameter $τ$ properly. In particular, denoting

k_{ℓ} = min {k \in Z : a_{k} \neq 0} and k_{r} = max {k \in Z : a_{k} \neq 0},

it is easy to prove that $φ$ has a symmetric support if and only if one of the following is true:

\begin{matrix} (p) τ = 0 and k_{ℓ} = - k_{r}; (d) τ = \frac{1}{2} and k_{ℓ} = 1 - k_{r} . \end{matrix}

(2)

Because of (2), univariate subdivision schemes are divided in two major families: primal schemes, satisfying $(p)$ , and dual schemes, satisfying $(d)$ .
By constructing the subdivision symbol

A (z) = \frac{1}{m} k_{r} \sum k = k_{ℓ} a_{k} z^{k}, z \in C, | z | = 1,

(3)

associated to the mask $a = {a_{k} \in R}_{k = k_{ℓ}, \dots, k_{r}}$ , it is well-known (see, e.g., MR2775138 ; MR2843037 ; MR3071114 ) that the shift parameter $τ$ satisfies

τ = A^{'} (1) .

The target of this work are interpolatory schemes, i.e., schemes with a basic limit function $φ$ satisfying

φ (n) = δ_{0, n}, n \in Z .

(4)

Within the class of interpolatory schemes, primal ones are characterized by a simple polynomial equation which involves the sub-symbols of the scheme, i.e., the Laurent polynomials

A_{n} (z) = \frac{1}{m} \sum k \in Z a_{m k + n} z^{m k + n}, n = 0, \dots, m - 1 satisfying A (z) = m - 1 \sum n = 0 A_{n} (z) .

(5)

In particular, a convergent subdivision scheme is primal interpolatory if and only if MR1397613

A_{0} (z) \equiv \frac{1}{m}, \forall z \in C, | z | = 1.

(6)

Due to (6), primal interpolatory schemes have the so called stepwise interpolation property, i.e., at each subdivision step they maintain the data of the previous one.
Beside the use of primal interpolatory schemes, there exist other approaches to interpolate points via subdivision, such as those that apply an approximating scheme after suitably preprocessing the data to be interpolated (see, e.g., Deng2010137 ; MR3921224 ; Zheng2006301 ). However, before LUCIA , none of the existing approaches took into account the possibility of constructing a native dual interpolatory scheme which does not have the property of retaining the initial data at each iteration, but achieves the interpolation in the sense that the initial data are still preserved in the limit function. All dual subdivision schemes we can find in literature are indeed not interpolatory MR3071114 , except for the family of dual quaternary schemes introduced in LUCIA and for the class of dual $(2 n)$ -point subdivision schemes proposed in DENG2019344 , which is shown to possess the interpolation property only when $n$ tends to infinity, i.e., when the subdivision mask has infinite length.
To fill this theoretical gap in the literature, in this paper we investigate dual interpolatory schemes with finite masks and present a complete algebraic characterization of their symbols, which covers every arity. Additionally, for any arbitrary arity $m$ greater than $2$

, this characterization is used as a general constructive method to produce a great amount of new interpolatory schemes with different features. In fact, the method we propose allows the user to tune a good amount of degrees of freedom: the arity

m

, the length of the mask

a

, the desired degree of polynomial reproduction and some samples of the resulting basic limit function

φ

The remainder of the paper is organized as follows. In Section 2 we state a necessary condition for a basic limit function to have prescribed values at lattices of the form $Z / T$ , for $τ \in [0, 1) \cap Q$ and $T \in N ∖ {0}$ satisfying $τ T \in N$ . In Section 3, we show an explicit algebraic characterization of dual interpolatory schemes in terms of polynomial equalities involving sub-symbols and other polynomials related to the evaluation of the basic limit function at $Z / 2$ . Exploiting this fact we are able to construct, in Section 4, new and interesting interpolatory schemes never considered in the literature, which possess comparable or even superior properties, in terms of regularity, length of the support and/or polynomial reproduction, with respect to their primal counterparts. Conclusions are drawn in Section 5.

2 Necessary Algebraic Condition for Refinability

We start proving a general necessary condition for a function $φ$ to be the basic limit function of a convergent subdivision scheme, having prescribed values over the lattice $Z / T$ , $T > 0$ . The proof exploits the notation

ˆ f (ω) = \int_{R} f (x) e^{- 2 π i x ω} d x, ω \in R

(7)

to refer to the Fourier transform of a function

f \in L^{1} (R)

, and is based on a fundamental result of harmonic analysis, i.e. the Poisson summation formula in its generalized form MR1420504 , which states:
if a function

f : R \to R

f \in L^{1} (R)

satisfies

| f (x) | + | ˆ f (x) | \leq C (1 + | x |)^{- 1 - ϵ}, x \in R,

(8)

for some $C, ϵ > 0$ , then,

\sum n \in Z ˆ f (ω + n T) = \frac{1}{T} \sum n \in Z f (\frac{n}{T}) e^{- 2 π i ω \frac{n}{T}}, ω \in R, T > 0.

(9)

Remark 2.1.

Condition (8) is easily satisfied by continuous compactly supported functions such as the basic limit functions of convergent subdivision schemes.

Theorem 2.2.

Let $φ$ be the basic limit function of a convergent subdivision scheme of arity $m \in N ∖ {0, 1}$ , compactly supported mask $a = {a_{ℓ}}_{ℓ \in Z}$ and sub-symbols $A_{k} (z)$ , $k \in Z$ , satisfying (1) with $τ \in [0, 1) \cap Q$ . Then, for every $T \in N ∖ {0}$ such that $τ T \in N$ , the following polynomial identity holds

m T - 1 \sum γ = 0 Φ_{T, γ} (z^{m}) = m z^{- τ T} m - 1 \sum β = 0 m T - 1 \sum \begin{matrix} γ = 0 γ + β T \equiv_{m} τ T \end{matrix} A_{β} (z^{T}) Φ_{T, γ} (z),

(10)

where

Φ_{T, n} (z) = \frac{1}{T} \sum k \in Z φ (m k + \frac{n}{} T) z^{m T k + n}, n \in Z,

(11)

with $z = e^{- 2 π i \frac{ω}{T}}$ .

Proof.

It is well known that, on the Fourier side, the refinement equation (1) reads as

ˆ φ (ω) = e^{2 π i \frac{ω}{m} τ} a (\frac{ω}{m}) ˆ φ (\frac{ω}{m}),

(12)

where

a (ω) = \frac{1}{m} \sum k \in Z a_{k} e^{- 2 π i ω k},

(13)

is the symbol associated to the refinement equation (1). Then, for every $n \in Z$ , we get

ˆ φ (m ω + n T) = e^{2 π i (ω + \frac{n T}{m}) τ} a (ω + \frac{n T}{m}) ˆ φ (ω + \frac{n T}{m}) .

Summing over $n \in Z$ , we obtain

\begin{matrix} \sum_{n \in Z} ˆ φ (m ω + n T) & = & \sum_{n \in Z} e^{2 π i (ω + \frac{n T}{m}) τ} a (ω + \frac{n T}{m}) ˆ φ (ω + \frac{n T}{m}) = & \sum_{α = 0}^{m - 1} \sum_{h \in Z} e^{2 π i (ω + h T + \frac{α T}{m}) τ} a (ω + h T + \frac{α T}{m}) ˆ φ (ω + h T + \frac{α T}{m}), \end{matrix}

where we set $n = m h + α$ . Since $τ T \in Z$ , we have

\begin{matrix} e^{2 π i (ω + h T + \frac{α T}{m}) τ} a (ω + h T + \frac{α T}{m}) & = & \frac{e^{2 π i (ω + \frac{α T}{m}) τ}}{m} \sum_{k \in Z} a_{k} e^{- 2 π i (ω + h T + \frac{α T}{m}) k} = & \frac{z^{- τ T} e^{2 π i \frac{α}{m} τ T}}{m} \sum_{k \in Z} e^{- 2 π i \frac{α}{m} k T} a_{k} {(e^{- 2 π i \frac{ω}{T}})}^{k T} = k = m n + β & \frac{z^{- τ T} e^{2 π i \frac{α}{m} τ T}}{} m \sum_{β = 0}^{m - 1} e^{- 2 π i \frac{α}{m} β T} \sum_{n \in Z} a_{m n + β} {(e^{- 2 π i \frac{ω}{T}})}^{(m n + β) T} = & z^{- τ T} e^{2 π i \frac{α}{m} τ T} \sum_{β = 0}^{m - 1} e^{- 2 π i \frac{α}{m} β T} A_{β} (z^{T}) . \end{matrix}

Thus,

\sum n \in Z ˆ φ (m ω + n T) = z^{- τ T} m - 1 \sum α = 0 e^{2 π i \frac{α}{m} τ T} ⎛ ⎝ m - 1 \sum β = 0 e^{- 2 π i \frac{α}{m} β T} A_{β} (z^{T}) ⎞ ⎠ \sum h \in Z ˆ φ (ω + h T + \frac{α T}{m}) .

(14)

At this point we are ready to apply Poisson summation formula to both sides of (14) obtaining, for the left-hand-side,

\begin{matrix} \sum_{n \in Z} ˆ φ (m ω + n T) & = & \frac{1}{T} \sum_{n \in Z} φ (\frac{n}{T}) {(e^{- 2 π i \frac{ω}{T} m})}^{n} = n = m T k + γ & \sum_{γ = 0}^{m T - 1} \frac{1}{T} \sum_{k \in Z} φ (m k + \frac{γ}{T}) (z^{m})^{m T k + γ} = & \sum_{γ = 0}^{m T - 1} Φ_{T, γ} (z^{m}) \end{matrix}

(15)

and, for the right-hand-side,

\begin{matrix} \sum_{h \in Z} ˆ φ (ω + h T + \frac{α T}{m}) & = & \frac{1}{T} \sum_{h \in Z} φ (\frac{h}{T}) e^{- 2 π i (ω + \frac{α T}{m}) \frac{h}{T}} = h = m T n + γ & \sum_{γ = 0}^{m T - 1} \frac{1}{T} \sum_{n \in Z} φ (m n + \frac{γ}{T}) e^{- 2 π i (ω + \frac{α T}{m}) \frac{m T n + γ}{T}} = & \sum_{γ = 0}^{m T - 1} e^{- 2 π i \frac{α}{m} γ} Φ_{T, γ} (z) . \end{matrix}

(16)

Combining (14) with (15) and (16) we obtain

\begin{matrix} \sum_{γ = 0}^{m T - 1} Φ_{T, γ} (z^{m}) & = & z^{- τ T} \sum_{α = 0}^{m - 1} e^{2 π i \frac{α}{m} τ T} (\sum_{β = 0}^{m - 1} e^{- 2 π i \frac{α}{m} β T} A_{β} (z^{T})) (\sum_{γ = 0}^{m T - 1} e^{- 2 π i \frac{α}{m} γ} Φ_{T, γ} (z)) = & z^{- τ T} \sum_{β = 0}^{m - 1} \sum_{γ = 0}^{m T - 1} A_{β} (z^{T}) Φ_{T, γ} (z) \sum_{α = 0}^{m - 1} e^{- 2 π i \frac{α}{m} (γ + β T - τ T)} . \end{matrix}

Since

m - 1 \sum α = 0 e^{- 2 π i \frac{α}{m} (γ + β T - τ T)} = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} m, & if γ + β T \equiv_{m} τ T, 0, & otherwise, \end{matrix}

we finally arrive at (10). ∎

Remark 2.3.

For $m = 2, τ = 0, T = 1$ , the identity (10) is equivalent to the algebraic condition stated in MR1790328 , equation $(2.7)$ , for a binary refinable function to have prescribed values at the integers, thus it can be seen as a generalization of it.

Remark 2.4.

In Theorem 2.2, once $m$ and $τ$ are fixed, the bigger is $T$ , the more evaluations of $φ$ we need to take into account. In particular, since the aim is to construct a refinable function given some of its values, it makes sense to consider

T = min {k \in N ∖ {0} : τ k \in N} .

(17)

Indeed, if there are no trigonometric polynomials ${A_{β}}_{β = 0}^{m - 1}$ satisfying (10) with $T$ as in (17) for a fixed set of values ${φ (k / T)}_{k \in Z}$ , there is no hope about the existence of solutions considering $k T$ , $k \in N ∖ {0, 1}$ , instead of $T$ , since $Z / T \subset Z / (k T)$ .

3 Algebraic Characterization

Let us consider the dual case, i.e., $(d)$ in (2). Since $τ = \frac{1}{2}$ , according to (17) we choose $T = 2$ . In Lemma 3.5 we start specializing the necessary condition of Theorem 2.2 to the dual interpolatory case. Then we split the characterization based on the arity $m$

being odd (Theorem

3.8) or even (Theorem 3.9).

Lemma 3.5.

Consider a convergent $m$ -ary dual interpolatory subdivision scheme with compactly supported mask $a = {a_{ℓ}}_{ℓ \in Z}$ , sub-symbols $A_{k} (z)$ , $k \in Z$ , and basic limit function $φ \in C^{0} (R)$ with compact symmetric support, that satisfies $φ (k) = δ_{0, k}$ , $k \in Z$ . Then

\frac{1}{2} + m - 1 \sum γ = 0 Φ_{2, 2 γ + 1} (z^{m}) = m z^{- 1} ⎛ ⎜ ⎜ ⎜ ⎝ m - 1 \sum \begin{matrix} β = 0 2 β \equiv_{m} 1 \end{matrix} \frac{A_{β} (z^{2})}{2} + m - 1 \sum \begin{matrix} β, γ = 0 2 (γ + β) \equiv_{m} 0 \end{matrix} A_{β} (z^{2}) Φ_{2, 2 γ + 1} (z) ⎞ ⎟ ⎟ ⎟ ⎠ .

(18)

Proof.

First we rewrite the necessary condition for refinability (10) with $τ = \frac{1}{2}$ and $T = 2$ , obtaining

2 m - 1 \sum γ = 0 Φ_{2, γ} (z^{m}) = m z^{- 1} m - 1 \sum β = 0 2 m - 1 \sum \begin{matrix} γ = 0 γ + 2 β \equiv_{m} 1 \end{matrix} A_{β} (z^{2}) Φ_{2, γ} (z) .

(19)

The fact that $φ (k) = δ_{0, k}$ , $k \in Z$ , together with the definition (11) implies that,

Φ_{2, 2 γ} (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} \frac{1}{2}, & if γ = 0, 0, & otherwise. \end{matrix}

Thus, dividing the sums over $γ$ in (19) into sums given by the indices $2 γ + j$ , $γ = 0, \dots, m - 1$ , $j = 0, 1$ , we obtain

2 m - 1 \sum γ = 0 Φ_{2, γ} (z^{m}) = \frac{1}{2} + m - 1 \sum γ = 0 Φ_{2, 2 γ + 1} (z),

and

m z^{- 1} m - 1 \sum β = 0 2 m - 1 \sum \begin{matrix} γ = 0 γ + 2 β \equiv_{m} 1 \end{matrix} A_{β} (z^{2}) Φ_{2, γ} (z) = m z^{- 1} ⎛ ⎜ ⎜ ⎜ ⎝ m - 1 \sum \begin{matrix} β = 0 2 β \equiv_{m} 1 \end{matrix} \frac{A_{β} (z^{2})}{2} + m - 1 \sum \begin{matrix} β, γ = 0 2 (γ + β) \equiv_{m} 0 \end{matrix} A_{β} (z^{2}) Φ_{2, 2 γ + 1} (z) ⎞ ⎟ ⎟ ⎟ ⎠ .

This completes the proof. ∎

As a consequence of Lemma 3.5, it is easy to prove that for the arity $m = 2$ there are no continuous refinable functions. This could be the reason why dual interpolatory schemes have not been investigated before.

Corollary 3.6.

For $m = 2$ there are no convergent dual interpolatory subdivision schemes.

Proof.

Consider the necessary condition (18) for $m = 2$ :

\frac{1}{2} + 1 \sum γ = 0 Φ_{2, 2 γ + 1} (z^{2}) = 2 z^{- 1} ⎛ ⎜ ⎜ ⎜ ⎝ 1 \sum \begin{matrix} β = 0 2 β \equiv_{2} 1 \end{matrix} \frac{A_{β} (z^{2})}{2} + 1 \sum \begin{matrix} β, γ = 0 2 (γ + β) \equiv_{2} 0 \end{matrix} A_{β} (z^{2}) Φ_{2, 2 γ + 1} (z) ⎞ ⎟ ⎟ ⎟ ⎠ .

In particular, the left-hand side can be rewritten as

\frac{1}{2} + 1 \sum γ = 0 Φ_{2, 2 γ + 1} (z^{2}) = \frac{1}{2} + Φ_{2, 1} (z^{2}) + Φ_{2, 3} (z^{2}) =: F (z^{2}) .

Then, recalling (5), we rewrite the right-hand side as

\begin{matrix} 2 z^{- 1} (\sum_{\begin{matrix} β = 0 2 β \equiv_{2} 1 \end{matrix}}^{1} \frac{A_{β} (z^{2})}{2} + \sum_{\begin{matrix} β, γ = 0 2 (γ + β) \equiv_{2} 0 \end{matrix}}^{1} A_{β} (z^{2}) Φ_{2, 2 γ + 1} (z)) = = 2 z^{- 1} (A_{0} (z^{2}) + A_{1} (z^{2})) (Φ_{2, 1} (z) + Φ_{2, 3} (z)) = \sum_{k \in Z} a_{k} z^{2 k - 1} (F (z) - \frac{1}{2}) . \end{matrix}

Thus, since

F (z) = \sum j \in Z f_{j} z^{j} with f_{j} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1}{2}, & j = 0, φ (2 k + \frac{1}{2}), & j = 4 k + 1, φ (2 k + \frac{3}{2}), & j = 4 k + 3, 0, & otherwise, \end{matrix}

we get

\sum j \in Z f_{j} z^{2 j} = \sum k \in Z a_{k} z^{2 k - 1} \sum j \in Z ∖ {0} f_{j} z^{j} .

(20)

Now both $a = {a_{k}}_{k \in Z}$ and $φ$ (and so $f = {f_{j}}_{j \in Z}$ ) are compactly supported. Then, because of (20), the first and the last non-zero elements of $a$ must be equal to $1$ . There follows that the associated subdivision scheme can not be convergent, since its difference scheme cannot be contractive CHOI2006351 ; MR1172120 ; MR2008967 . ∎

Remark 3.7.

Actually a dual interpolatory scheme was almost known since 1884. The scheme is the one with arity $m = 3$ and mask $a = {\frac{1}{2}, 1, 1, \frac{1}{2}}$ , whose basic limit function is related to the Cantor function (see, e.g., MR2195181 ). Indeed, the resulting basic limit function $φ$ , Figure 1, has $s u p p (φ) = [- \frac{3}{4}, \frac{3}{4}]$ and is divided into three parts: over $[- \frac{3}{4}, - \frac{1}{4}]$ it is exactly the well-known ascending Cantor function, over $[- \frac{1}{4}, \frac{1}{4}]$ it is constant equal to $1$ and over $[\frac{1}{4}, \frac{3}{4}]$ it is equal to the descending Cantor function. This scheme however reproduces only constants and $φ \in C^{{log}_{3} (2)} (R)$ . So it is not really useful for applications. This is the shortest possible basic limit function obtained by a converging dual interpolatory scheme.

Figure 1: The basic limit function of the ternary scheme related to the Cantor function, which is supported in $[- 0.75, 0.75]$ , reproduces constants and belongs to $C^{{log}_{3} (2)} (R)$ .

At this point, to proceed with the algebraic characterization of dual interpolatory schemes we have to split computations into two cases: the case with $m$ odd (Theorem 3.8) and the case with $m$ even (Theorem 3.9). The changes are due to the equivalences that the indices of the sums in the right-hand-side of (18) must satisfy, which depend on $m$ . We thus treat the two cases separately.

Theorem 3.8.

Let $m = 2 λ + 1, λ \in N ∖ {0}$ . A convergent $m$ -ary subdivision scheme is a dual interpolatory scheme if and only if

\frac{1}{2} + m - 1 \sum γ = 0 Φ_{2, 2 γ + 1} (z^{m}) = m z^{- 1} ⎛ ⎜ ⎝ \frac{A_{\frac{m + 1}{2}} (z^{2})}{2} + m - 1 \sum γ = 0 A_{m - γ} (z^{2}) Φ_{2, 2 γ + 1} (z) ⎞ ⎟ ⎠ .

(21)

Proof.

We start from (18) and we need to further study the right-hand-side. We observe that the only $β \in {0, \dots, m - 1}$ satisfying $2 β \equiv_{m} 1$ is $β = \frac{m + 1}{2}$ . On the other hand, the only possibility to satisfy $2 (β + γ) \equiv_{m} 0$ for $β, γ \in {0, \dots, m - 1}$ is that $β + γ \in {0, m}$ , i.e., $β = m - γ$ if $β, γ \neq 0$ , and $β = γ = 0$ . Thus, the expressions in the right-hand side of (18) can be rewritten as

\begin{matrix} \sum_{\begin{matrix} β = 0 2 β \equiv_{m} 1 \end{matrix}}^{m - 1} \frac{A_{β} (z^{2})}{2} + \sum_{\begin{matrix} β, γ = 0 2 (γ + β) \equiv_{m} 0 \end{matrix}}^{m - 1} A_{β} (z^{2}) Φ_{2, 2 γ + 1} (z) = = \frac{A_{\frac{m + 1}{2}} (z^{2})}{2} + A_{0} (z^{2}) Φ_{2, 1} (z) + \sum_{γ = 1}^{m - 1} A_{m - γ} (z^{2}) Φ_{2, 2 γ + 1} (z) = \frac{A_{\frac{m + 1}{2}} (z^{2})}{2} + \sum_{γ = 0}^{m - 1} A_{m - γ} (z^{2}) Φ_{2, 2 γ + 1} (z), \end{matrix}

where the last equality holds since $A_{0} (z) = A_{m} (z)$ . Thus, equation (21) follows from (18).

Conversely, if (21) holds, we can subtract it from the necessary condition for refinability (10) with $τ = \frac{1}{2}$ and $T = 2$ , getting, for the left-hand-side,

2 m - 1 \sum γ = 0 Φ_{2, γ} (z^{m}) - \frac{1}{2} - m - 1 \sum γ = 0 Φ_{2, 2 γ + 1} (z^{m}) = m - 1 \sum γ = 0 Φ_{2, 2 γ} (z^{m}) - \frac{1}{2},

(22)

and, for the right-hand-side,

\begin{matrix} m z^{- 1} \sum_{β = 0}^{m - 1} \sum_{\begin{matrix} γ = 0 γ + 2 β \equiv_{m} 1 \end{matrix}}^{2 m - 1} A_{β} (z^{2}) Φ_{2, γ} (z) - m z^{- 1} ⎛ ⎝ \frac{A_{\frac{m + 1}{2}} (z^{2})}{2} + \sum_{γ = 0}^{m - 1} A_{m - γ} (z^{2}) Φ_{2, 2 γ + 1} (z) ⎞ ⎠ = = m z^{- 1} (\sum_{\begin{matrix} β, γ = 0 2 (γ + β) \equiv_{m} 1 \end{matrix}}^{m - 1} A_{β} (z^{2}) Φ_{2, 2 γ} (z) + \sum_{\begin{matrix} β, γ = 0 2 (γ + β) \equiv_{m} 0 \end{matrix}}^{m - 1} A_{β} (z^{2}) Φ_{2, 2 γ + 1} (z) - \frac{A_{\frac{m + 1}{2}} (z^{2})}{2} - \sum_{γ = 0}^{m - 1} A_{m - γ} (z^{2}) Φ_{2, 2 γ + 1} (z)) = m z^{- 1} ⎛ ⎝ \sum_{γ = 0}^{\frac{m + 1}{2}} A_{\frac{m + 1}{2} - γ} (z^{2}) Φ_{2, 2 γ} (z) + \sum_{γ = \frac{m + 3}{2}}^{m - 1} A_{\frac{3 m + 1}{2} - γ} (z^{2}) Φ_{2, 2 γ} (z) - \frac{A_{\frac{m + 1}{2}} (z^{2})}{2} + A_{0} (z^{2}) Φ_{2, 1} (z) + \sum_{γ = 1}^{m - 1} A_{m - γ} (z^{2}) Φ_{2, 2 γ + 1} (z) - \sum_{γ = 0}^{m - 1} A_{m - γ} (z^{2}) Φ_{2, 2 γ + 1} (z)) = m z^{- 1} ⎛ ⎝ \sum_{γ = 0}^{m - 1} A_{\frac{m + 1}{2} - γ} (z^{2}) Φ_{2, 2 γ} (z) - \frac{A_{\frac{m + 1}{2}} (z^{2})}{2} ⎞ ⎠, \end{matrix}

(23)

where we used the fact that $A_{\frac{3 m + 1}{2} - γ} (z) = A_{\frac{m + 1}{2} - γ} (z)$ . Combining (22) and (23), we obtain

\begin{matrix} \frac{1}{2} & = & \sum_{γ = 0}^{m - 1} Φ_{2, 2 γ} (z^{m}) + m z^{- 1} ⎛ ⎝ \frac{A_{\frac{m + 1}{2}} (z^{2})}{2} - \sum_{γ = 0}^{m - 1} A_{\frac{m + 1}{2} - γ} (z^{2}) Φ_{2, 2 γ} (z) ⎞ ⎠ . \end{matrix}

(24)

Recalling (5) and (11), one realizes that all the powers of $z$ of the second term of the right-hand-side of (24) are odd, and thus, since the left-hand side $\frac{1}{2}$ presents a unique power of $z$ which is even, it must hold that

\frac{1}{2} = m - 1 \sum γ = 0 Φ_{2, 2 γ} (z^{m}) = \frac{1}{2} m - 1 \sum γ = 0 \sum k \in Z φ (m k + γ) z^{2 m (m k + γ)},

(25)

which implies $φ (k) = δ_{0, k}$ , $k \in Z$ , and this concludes the proof. ∎

Theorem 3.9.

Let $m = 2 λ, λ \in N ∖ {0, 1}$ . A convergent $m$ -ary subdivision scheme is a dual interpolatory scheme if and only if

\frac{1}{2} + m - 1 \sum γ = 0 Φ_{2, 2 γ + 1} (z^{m}) = m z^{- 1} m - 1 \sum γ = 0 (A_{\frac{m}{2} - γ} (z^{2}) + A_{m - γ} (z^{2})) Φ_{2, 2 γ + 1} (z) .

(26)

Proof.

As for the proof of Theorem 3.8, we start from (18) and we need to further study the right-hand-side. We first observe that no $β \in {0, \dots, m - 1}$ satisfies $2 β \equiv_{m} 1$ , being $m$ even. On the other hand, the only possibility to satisfy $2 (β + γ) \equiv_{m} 0$ for $β, γ \in {0, \dots, m - 1}$ is that $β + γ \in {0, \frac{m}{2}, m, \frac{3 m}{2}}$ , namely $β = γ = 0$ or $β = \frac{m}{2} - γ$ , $β = m - γ$ , $β = \frac{3 m}{2} - γ$ . Thus,

\begin{matrix} \sum_{\begin{matrix} β = 0 2 β \equiv_{m} 1 \end{matrix}}^{m - 1} \frac{A_{β} (z^{2})}{2} + \sum_{\begin{matrix} β, γ = 0 2 (γ + β) \equiv_{m} 0 \end{matrix}}^{m - 1} A_{β} (z^{2}) Φ_{2, 2 γ + 1} (z) = = A_{0} (z^{2}) Φ_{2, 1} (z) + \sum_{γ = 0}^{\frac{m}{2}} A_{\frac{m}{2} - γ} (z^{2}) Φ_{2, 2 γ + 1} (z) + \sum_{γ = 1}^{m - 1} A_{m - γ} (z^{2}) Φ_{2, 2 γ + 1} (z) + \sum_{γ = \frac{m}{2} + 1}^{m - 1} A_{\frac{3 m}{2} - γ} (z^{2}) Φ_{2, 2 γ + 1} (z) = \sum_{γ = 0}^{m - 1} (A_{\frac{m}{2} - γ} (z^{2}) + A_{m - γ} (z^{2})) Φ_{2, 2 γ + 1} (z), \end{matrix}

where the last equality holds since $A_{0} (z) = A_{m} (z)$ and $A_{\frac{3 m}{2} - γ} (z) = A_{\frac{m}{2} - γ} (z)$ . Equation (26) follows then from (18).

Conversely, if (26) holds, we can subtract it from the necessary condition for refinability (10) with $τ = \frac{1}{2}$ and $T = 2$ , getting again (22), while, for the right-hand-side,

\begin{matrix} m z^{- 1} \sum_{β = 0}^{m - 1} \sum_{\begin{matrix} γ = 0 γ + 2 β \equiv_{m} 1 \end{matrix}}^{2 m - 1} A_{β} (z^{2}) Φ_{2, γ} (z) - m z^{- 1} \sum_{γ = 0}^{m - 1} (A_{\frac{m}{2} - γ} (z^{2}) + A_{m - γ} (z^{2})) Φ_{2, 2 γ + 1} (z) = = m z^{- 1} (\sum_{\begin{matrix} β, γ = 0 2 (γ + β) \equiv_{m} 1 \end{matrix}}^{m - 1} A_{β} (z^{2}) Φ_{2, 2 γ} (z) + \sum_{\begin{matrix} β, γ = 0 2 (γ + β) \equiv_{m} 0 \end{matrix}}^{m - 1} A_{β} (z^{2}) Φ_{2, 2 γ + 1} (z) - \sum_{γ = 0}^{m - 1} (A_{\frac{m}{2} - γ} (z^{2}) + A_{m - γ} (z^{2})) Φ_{2, 2 γ + 1} (z)) = 0. \end{matrix}

(27)

Thus we obtain again (25) which leads to $φ (k) = δ_{0, k}$ , $k \in Z$ , so concluding the proof. ∎

4 Constructive Examples

In this section we show a linear algebra approach exploiting (21) and (26) for the construction of dual interpolatory schemes with arity $3$ , $4$ and $5$ , pointing out the pros and cons with respect to known primal interpolatory schemes. Exploiting the various degrees of freedom, this method can be used to search for dual interpolatory schemes with given wanted properties such as given arity, values of the basic limit function ${φ (k + \frac{1}{2})}_{k \in Z}$ , specific support length of the mask/basic limit function and/or degree of polynomial reproduction. The cases when the resulting linear system has no solution, are equivalent to the non-existence of a dual interpolatory scheme that satisfies the required properties.

In what follows matrices and vectors are considered to be bi-infinite where not specified and are labeled by bold uppercase and lowercase letters respectively. When useful, subsets of those matrices and vectors will be denoted using the convenient MatLab notation. Furthermore, we let the reader know that the given estimates of the regularity of the proposed schemes are obtained via joint spectral radius techniques. For the approximation of the joint spectral radius, the MatLab package

t-toolboxes, with parameter

δ = 1 - 1^{- 12}

, has been used. This package implements the modified invariant polytope method by Guglielmi, Mejstrik and Protasov (see MR3886713 ; MR3009529 ; THOMAS2 ). The Hölder regularities are given with a precision of

10^{- 4}

4.1 General Strategy

The linear system to be solved arises from matching the coefficients of the Laurent polynomial in the left- and right-hand side of (21) (when $m$ is odd) or (26) (when $m$ is even). The result is described in the following Proposition.

Proposition 4.10.

Let $m \in N$ , $m > 3$ . Necessary condition for a bi-infinite vector $a$ to be the mask of an $m$ -ary dual interpolatory scheme with basic limit function $φ$ with prescribed values ${{φ (2 k +}$

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2 Necessary Algebraic Condition for Refinability

Remark 2.1.

Theorem 2.2.

Proof.

Remark 2.3.

Remark 2.4.

3 Algebraic Characterization

Lemma 3.5.

Proof.

Corollary 3.6.

Proof.

Remark 3.7.

Theorem 3.8.

Proof.

Theorem 3.9.

Proof.

4 Constructive Examples

4.1 General Strategy

Proposition 4.10.

Dual Univariate Interpolatory Subdivision of Every Arity: Algebraic Characterization and Construction

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1 Introduction

2 Necessary Algebraic Condition for Refinability

Remark 2.1.

Theorem 2.2.

Proof.

Remark 2.3.

Remark 2.4.

3 Algebraic Characterization

Lemma 3.5.

Proof.

Corollary 3.6.

Proof.

Remark 3.7.

Theorem 3.8.

Proof.

Theorem 3.9.

Proof.

4 Constructive Examples

4.1 General Strategy

Proposition 4.10.