# A Family of Binary Univariate Nonstationary Quasi-Interpolatory Subdivision Reproducing Exponential Polynomials

Authors: Baoxing Zhang; Hongchan Zheng; Lulu Pan; Guohua Peng; Weijie Song
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, by suitably using the so-called push-back operation, a connection between the approximating and interpolatory subdivision, a new family of nonstationary subdivision schemes is presented. Each scheme of this family is a quasi-interpolatory scheme and reproduces a certain space of exponential polynomials. This new family of schemes unifies and extends quite a number of the existing interpolatory schemes reproducing exponential polynomials and noninterpolatory schemes like the cubic exponential B-spline scheme. For these new schemes, we investigate their convergence, smoothness, and accuracy and show that they can reach higher smoothness orders than the interpolatory schemes with the same reproduction property and better accuracy than the exponential B-spline schemes. Several examples are given to illustrate the performance of these new schemes.


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## Body

## 1. Introduction

Subdivision schemes are efficient tools to generate smooth curves/surfaces from a given set of discrete control points. Over the last decades, they are shown to be important tools in many fields like CAD/CAM [1, 2], wavelets [3, 4], biomedical imaging [5], and isogeometric analysis [6]. According to whether the refinement rules depend on the recursion level, subdivision schemes can be divided into stationary and nonstationary schemes. Stationary schemes have been extensively studied, for example, in [7, 8]. It is known that stationary schemes generate algebraic polynomials. The nonstationary schemes, however, can generate the richer function spaces, i.e., the exponential polynomial spaces and special curves/surfaces such as hyperbolas/spheres, which cannot be done using stationary schemes (see, e.g., [9–11]). Therefore, there have been continuous works on nonstationary subdivision schemes generating exponential polynomials.In connection with the construction of such nonstationary subdivision, Romani [12] converted three exponential B-spline schemes into interpolatory schemes without changing the generation property. Conti et al. [13] transformed the nonstationary approximating schemes into interpolatory ones with the same generation property. All of these works can be seen as performed using the polynomial correction, which actually operates by taking the convex combination of the approximating subdivision masks to derive new subdivision masks, including the interpolatory ones. For other references on this method, refer [14–17] and the references therein.Apart from polynomial correction, the push-back operation [18], a connection between the approximating and interpolatory subdivision, can also be used to derive interpolatory schemes from the approximating ones. Using this connection, Lin et al. [19] obtained interpolatory surface subdivision from the approximating subdivision. Luo and Qi [20] analyzed the interpolatory subdivision obtained by this connection systematically in the univariate case. For other references on this connection, see also [21–23] and the references therein. Yet, unlike the polynomial correction, most of the works related to the push-back operation, except the work in [24], are restricted to the stationary case, and the reproduction property of the obtained interpolatory schemes depends largely on that of the original approximating schemes [20]. In [24], the authors presented a nonstationary combined subdivision generating/reproducing different exponential polynomials using the push-back operation in a suitable way. However, when reproducing exponential polynomials, this combined scheme only reduces to the existing interpolatory schemes. Since interpolatory schemes are usually less smooth than the approximating ones, in this paper, we aim to give a different try on the use of the push-back operation to construct new nonstationary subdivision schemes with better properties such as reproduction of exponential polynomials with higher smoothness orders. As a by-product, this shows that, just like the polynomial correction, the push-back operation can also be used to construct a nonstationary subdivision with satisfactory properties.In this paper, similar to [24], the displacements in the push-back operation are decomposed into small ones. Yet, to obtain the desired new nonstationary schemes, we use these small displacements in a different way. We point out that each one of these new schemes is a nonstationary quasi-interpolatory scheme. Here, a nonstationary quasi-interpolatory scheme refers to a nonstationary subdivision scheme reproducing a certain space of exponential polynomials. In fact, the nonstationary quasi-interpolatory schemes in this paper also combined approximating/interpolatory subdivision. Thus, they unify and extend quite a number of the existing interpolatory and noninterpolatory schemes reproducing exponential polynomials, including the nonstationary interpolatory 4-point scheme [25], the cubic exponential B-spline scheme. For these new schemes, we show that their asymptotical similar schemes are the schemes S2M with M∈ℤ+ in [26]. Then, based on this result, the convergence, smoothness, and accuracy can be investigated. Each one of these new schemes owns a parameter and can be seen as a parameter-dependent subdivision (see [27]). With suitable choices of this parameter, each new scheme provides a better smoothness property at the expense of slightly larger support than the interpolatory one with the same reproduction property. Besides, this parameter also provides flexibility in curve design. Moreover, each one of these new schemes owns a good approximation order due to the reproduction property. Thus, a nonstationary quasi-interpolatory scheme in this paper is also a nonstationary scheme with a good approximation order. In this way, these new schemes can own higher smoothness orders than the interpolatory ones with the same reproduction property and better accuracy than the exponential B-spline schemes. Several examples are given to illustrate the performance of these new schemes.The rest of this paper is organized as follows: in Section2, we recall some basic knowledge about subdivision schemes. A new family of nonstationary quasi-interpolatory schemes is constructed in Section 3, while the properties of convergence, smoothness, and accuracy are investigated in Section 4. In Section 5, we present some examples and make a comparison with the existing schemes to illustrate the performance of these new schemes. Section 6 concludes this paper.


## 2. Background

In this section, let us recall some basic definitions and results about the subdivision which form the basis of the rest of this paper. Letl0ℤ denote the linear space of real sequences with finite support. Given a sequence of initial control points q0=qj0,j∈ℤ∈l0ℤ, we consider the binary nonstationary subdivision scheme(1)qk+1i=Sakqki≔∑jai−2jkqjk,k∈ℕ0≔ℕ∪0,where Sak is the k-level subdivision operator mapping l0ℤ to l0ℤ, the sequence ak=aik,i∈ℤ∈l0ℤ is the k-level mask with finite support, and we denote this subdivision scheme by Sakk≥0. The k-level symbol corresponding to the mask ak is akz=∑i∈ℤaikzi with the subsymbols a0kz=∑i∈ℤa2ikz2i and a1kz=∑i∈ℤa2i+1kz2i+1 satisfying akz=a0kz+a1kz.Following [28], now we give the definition of convergence of the nonstationary subdivision.Definition 1 (see [28]).
The subdivisionSakk≥0 is termed uniformly convergent if, for an initial control sequence q0, there exists a continuous function fq0∈Cℝ such that (2)limk⟶∞maxi∈ℤ∩​Kfq0i2k−qki=0,where K is any compact set in ℝ and fq0 is nontrivial for at least one initial data sequence. The subdivision Sakk≥0 is termed Cl convergent if the limit function fq0 has continuous derivatives up to order l with l∈ℤ+.
The following definitions and results are needed to investigate the convergence and smoothness of nonstationary subdivision schemes.Definition 2 (see [29]).
A nonstationary subdivision schemeSakk≥0 is said to be asymptotic similar to the stationary subdivision scheme Sa if the masks akk≥0 and a have the same support U (i.e., aik=ai=0 for i∉U) and satisfy (3)limk⟶∞aik=ai,i∈U.Definition 3 (see [30]).
LetDn be the n-th order differential operator. A binary nonstationary subdivision scheme Sakk≥0 is said to satisfy approximate sum rules of order r+1, r∈ℕ0, if (4)μk≔ak1−2,δk≔max0≤η≤r2−kηDηak−1,satisfying(5)∑kμk<∞,∑k2krδk<∞.Theorem 1 (see [30]).
Assume that the nonstationary subdivision schemeSakk≥0 satisfies approximate sum rules of order r+1, r∈ℕ0 and is asymptotic similar to a Cr-convergent stationary scheme Sa. Then, the nonstationary scheme Sakk≥0 is also Cr convergent.Apart from convergence and smoothness, the property of exponential polynomial generation and reproduction (see, for example, [31] for their details) is also important for subdivision schemes due to the close relationship with approximation order and the use in modeling objects of different shapes. Thus, we now review the definition of exponential polynomial spaces and related results as follows:Definition 4 (see [31]).
LetT∈ℤ+, and let γ=γ0,γ1,…,γT with γT≠0 be a finite set of real or imaginary numbers. The space of exponential polynomials VT,γ is defined as (6)VT,γ≔f:ℝ⟶ℂ,f∈CTℝ:∑j=0TγjDjf=0.
The exponential polynomial spaceVT,γ can be characterized by the following lemma:Lemma 1 (see [31]).
Letγz=∑j=0Tγjzj and denote by θi,τii=0,…,N the set of zeros with multiplicity, satisfying(7)Drγθi=0,r=0,…,τi−1,i=0,…,N.Then,(8)T=∑i=0Nτi,VT,γ≔spanxreθix,r=0,…,τi−1,i=0,…,N.The following two theorems, which can be deduced from Proposition 4.2 and Theorem 4.4 in [32], will be useful in discussing the generation/reproduction property of nonstationary subdivision schemes.Theorem 2.
A nonstationary binary subdivision scheme associated with symbolsakzk≥0 generates VT,γ if (9)Drak−zik=0,r=0,…,τi−1,forzik≔e−θi/2k+1,i=1,…,N.Theorem 3.
Withzik=eθi/2k+1,i=1,…,N, a binary nonstationary subdivision scheme associated with symbols akzk≥0 reproduces VT,γ if it generates VT,γ and there exists a shift parameter p such that for each k≥0(10)Drakzik=2zikp−rqrp,qrp=∏j=0r−1p−j,r=1,…,τi−1,1,r=0.


## 3. Nonstationary Quasi-Interpolatory Schemes Reproducing Exponential Polynomials

This section is devoted to the construction of the new family of nonstationary subdivision schemes reproducing exponential polynomials. Before that, we first give an exponential polynomial space, which specializes the space in Lemma 1 and generalizes the space in ([24], Section 6).Givenn,l∈ℤ+, let the set of zeros with multiplicity in Lemma 1 be given as(11)ΓΛn≔0,2τ0,±jt,τj:j=1,…,l,τ0≤2,τ0,τ1≥1,τ0+∑j=1lτj=n+1,where t∈ℝ+∪ι0,π and ι2=−1. In other words, we assume γz in Lemma 1 has N=2l+1 pairwise distinct zeros, and its total number of zeros is T=2∑j=0lτj=2n+2. Then, the exponential polynomial space in Lemma 1, which is actually a 2n+2 dimensional space, can now be rewritten as(12)EPΓΛn≔span1,x,…,x2τ0−1⋃spanxre±jtx:0≤r≤τj−1,±jt,τj∈ΓΛn,j=1,…,l.Here, whenn=0, let ΓΛ0≔0,2 and we get EPΓΛ0≔span1,x. When n=1, we have τ0+∑j=1lτj=2. Since τ0,τ1≥1, we have l=1, τ0=τ1=1 and then ΓΛ1=0,2,±t,1. Thus, EPΓΛ1=span1,x,e±tx. Note that compared with the space in ([24] Section 6), this new one contains more exponentials like xre±tx, r≥2.Now, we try to construct the desired nonstationary subdivision schemes, which are, in fact, quasi-interpolatory ones reproducing exponential polynomials inEPΓΛn. In order to describe the construction clearly, we first derive the schemes reproducing EPΓΛ0 and EPΓΛ1. Then, by taking a generalization, we obtain the general scheme reproducing EPΓΛn.


### 3.1. The Nonstationary Quasi-Interpolatory Schemes ReproducingEPΓΛ0 and EPΓΛ1

Now, we construct the desired nonstationary schemes reproducingEPΓΛ0 and EPΓΛ1.


#### 3.1.1. The Cubic Exponential B-Spline Scheme and the Nonstationary Quasi-Interpolatory Scheme ReproducingEPΓΛ0

We start from the cubic exponential B-spline scheme. As it is known, fork≥0, the cubic exponential B-spline scheme generates the refined data sequence Pik+1i∈ℤ from the coarser data sequence piki∈ℤ through the refinement rules(13)P2ik+1=141+vk+1Pi−1k+Pi+1k+1+2vk+121+vk+1pik,P2i+1k+1=12pik+Pi+1k,where(14)vk+1≔etk+1+e−tk+12,tk+1=t2k+1,t∈ℝ+∪ι0,π.The cubic exponential B-spline scheme (1) is known to generate the spaceEPΓΛ1. Based on this scheme, by making a suitable modification to its refinement rules [24], a nonstationary combined subdivision scheme can be obtained as(15)P2ik+1=P˜2ik+1+α0kΔpik,P2i+1k+1=P˜2i+1k+1−β0kΔpik+ΔPi+1k,where α0k and β0k are constants depending on k, Δpik≔−Pi−1k−2pik+Pi+1k and P˜2ik+1=1/41+vk+1Pi−1k+Pi+1k+1+2vk+1/21+vk+1pik, P˜2i+1k+1=1/2pik+Pi+1k are the two points provided by the cubic exponential B-spline scheme (1).The nonstationary combined scheme (3) can be seen as obtained by moving the pointsP˜2ik+1 and P˜2i+1k+1 to new positions according to the displacements α0kΔpik and −β0kΔpik+ΔPi+1k, respectively. From [24], scheme (3) unifies several existing subdivision schemes, including the nonstationary interpolatory 4-point scheme [25] and the cubic exponential B-spline scheme (1). In particular, when β0k=0, the combined scheme (3) can be rewritten as(16)P2ik+1=141+vk+1−α0kPi−1k+Pi+1k+1+2vk+121+vk+1+2α0kpik,P2i+1k+1=12pik+Pi+1k.By writing down the symbols of the scheme (4), from Theorem 3, it can be seen that the scheme (4), if convergent, reproducesEPΓΛ0. Besides, when α0k=0, the scheme (4) reduces to the cubic exponential B-spline scheme (1), and when α0k=1/41+vk+1, it becomes the D-D 2-point scheme. In this way, the scheme (4) is a nonstationary scheme reproducing EPΓΛ0, which combines the D-D 2-point scheme and the cubic exponential B-spline scheme (1).


#### 3.1.2. The Nonstationary Quasi-Interpolatory Scheme ReproducingEPΓΛ1

Now, we try to obtain the quasi-interpolatory subdivision reproducingEPΓΛ1, which reproduces more exponential polynomials than the scheme (4).In fact, we modify both rules of the scheme (4) by adding the term−β0kΔpik+ΔPi+1k to the second rule of 4 and adding additionally the term α1kΔPi−1k+ΔPi+1k to its first rule to obtain the following new scheme:(17)P2ik+1=P˜2ik+α0kΔpik+α1kΔPi−1k+ΔPi+1k,P2i+1k+1=P˜2i+1k−β0kΔpik+ΔPi+1k.The scheme (5) can be seen as obtained by moving the pointsP˜2ik+1 and P˜2i+1k+1, generated by the cubic exponential B-spline scheme (1), according to the displacements α0kΔpik+α1kΔPi−1k+ΔPi+1k and −β0kΔpik+ΔPi+1k, respectively. In this way, the corresponding k-level symbol can be written as(18)akz=a0kz+a1kz,where(19)a0kz=−α1kz4+z−4+141+vk+1−α0k+2α1kz2+z−2+2vk+1+121+vk+1+2α0k−2α1k,a1kz=β0kz3+z−3+12−β0kz+z−1.For the scheme (5), from Theorem 3, by solving the linear system,(20)Diak−1=0,Diak1=2δi,0,ak−e±tk+1=0,ake±tk+1=2,i=0,1.We can find its solution containing a free parameter, which we denote byωk. Here, we set ωk≔α1k. Then, with the parameters in this solution, the scheme (5), if convergent, reproduces EPΓΛ1. We denote this new scheme reproducing EPΓΛ1 by SaΓΛ1,ωkkk≥0, and the corresponding k-level symbol can be written as(21)aΓΛ1,ωkkz=−ωkz4+z−4−18vk+11+vk+1z3+z−3+4ωkvk+12z2+z−2+1+2vk+128vk+11+vk+1z+z−1+1+2ωk1−4vk+12.In addition, Theorem 2 implies that the schemeSaΓΛ1,ωkkk≥0 also generates the space(22)EPΓΛ2≔1,x,x2,x3,e±txwithΓΛ2=0,4,±t,1,when ωk=vk+1+2/32vk+11+vk+12. Note that, when ωk=0, the scheme SaΓΛ1,ωkkk≥0 reduces to the nonstationary interpolatory 4-point scheme [25], and when ωk=vk+1+2/32vk+11+vk+12, it actually becomes the exponential pseudospline scheme generating EPΓΛ2 (22) and reproducing EPΓΛ1 [33]. Here, we denote by eΓΛ1,1kz and eΓΛ1,2kz the k-level symbols of the nonstationary interpolatory 4-point scheme [25] and the aforementioned exponential pseudospline scheme, which can be written as(23)eΓΛ1,1kz=−18vk+11+vk+1z3+z−3+1+2vk+128vk+11+vk+1z+z−1+1,eΓΛ1,2kz=1+z48z2+2vk+1z+121+vk+1−vk+1+22vk+1vk+1+11+z2+vk+12+2vk+1+2vk+11+vk+1zz−4,respectively. Now, let θk=32vk+11+vk+12/vk+1+2ωk, and then the k-level symbol aΓΛ1,ωkkz can be rewritten as(24)aΓΛ1,ωkkz=1−θkeΓΛ1,1kz+θkeΓΛ1,2kz.In this way, we obtain a new nonstationary scheme reproducingEPΓΛ1, which combines the nonstationary interpolatory 4-point scheme as shown in [25] and the noninterpolatory one with the k-level symbol eΓΛ1,2kz. Note that the k-level symbol in (24) is actually a convex combination of the symbols eΓΛ1,1kz and eΓΛ1,2kz for 0≤θk≤1.Figure1 shows the basic limit functions of this new scheme reproducing EPΓΛ1 with v0=cosπ/4 and ωk=−0.015,0,0.015,0.03. Figure 2 shows the curves generated by this new scheme reproducing EPΓΛ1 with different values of ωk and v0. In particular, when v0=cosπ/4 (solid line in the left column), 1 (solid line in the middle column), and cosh3/5 (solid line in the right column), Figure 2 shows the reproduction of the circle, parabola, and hyperbola, respectively. From Figures 1 and 2, we can also see the effect of the parameter ωk on the shape of the limit functions of this new scheme.Figure 1
Basic limit functions of the newly obtained scheme reproducingEPΓΛ1 with ωk=−0.015,0,0.015,0.03 and v0=cosπ/4.Figure 2
Curves generated by the newly obtained scheme reproducingEPΓΛ1 with ωk=0.14 (top row) and −0.01 (bottom row); v0=0 (dashed line), 0.3 (dotted line), and cosπ/4 (solid line in the left column); v0=1 (solid line in the middle column); and v0=cosh3/5 (solid line in the right column).


### 3.2. Nonstationary Quasi-Interpolatory Schemes ReproducingEPΓΛn

As it can be seen from Section3.1, the two newly derived schemes reproducing EPΓΛ0 and EPΓΛ1 are actually obtained by suitably using the push-back operation. Now, we follow this method and take a generalization to derive nonstationary schemes reproducing general exponential polynomials, which are in EPΓΛn.Specifically speaking, we use more terms likeα1kΔPi−1k+ΔPi+1k and β0kΔpik+ΔPi+1k to modify the first and second rules of the scheme (4) to get(25)P2ik+1=P˜2ik+1+α0kΔpik+Δ2i,nkPk,P2i+1k+1=P˜2i+1k+1−Δ2i+1,nkPk,where(26)Δ2i,nkPk=∑j=1NαjkΔPi−jk+ΔPi+jk,Δ2i+1,nkPk=∑j=0n−1βjkΔPi−jk+ΔPi+1+jk,n≥0,with the k-level symbol(27)akz=a0kz+a1kz,where(28)a0kz=1+2vk+121+vk+1+2α0k−2α1k+141+vk+1z2+z−2+∑j=1n−1−αj−1k+2αjk−αj+1kz2j+z−2j+2αnk−αn−1kz2n+z−2n−αnkz2n+2+z−2n−2,a1kz=12−β0k+β1kz+z−1+∑j=1n−2βj−1k−2βjk+βj+1kz2j+1+z−2j−1+βn−2k−2βn−1kz2n−1+z−2n+1+βn−1kz2n+1+z−2n−1.In other words, the scheme (10) is obtained by moving the pointsP˜2ik+1 and P˜2i+1k+1, generated by the cubic exponential B-spline scheme (1), to new positions according to the displacements α0kΔpik+Δ2i,nk and −Δ2i+1,nk, respectively. In particular, when n=0, we have Δ2i,0k=Δ2i+1,0k=0 and the scheme (10) reduces to the scheme (4). Besides, when α1k=⋯=αnk=0, the scheme (10) becomes the generalized combined scheme ([24], formula (25)).Now, we focus on the reproduction/generation property of the scheme (10) and show that by suitably choosing the parameters, the desired scheme reproducingEPΓΛn can be obtained. In fact, we have the following result.Theorem 4.
Given the setΓΛn with n≥1, with suitable choices of the parameters α0k,…,αnk, β0k,…,βn−1k, the scheme (10), if convergent, reproduces EPΓΛn.Proof.
Since the case ofn=1 has been investigated in Section 3.1.2, here we only need to investigate the case of n≥2. From Theorem 3, the scheme (10) reproduces EPΓΛn, if for p=0,1,…,2τ0−1 and q=0,1,…,τj−1, j=1,…,l, the corresponding k-level symbol in (27) satisfies(29)Dpak−1=0,Dqak−e±jtk+1=0,Dpak1=2δp,0,Dqake±jtk+1=2δq,0.
The two subsymbolsa0kz and a1kz satisying akz=a0kz+a1kz satisfy(30)a0kz=a0k−z,a1k−z=−a1kz,respectively. Thus, for r∈ℕ0, Dra0kz is an even function if r is an even number and odd function if r is an odd number, while Dra1kz is an even function if r is an odd number and odd function if r is an even number. Therefore, for r∈ℕ0, the subsymbols a0kz and a1kz satisfy(31)Dra0k−e±νtk+1=−1rDra0ke±νtk+1,Dra1k−e±νtk+1=−1r+1Dra1ke±νtk+1,where tk+1 is defined as in (14). In this way, from (29), for p=0,1,…,2τ0−1, we have(32)Dpak−1=Dpa0k−1+Dpa1k−1=−1pDpa0k1+−1p+1Dpa1k1=0,Dpak1=Dpa0k1+Dpa1k1=2δp,0.
This leads to the result thatDpa0k1=Dpa1k1=δp,0. The other two equations in (29) can be dealt with in a similar way. In this way, the linear system (29) is actually equivalent to the following two ones:(33)Dpa0k1=δp,0,Dqa0ke±jtk+1=δq,0,Dpa1k1=δp,0,Dqa1ke±jtk+1=δq,0,withp=0,1,…,2τ0−1,q=0,1,…,τj−1,j=1,…,l.
Now, let us find the desiredα0k,…,αnk, β0k,…,βn−1k by solving the linear systems in (33). For the first linear system in (33), we make a substitution as follows:(34)h0k=21+2vk+141+vk+1+α0k−α1k,h1k=141+vk+1−α0k+2α1k−α2k,hjk=−αj−1k+2αjk−αj+1kj=2,…,n−1,…,hnk=2αnk−αn−1k,hn+1k=−αnk.
Then, the first linear system in (33) can be rewritten as(35)Akhk=g.
Here,Ak is the 2n+2×n+2 matrix Ak=A0k⊤,A1k⊤,A¯1k⊤,…,Alk⊤,A¯lk⊤⊤ with(36)A0k=1D0z2+z−21…D0z2n+1+z−2n+110D1z2+z−21…D1z2n+1+z−2n+11⋮⋮⋮⋮0D2τ0−1z2+z−21…D2τ0−1z2n+1+z−2n+11,Ajk=1D0z2+z−2ejtk+1…D0z2n+1+z−2n+1ejtk+10D1z2+z−2ejtk+1…D1z2n+1+z−2n+1ejtk+1⋮⋮⋮⋮0Dτj−1z2+z−2ejtk+1…Dτj−1z2n+1+z−2n+1ejtk+1,j=1,…,l,Ajk=1D0z2+z−2e−jtk+1…D0z2n+1+z−2n+1e−jtk+10D1z2+z−2e−jtk+1…D1z2n+1+z−2n+1e−jtk+1⋮⋮⋮⋮0Dτj−1z2+z−2e−jtk+1…Dτj−1z2n+1+z−2n+1e−jtk+1,j=1,…,l,where hk=h0k,…,hn+1k⊤ and g is the 2n+2×1 vector g=1,0,…,0︷2τ0−1−times,1,0,…,0︷τ1−1−times,1,0,…,0︷τ1−1−times,…,1,0,…,0︷τl−1−times,1,0,…,0︷τl−1−times. Note that with the substitution in (34), a0kz satisfies(37)D1a0k1=0,D3a0k1=−3D2a0k1.
Besides, due to the symmetry ofa0kz (symmetric about z0), if q∈ℤ+, we have(38)Dqa0kejtk+1=0⟺Dqa0ke−jtk+1=0.
Thus, by a process of row operation (Gaussian elimination), it can be seen that the rank ofAk and Ak∣g is τ0+∑j=1lτj=n+1. Since the unknown hk is a n+2×1 vector, the linear system (35) can be solved containing a free parameter.
Sincehk has been found, let us now fix the desired parameters α0k,…,αnk from hk. In fact, through the substitution in (34), we obtain a linear system A0A0k=h¯k, where A0 is the coefficient matrix, which is a n+2×n+1 matrix, A0k=α0k,…,αnk⊤, h¯k=h0k−1+2vk+1/21+vk+1,h1k−9,h2k,…,hn+1k⊤. Note that h0k,h1k,…,hn+1k satisfy h0k+∑j=1n+12hjk+1=1. Then, by the row operation (Gaussian elimination), this linear system is uniquely solvable. In this way, the desired parameters α0k,…,αnk are found containing a free parameter. Similarly, through the second linear system in (33), the parameters β0k,…,βn−1k can also be uniquely determined. Then, with such chosen parameters, the scheme (10), if convergent, reproduces EPΓΛn.Remark 1.
We denote byωk the free parameter obtained in the proof of Theorem 4. For this free parameter, we set ωk=−1n+1αnk. Then, if ωk is chosen to be 0, the first linear system in (33) can be uniquely solved and the solution can be found with α0k=1/41+vk+1, α1k=⋯=αnk=0. This solution forces the subsymbol a0kz to be a0kz=1, which implies that the obtained scheme in Theorem 4 is an interpolatory scheme. Besides, with Theorem 2, it can be shown that, with another suitable choice of ωk, this obtained scheme in Theorem 4 also generates EPΓΛn+1 with EPΓΛn⊂EPΓΛn+1.Remark 2.
For the casen=0, the desired scheme reproducing EPΓΛ0 is actually the scheme (4). From Section 3.1.1, by setting ωk=−α0k−1/41+vk+1, it reduces to the D-D 2-point scheme reproducing EPΓΛ0 when ωk=0 and becomes the cubic exponential B-spline scheme (1) generating EPΓΛ1 and reproducing EPΓΛ0 when ωk=1/41+vk+1. If, in addition, θk=4ωk1+vk+1, the corresponding k-level symbol can be rewritten as aΓΛ0,ωkkz=1−θkeΓΛ0,1z+θkeΓΛ0,2kz, where eΓΛ0,1z denotes the symbol of the D-D 2-point scheme and eΓΛ0,2kz denotes the k-level symbol of the cubic exponential B-spline scheme (1).
In this way, given the setΓΛn with n≥0, we can derive the desired nonstationary scheme reproducing EPΓΛn, which, in fact, is a quasi-interpolatory scheme combining the interpolatory and noninterpolatory schemes with the same reproduction property. We denote by SaΓΛn,ωkkk≥0 this newly obtained scheme and by aΓΛn,ωkkz the corresponding k-level symbol. In particular, when n=0, SaΓΛ0,ωkkk≥0 refers to the equivalent form of the scheme (4) obtained using the substitution in Remark 2. From the proof of Theorem 4, given a nonstationary scheme with the k-level symbol hkz, which is supported on −2n−2,2n+2 and reproduces EPΓΛn, we can find the parameters α0k,…,αnk, β0k,…,βn−1k such that akz in (27) satifies (29) (akz actually becomes hkz), and thus the corresponding nonstationary scheme reproduces EPΓΛn. This means that the nonstationary scheme with the k-level symbol hkz falls into the family of the nonstationary schemes in this paper, and these new schemes in this paper can actually unify the schemes which are supported on −2n−2,2n+2 and reproduce EPΓΛn.
We point out that the aforementioned process to construct these nonstationary quasi-interpolatory schemes can be generalized to the case of any arity. Note that by the generating function approach, Muntingh [34] derived the expressions for the symbols of the symmetric pseudosplines of any arity. Here, in a similar way, we can also obtain the generalization of these new nonstationary schemes to the case of any arity. Besides, by the same method, we can also get the nonstationary generalization of the work in [34], i.e., the exponential pseudosplines of any arity.


## 3.1. The Nonstationary Quasi-Interpolatory Schemes ReproducingEPΓΛ0 and EPΓΛ1

Now, we construct the desired nonstationary schemes reproducingEPΓΛ0 and EPΓΛ1.


### 3.1.1. The Cubic Exponential B-Spline Scheme and the Nonstationary Quasi-Interpolatory Scheme ReproducingEPΓΛ0

We start from the cubic exponential B-spline scheme. As it is known, fork≥0, the cubic exponential B-spline scheme generates the refined data sequence Pik+1i∈ℤ from the coarser data sequence piki∈ℤ through the refinement rules(13)P2ik+1=141+vk+1Pi−1k+Pi+1k+1+2vk+121+vk+1pik,P2i+1k+1=12pik+Pi+1k,where(14)vk+1≔etk+1+e−tk+12,tk+1=t2k+1,t∈ℝ+∪ι0,π.The cubic exponential B-spline scheme (1) is known to generate the spaceEPΓΛ1. Based on this scheme, by making a suitable modification to its refinement rules [24], a nonstationary combined subdivision scheme can be obtained as(15)P2ik+1=P˜2ik+1+α0kΔpik,P2i+1k+1=P˜2i+1k+1−β0kΔpik+ΔPi+1k,where α0k and β0k are constants depending on k, Δpik≔−Pi−1k−2pik+Pi+1k and P˜2ik+1=1/41+vk+1Pi−1k+Pi+1k+1+2vk+1/21+vk+1pik, P˜2i+1k+1=1/2pik+Pi+1k are the two points provided by the cubic exponential B-spline scheme (1).The nonstationary combined scheme (3) can be seen as obtained by moving the pointsP˜2ik+1 and P˜2i+1k+1 to new positions according to the displacements α0kΔpik and −β0kΔpik+ΔPi+1k, respectively. From [24], scheme (3) unifies several existing subdivision schemes, including the nonstationary interpolatory 4-point scheme [25] and the cubic exponential B-spline scheme (1). In particular, when β0k=0, the combined scheme (3) can be rewritten as(16)P2ik+1=141+vk+1−α0kPi−1k+Pi+1k+1+2vk+121+vk+1+2α0kpik,P2i+1k+1=12pik+Pi+1k.By writing down the symbols of the scheme (4), from Theorem 3, it can be seen that the scheme (4), if convergent, reproducesEPΓΛ0. Besides, when α0k=0, the scheme (4) reduces to the cubic exponential B-spline scheme (1), and when α0k=1/41+vk+1, it becomes the D-D 2-point scheme. In this way, the scheme (4) is a nonstationary scheme reproducing EPΓΛ0, which combines the D-D 2-point scheme and the cubic exponential B-spline scheme (1).


### 3.1.2. The Nonstationary Quasi-Interpolatory Scheme ReproducingEPΓΛ1

Now, we try to obtain the quasi-interpolatory subdivision reproducingEPΓΛ1, which reproduces more exponential polynomials than the scheme (4).In fact, we modify both rules of the scheme (4) by adding the term−β0kΔpik+ΔPi+1k to the second rule of 4 and adding additionally the term α1kΔPi−1k+ΔPi+1k to its first rule to obtain the following new scheme:(17)P2ik+1=P˜2ik+α0kΔpik+α1kΔPi−1k+ΔPi+1k,P2i+1k+1=P˜2i+1k−β0kΔpik+ΔPi+1k.The scheme (5) can be seen as obtained by moving the pointsP˜2ik+1 and P˜2i+1k+1, generated by the cubic exponential B-spline scheme (1), according to the displacements α0kΔpik+α1kΔPi−1k+ΔPi+1k and −β0kΔpik+ΔPi+1k, respectively. In this way, the corresponding k-level symbol can be written as(18)akz=a0kz+a1kz,where(19)a0kz=−α1kz4+z−4+141+vk+1−α0k+2α1kz2+z−2+2vk+1+121+vk+1+2α0k−2α1k,a1kz=β0kz3+z−3+12−β0kz+z−1.For the scheme (5), from Theorem 3, by solving the linear system,(20)Diak−1=0,Diak1=2δi,0,ak−e±tk+1=0,ake±tk+1=2,i=0,1.We can find its solution containing a free parameter, which we denote byωk. Here, we set ωk≔α1k. Then, with the parameters in this solution, the scheme (5), if convergent, reproduces EPΓΛ1. We denote this new scheme reproducing EPΓΛ1 by SaΓΛ1,ωkkk≥0, and the corresponding k-level symbol can be written as(21)aΓΛ1,ωkkz=−ωkz4+z−4−18vk+11+vk+1z3+z−3+4ωkvk+12z2+z−2+1+2vk+128vk+11+vk+1z+z−1+1+2ωk1−4vk+12.In addition, Theorem 2 implies that the schemeSaΓΛ1,ωkkk≥0 also generates the space(22)EPΓΛ2≔1,x,x2,x3,e±txwithΓΛ2=0,4,±t,1,when ωk=vk+1+2/32vk+11+vk+12. Note that, when ωk=0, the scheme SaΓΛ1,ωkkk≥0 reduces to the nonstationary interpolatory 4-point scheme [25], and when ωk=vk+1+2/32vk+11+vk+12, it actually becomes the exponential pseudospline scheme generating EPΓΛ2 (22) and reproducing EPΓΛ1 [33]. Here, we denote by eΓΛ1,1kz and eΓΛ1,2kz the k-level symbols of the nonstationary interpolatory 4-point scheme [25] and the aforementioned exponential pseudospline scheme, which can be written as(23)eΓΛ1,1kz=−18vk+11+vk+1z3+z−3+1+2vk+128vk+11+vk+1z+z−1+1,eΓΛ1,2kz=1+z48z2+2vk+1z+121+vk+1−vk+1+22vk+1vk+1+11+z2+vk+12+2vk+1+2vk+11+vk+1zz−4,respectively. Now, let θk=32vk+11+vk+12/vk+1+2ωk, and then the k-level symbol aΓΛ1,ωkkz can be rewritten as(24)aΓΛ1,ωkkz=1−θkeΓΛ1,1kz+θkeΓΛ1,2kz.In this way, we obtain a new nonstationary scheme reproducingEPΓΛ1, which combines the nonstationary interpolatory 4-point scheme as shown in [25] and the noninterpolatory one with the k-level symbol eΓΛ1,2kz. Note that the k-level symbol in (24) is actually a convex combination of the symbols eΓΛ1,1kz and eΓΛ1,2kz for 0≤θk≤1.Figure1 shows the basic limit functions of this new scheme reproducing EPΓΛ1 with v0=cosπ/4 and ωk=−0.015,0,0.015,0.03. Figure 2 shows the curves generated by this new scheme reproducing EPΓΛ1 with different values of ωk and v0. In particular, when v0=cosπ/4 (solid line in the left column), 1 (solid line in the middle column), and cosh3/5 (solid line in the right column), Figure 2 shows the reproduction of the circle, parabola, and hyperbola, respectively. From Figures 1 and 2, we can also see the effect of the parameter ωk on the shape of the limit functions of this new scheme.Figure 1
Basic limit functions of the newly obtained scheme reproducingEPΓΛ1 with ωk=−0.015,0,0.015,0.03 and v0=cosπ/4.Figure 2
Curves generated by the newly obtained scheme reproducingEPΓΛ1 with ωk=0.14 (top row) and −0.01 (bottom row); v0=0 (dashed line), 0.3 (dotted line), and cosπ/4 (solid line in the left column); v0=1 (solid line in the middle column); and v0=cosh3/5 (solid line in the right column).


## 3.1.1. The Cubic Exponential B-Spline Scheme and the Nonstationary Quasi-Interpolatory Scheme ReproducingEPΓΛ0

We start from the cubic exponential B-spline scheme. As it is known, fork≥0, the cubic exponential B-spline scheme generates the refined data sequence Pik+1i∈ℤ from the coarser data sequence piki∈ℤ through the refinement rules(13)P2ik+1=141+vk+1Pi−1k+Pi+1k+1+2vk+121+vk+1pik,P2i+1k+1=12pik+Pi+1k,where(14)vk+1≔etk+1+e−tk+12,tk+1=t2k+1,t∈ℝ+∪ι0,π.The cubic exponential B-spline scheme (1) is known to generate the spaceEPΓΛ1. Based on this scheme, by making a suitable modification to its refinement rules [24], a nonstationary combined subdivision scheme can be obtained as(15)P2ik+1=P˜2ik+1+α0kΔpik,P2i+1k+1=P˜2i+1k+1−β0kΔpik+ΔPi+1k,where α0k and β0k are constants depending on k, Δpik≔−Pi−1k−2pik+Pi+1k and P˜2ik+1=1/41+vk+1Pi−1k+Pi+1k+1+2vk+1/21+vk+1pik, P˜2i+1k+1=1/2pik+Pi+1k are the two points provided by the cubic exponential B-spline scheme (1).The nonstationary combined scheme (3) can be seen as obtained by moving the pointsP˜2ik+1 and P˜2i+1k+1 to new positions according to the displacements α0kΔpik and −β0kΔpik+ΔPi+1k, respectively. From [24], scheme (3) unifies several existing subdivision schemes, including the nonstationary interpolatory 4-point scheme [25] and the cubic exponential B-spline scheme (1). In particular, when β0k=0, the combined scheme (3) can be rewritten as(16)P2ik+1=141+vk+1−α0kPi−1k+Pi+1k+1+2vk+121+vk+1+2α0kpik,P2i+1k+1=12pik+Pi+1k.By writing down the symbols of the scheme (4), from Theorem 3, it can be seen that the scheme (4), if convergent, reproducesEPΓΛ0. Besides, when α0k=0, the scheme (4) reduces to the cubic exponential B-spline scheme (1), and when α0k=1/41+vk+1, it becomes the D-D 2-point scheme. In this way, the scheme (4) is a nonstationary scheme reproducing EPΓΛ0, which combines the D-D 2-point scheme and the cubic exponential B-spline scheme (1).


## 3.1.2. The Nonstationary Quasi-Interpolatory Scheme ReproducingEPΓΛ1

Now, we try to obtain the quasi-interpolatory subdivision reproducingEPΓΛ1, which reproduces more exponential polynomials than the scheme (4).In fact, we modify both rules of the scheme (4) by adding the term−β0kΔpik+ΔPi+1k to the second rule of 4 and adding additionally the term α1kΔPi−1k+ΔPi+1k to its first rule to obtain the following new scheme:(17)P2ik+1=P˜2ik+α0kΔpik+α1kΔPi−1k+ΔPi+1k,P2i+1k+1=P˜2i+1k−β0kΔpik+ΔPi+1k.The scheme (5) can be seen as obtained by moving the pointsP˜2ik+1 and P˜2i+1k+1, generated by the cubic exponential B-spline scheme (1), according to the displacements α0kΔpik+α1kΔPi−1k+ΔPi+1k and −β0kΔpik+ΔPi+1k, respectively. In this way, the corresponding k-level symbol can be written as(18)akz=a0kz+a1kz,where(19)a0kz=−α1kz4+z−4+141+vk+1−α0k+2α1kz2+z−2+2vk+1+121+vk+1+2α0k−2α1k,a1kz=β0kz3+z−3+12−β0kz+z−1.For the scheme (5), from Theorem 3, by solving the linear system,(20)Diak−1=0,Diak1=2δi,0,ak−e±tk+1=0,ake±tk+1=2,i=0,1.We can find its solution containing a free parameter, which we denote byωk. Here, we set ωk≔α1k. Then, with the parameters in this solution, the scheme (5), if convergent, reproduces EPΓΛ1. We denote this new scheme reproducing EPΓΛ1 by SaΓΛ1,ωkkk≥0, and the corresponding k-level symbol can be written as(21)aΓΛ1,ωkkz=−ωkz4+z−4−18vk+11+vk+1z3+z−3+4ωkvk+12z2+z−2+1+2vk+128vk+11+vk+1z+z−1+1+2ωk1−4vk+12.In addition, Theorem 2 implies that the schemeSaΓΛ1,ωkkk≥0 also generates the space(22)EPΓΛ2≔1,x,x2,x3,e±txwithΓΛ2=0,4,±t,1,when ωk=vk+1+2/32vk+11+vk+12. Note that, when ωk=0, the scheme SaΓΛ1,ωkkk≥0 reduces to the nonstationary interpolatory 4-point scheme [25], and when ωk=vk+1+2/32vk+11+vk+12, it actually becomes the exponential pseudospline scheme generating EPΓΛ2 (22) and reproducing EPΓΛ1 [33]. Here, we denote by eΓΛ1,1kz and eΓΛ1,2kz the k-level symbols of the nonstationary interpolatory 4-point scheme [25] and the aforementioned exponential pseudospline scheme, which can be written as(23)eΓΛ1,1kz=−18vk+11+vk+1z3+z−3+1+2vk+128vk+11+vk+1z+z−1+1,eΓΛ1,2kz=1+z48z2+2vk+1z+121+vk+1−vk+1+22vk+1vk+1+11+z2+vk+12+2vk+1+2vk+11+vk+1zz−4,respectively. Now, let θk=32vk+11+vk+12/vk+1+2ωk, and then the k-level symbol aΓΛ1,ωkkz can be rewritten as(24)aΓΛ1,ωkkz=1−θkeΓΛ1,1kz+θkeΓΛ1,2kz.In this way, we obtain a new nonstationary scheme reproducingEPΓΛ1, which combines the nonstationary interpolatory 4-point scheme as shown in [25] and the noninterpolatory one with the k-level symbol eΓΛ1,2kz. Note that the k-level symbol in (24) is actually a convex combination of the symbols eΓΛ1,1kz and eΓΛ1,2kz for 0≤θk≤1.Figure1 shows the basic limit functions of this new scheme reproducing EPΓΛ1 with v0=cosπ/4 and ωk=−0.015,0,0.015,0.03. Figure 2 shows the curves generated by this new scheme reproducing EPΓΛ1 with different values of ωk and v0. In particular, when v0=cosπ/4 (solid line in the left column), 1 (solid line in the middle column), and cosh3/5 (solid line in the right column), Figure 2 shows the reproduction of the circle, parabola, and hyperbola, respectively. From Figures 1 and 2, we can also see the effect of the parameter ωk on the shape of the limit functions of this new scheme.Figure 1
Basic limit functions of the newly obtained scheme reproducingEPΓΛ1 with ωk=−0.015,0,0.015,0.03 and v0=cosπ/4.Figure 2
Curves generated by the newly obtained scheme reproducingEPΓΛ1 with ωk=0.14 (top row) and −0.01 (bottom row); v0=0 (dashed line), 0.3 (dotted line), and cosπ/4 (solid line in the left column); v0=1 (solid line in the middle column); and v0=cosh3/5 (solid line in the right column).


## 3.2. Nonstationary Quasi-Interpolatory Schemes ReproducingEPΓΛn

As it can be seen from Section3.1, the two newly derived schemes reproducing EPΓΛ0 and EPΓΛ1 are actually obtained by suitably using the push-back operation. Now, we follow this method and take a generalization to derive nonstationary schemes reproducing general exponential polynomials, which are in EPΓΛn.Specifically speaking, we use more terms likeα1kΔPi−1k+ΔPi+1k and β0kΔpik+ΔPi+1k to modify the first and second rules of the scheme (4) to get(25)P2ik+1=P˜2ik+1+α0kΔpik+Δ2i,nkPk,P2i+1k+1=P˜2i+1k+1−Δ2i+1,nkPk,where(26)Δ2i,nkPk=∑j=1NαjkΔPi−jk+ΔPi+jk,Δ2i+1,nkPk=∑j=0n−1βjkΔPi−jk+ΔPi+1+jk,n≥0,with the k-level symbol(27)akz=a0kz+a1kz,where(28)a0kz=1+2vk+121+vk+1+2α0k−2α1k+141+vk+1z2+z−2+∑j=1n−1−αj−1k+2αjk−αj+1kz2j+z−2j+2αnk−αn−1kz2n+z−2n−αnkz2n+2+z−2n−2,a1kz=12−β0k+β1kz+z−1+∑j=1n−2βj−1k−2βjk+βj+1kz2j+1+z−2j−1+βn−2k−2βn−1kz2n−1+z−2n+1+βn−1kz2n+1+z−2n−1.In other words, the scheme (10) is obtained by moving the pointsP˜2ik+1 and P˜2i+1k+1, generated by the cubic exponential B-spline scheme (1), to new positions according to the displacements α0kΔpik+Δ2i,nk and −Δ2i+1,nk, respectively. In particular, when n=0, we have Δ2i,0k=Δ2i+1,0k=0 and the scheme (10) reduces to the scheme (4). Besides, when α1k=⋯=αnk=0, the scheme (10) becomes the generalized combined scheme ([24], formula (25)).Now, we focus on the reproduction/generation property of the scheme (10) and show that by suitably choosing the parameters, the desired scheme reproducingEPΓΛn can be obtained. In fact, we have the following result.Theorem 4.
Given the setΓΛn with n≥1, with suitable choices of the parameters α0k,…,αnk, β0k,…,βn−1k, the scheme (10), if convergent, reproduces EPΓΛn.Proof.
Since the case ofn=1 has been investigated in Section 3.1.2, here we only need to investigate the case of n≥2. From Theorem 3, the scheme (10) reproduces EPΓΛn, if for p=0,1,…,2τ0−1 and q=0,1,…,τj−1, j=1,…,l, the corresponding k-level symbol in (27) satisfies(29)Dpak−1=0,Dqak−e±jtk+1=0,Dpak1=2δp,0,Dqake±jtk+1=2δq,0.
The two subsymbolsa0kz and a1kz satisying akz=a0kz+a1kz satisfy(30)a0kz=a0k−z,a1k−z=−a1kz,respectively. Thus, for r∈ℕ0, Dra0kz is an even function if r is an even number and odd function if r is an odd number, while Dra1kz is an even function if r is an odd number and odd function if r is an even number. Therefore, for r∈ℕ0, the subsymbols a0kz and a1kz satisfy(31)Dra0k−e±νtk+1=−1rDra0ke±νtk+1,Dra1k−e±νtk+1=−1r+1Dra1ke±νtk+1,where tk+1 is defined as in (14). In this way, from (29), for p=0,1,…,2τ0−1, we have(32)Dpak−1=Dpa0k−1+Dpa1k−1=−1pDpa0k1+−1p+1Dpa1k1=0,Dpak1=Dpa0k1+Dpa1k1=2δp,0.
This leads to the result thatDpa0k1=Dpa1k1=δp,0. The other two equations in (29) can be dealt with in a similar way. In this way, the linear system (29) is actually equivalent to the following two ones:(33)Dpa0k1=δp,0,Dqa0ke±jtk+1=δq,0,Dpa1k1=δp,0,Dqa1ke±jtk+1=δq,0,withp=0,1,…,2τ0−1,q=0,1,…,τj−1,j=1,…,l.
Now, let us find the desiredα0k,…,αnk, β0k,…,βn−1k by solving the linear systems in (33). For the first linear system in (33), we make a substitution as follows:(34)h0k=21+2vk+141+vk+1+α0k−α1k,h1k=141+vk+1−α0k+2α1k−α2k,hjk=−αj−1k+2αjk−αj+1kj=2,…,n−1,…,hnk=2αnk−αn−1k,hn+1k=−αnk.
Then, the first linear system in (33) can be rewritten as(35)Akhk=g.
Here,Ak is the 2n+2×n+2 matrix Ak=A0k⊤,A1k⊤,A¯1k⊤,…,Alk⊤,A¯lk⊤⊤ with(36)A0k=1D0z2+z−21…D0z2n+1+z−2n+110D1z2+z−21…D1z2n+1+z−2n+11⋮⋮⋮⋮0D2τ0−1z2+z−21…D2τ0−1z2n+1+z−2n+11,Ajk=1D0z2+z−2ejtk+1…D0z2n+1+z−2n+1ejtk+10D1z2+z−2ejtk+1…D1z2n+1+z−2n+1ejtk+1⋮⋮⋮⋮0Dτj−1z2+z−2ejtk+1…Dτj−1z2n+1+z−2n+1ejtk+1,j=1,…,l,Ajk=1D0z2+z−2e−jtk+1…D0z2n+1+z−2n+1e−jtk+10D1z2+z−2e−jtk+1…D1z2n+1+z−2n+1e−jtk+1⋮⋮⋮⋮0Dτj−1z2+z−2e−jtk+1…Dτj−1z2n+1+z−2n+1e−jtk+1,j=1,…,l,where hk=h0k,…,hn+1k⊤ and g is the 2n+2×1 vector g=1,0,…,0︷2τ0−1−times,1,0,…,0︷τ1−1−times,1,0,…,0︷τ1−1−times,…,1,0,…,0︷τl−1−times,1,0,…,0︷τl−1−times. Note that with the substitution in (34), a0kz satisfies(37)D1a0k1=0,D3a0k1=−3D2a0k1.
Besides, due to the symmetry ofa0kz (symmetric about z0), if q∈ℤ+, we have(38)Dqa0kejtk+1=0⟺Dqa0ke−jtk+1=0.
Thus, by a process of row operation (Gaussian elimination), it can be seen that the rank ofAk and Ak∣g is τ0+∑j=1lτj=n+1. Since the unknown hk is a n+2×1 vector, the linear system (35) can be solved containing a free parameter.
Sincehk has been found, let us now fix the desired parameters α0k,…,αnk from hk. In fact, through the substitution in (34), we obtain a linear system A0A0k=h¯k, where A0 is the coefficient matrix, which is a n+2×n+1 matrix, A0k=α0k,…,αnk⊤, h¯k=h0k−1+2vk+1/21+vk+1,h1k−9,h2k,…,hn+1k⊤. Note that h0k,h1k,…,hn+1k satisfy h0k+∑j=1n+12hjk+1=1. Then, by the row operation (Gaussian elimination), this linear system is uniquely solvable. In this way, the desired parameters α0k,…,αnk are found containing a free parameter. Similarly, through the second linear system in (33), the parameters β0k,…,βn−1k can also be uniquely determined. Then, with such chosen parameters, the scheme (10), if convergent, reproduces EPΓΛn.Remark 1.
We denote byωk the free parameter obtained in the proof of Theorem 4. For this free parameter, we set ωk=−1n+1αnk. Then, if ωk is chosen to be 0, the first linear system in (33) can be uniquely solved and the solution can be found with α0k=1/41+vk+1, α1k=⋯=αnk=0. This solution forces the subsymbol a0kz to be a0kz=1, which implies that the obtained scheme in Theorem 4 is an interpolatory scheme. Besides, with Theorem 2, it can be shown that, with another suitable choice of ωk, this obtained scheme in Theorem 4 also generates EPΓΛn+1 with EPΓΛn⊂EPΓΛn+1.Remark 2.
For the casen=0, the desired scheme reproducing EPΓΛ0 is actually the scheme (4). From Section 3.1.1, by setting ωk=−α0k−1/41+vk+1, it reduces to the D-D 2-point scheme reproducing EPΓΛ0 when ωk=0 and becomes the cubic exponential B-spline scheme (1) generating EPΓΛ1 and reproducing EPΓΛ0 when ωk=1/41+vk+1. If, in addition, θk=4ωk1+vk+1, the corresponding k-level symbol can be rewritten as aΓΛ0,ωkkz=1−θkeΓΛ0,1z+θkeΓΛ0,2kz, where eΓΛ0,1z denotes the symbol of the D-D 2-point scheme and eΓΛ0,2kz denotes the k-level symbol of the cubic exponential B-spline scheme (1).
In this way, given the setΓΛn with n≥0, we can derive the desired nonstationary scheme reproducing EPΓΛn, which, in fact, is a quasi-interpolatory scheme combining the interpolatory and noninterpolatory schemes with the same reproduction property. We denote by SaΓΛn,ωkkk≥0 this newly obtained scheme and by aΓΛn,ωkkz the corresponding k-level symbol. In particular, when n=0, SaΓΛ0,ωkkk≥0 refers to the equivalent form of the scheme (4) obtained using the substitution in Remark 2. From the proof of Theorem 4, given a nonstationary scheme with the k-level symbol hkz, which is supported on −2n−2,2n+2 and reproduces EPΓΛn, we can find the parameters α0k,…,αnk, β0k,…,βn−1k such that akz in (27) satifies (29) (akz actually becomes hkz), and thus the corresponding nonstationary scheme reproduces EPΓΛn. This means that the nonstationary scheme with the k-level symbol hkz falls into the family of the nonstationary schemes in this paper, and these new schemes in this paper can actually unify the schemes which are supported on −2n−2,2n+2 and reproduce EPΓΛn.
We point out that the aforementioned process to construct these nonstationary quasi-interpolatory schemes can be generalized to the case of any arity. Note that by the generating function approach, Muntingh [34] derived the expressions for the symbols of the symmetric pseudosplines of any arity. Here, in a similar way, we can also obtain the generalization of these new nonstationary schemes to the case of any arity. Besides, by the same method, we can also get the nonstationary generalization of the work in [34], i.e., the exponential pseudosplines of any arity.


## 4. Analysis of Smoothness and Approximation Order

This section is devoted to the analysis of the convergence, smoothness, and approximation power of the new schemeSaΓΛn,ωkkk≥0. Before that, we need to derive the corresponding asymptotical similar scheme first.In order to derive the asymptotical similar scheme of the schemeSaΓΛn,ωkkk≥0, we try to get their symbols. From Remark 2 and Section 3, it can be seen that the symbols of the nonstationary quasi-interpolatory subdivision reproducing EPΓΛ0 and EPΓΛ1 are actually convex combinations of the symbols of exponential pseudospline schemes. Now, we show that this is true for general n. In fact, recall that from Remark 1, the scheme SaΓΛn,ωkkk≥0 reduces to the interpolatory scheme reproducing EPΓΛn when ωk=0 and becomes a noninterpolatory one generating EPΓΛn+1 and reproducing EPΓΛn with ΓΛn⊂ΓΛn+1 when ωk is chosen to be another suitable value. For these two schemes, let eΓΛn,1k=e1,−2n−1k,0,…,0,e1,−1k,1,e1,1k,0...,0,e1,2n+1k and eΓΛn,2k=e1,−2n−2k,e1,−2n−1k,…,e1,2n+1k,e1,2n+2k denote their k-level masks, and by eΓΛn,1kz and eΓΛn,2kz we denote their corresponding k-level symbols. In particular, when n=1, these two k-level symbols are just the pair of symbols in (23). Then, from ([33], Section 5), these two schemes are just the odd symmetric exponential pseudospline schemes (see [33] for the way to compute their symbols). In this way, the corresponding k-level symbol aΓΛn,ωkkz is actually a convex combination of eΓΛn,1kz and eΓΛn,2kz. In other words, we have the following result.Lemma 2.
Given the setΓΛn, the k-level symbol aΓΛn,ωkkz of the new scheme SaΓΛn,ωkkk≥0 can be written as (39)aΓΛn,ωkkz=1−θkeΓΛn,1kz+θkeΓΛn,2kz,where θk is a free parameter depending on k.Remark 3.
Since the exponential pseudospline schemes in [33] are derived using polynomial correction, Theorem 4 and Lemma 2 imply that quite a number of the nonstationary subdivision schemes obtained using the polynomial correction, including the existing interpolatory schemes reproducing exponential polynomials, such as the ones in [12, 25], can also be derived by suitably using the push-back operation.
Lemma 2 also implies that these nonstationary quasi-interpolatory subdivision schemes are also parameter-dependent schemes (see also [27]). Now, we let θk=ωk/e2,2n+2k=ωk/e2,−2n−2k in Lemma 2. As a result, we choose ωk=e2,2n+2k in the new scheme SaΓΛn,ωkkk≥0 so that it also generates EPΓΛn+1. Now, we make a mild condition on the free parameter ωk. We assume that the limit of ωk as k tends to infinity exists and we denote its limit by ω, that is to say we assume limk⟶∞ωk≔ω∈ℝ exists. Then, together with Lemma 2, we can show that the asymptotical similar scheme of SaΓΛn,ωkkk≥0 is the scheme S2M with M=n+1 in [26]. This is shown in the following result.Proposition 1.
Assumelimk⟶∞ωk≔ω∈ℝ. Then, the asymptotical similar scheme of the new nonstationary scheme SaΓΛn,ωkkk≥0 reproducing EPΓΛn is the stationary scheme S2M with M=n+1.Proof.
Due to the fact that the asymptotical similar counterparts of the exponential pseudosplines are the pseudosplines [33], we have (40)limk⟶∞eΓΛn,1kz=an+1nz,limk⟶∞eΓΛn,2kz=an+2nz,where(41)amlz=21+z24zm∑i=0lm+li−1−z24zi1+z24zl−i,m,l∈ℕ0,denotes the pseudosplines of type II [35]. In this way, for the leading term of an+2nz, the absolute value of its coefficient is limk⟶∞e2,2n+2k=2n+2n+1/16n+1. Since θk=ωk/e2,2n+2k, it can be seen that limk⟶∞θk=ω16n+1/2n+2n+1 and we denote this limit by θ. Thus, we have(42)an,ωz≔limk⟶∞aΓΛn,ωkkz=1−θan+1nz+θan+2nz.
Therefore,an,ωz is just the symbol of the scheme S2M with M=n+1 in [26] (see [35]). Thus, the scheme S2M with M=n+1 in [26] is the asymptotical similar scheme of SaΓΛn,ωkkk≥0.Based on Proposition 1, now we investigate theCr convergence of the scheme SaΓΛn,ωkkk≥0. In fact, we have the following result.Theorem 5.
Given the setΓΛn, for the new scheme SaΓΛn,ωkkk≥0, it is Cr convergent with r<2n+4, if the free parameter ωk is such that the asymptotical similar counterpart is Cr convergent.Proof.
From Theorem 4 and Remark 1, the new schemeSaΓΛn,ωkkk≥0 reproduces EPΓΛn and generates EPΓΛn+1 with ΓΛn⊂ΓΛn+1. Then, according to [36], Corollary 14, this new nonstationary scheme satisfies approximate sum rules of order r˜ with r˜=2τ0+2∑​j=1lτl=2n+1+1=2n+4. Then, from Theorem 1, this new nonstationary scheme is Cr convergent, if the free parameter ωk is such that the asymptotical similar subdivision is Cr convergent with r<2n+4.Now, let us discuss the approximation order of the schemeSaΓΛn,ωkkk≥0. Let f be a function satisfying DjfL∞K<∞ with j=0,…,L, L=2n+2 and K being a compact set in ℝ. Assume that the initial data is of the form fκ≔fjκ=fj2−κ,j∈ℤ for some κ∈ℤ+. Then, for the scheme SaΓΛn,ωkkk≥0, according to [36], Theorem 21 (see also [37]), we have(43)f∞−fL∞K≤cf2−κL,where f∞ denotes the limit of the scheme SaΓΛn,ωkkk≥0 obtained from fκ and cf is a positive constant only dependent on f. This implies that this newly obtained scheme SaΓΛn,ωkkk≥0 has the same approximation order as its stationary counterpart (see [26] for the approximation order of the corresponding stationary counterpart).


## 5. Examples and Comparison

This section is devoted to the comparison with the exponential pseudospline schemes and several examples and to illustrate the performance of this new family of nonstationary schemes.Note that the exponential pseudospline schemes contain the interpolatory exponentials reproducing schemes and the exponential B-spline schemes as special subclasses. Thus, now we first compare the new nonstationary quasi-interpolatory schemes with the interpolatory ones reproducing the same exponential polynomials. Given the setΓΛn, recall that the new scheme SaΓΛn,ωkkk≥0 reproduces EPΓΛn and is supported on −2n−2,2n+2. From Remark 1, when ωk=0, this scheme becomes the interpolatory subdivision with the same reproduction property, which is actually supported on −2n−1,2n+1. Besides, Theorem 6 implies that this newly obtained scheme has the same smoothness order as its stationary counterpart. Thus, we can choose ωk away from 0 to increase its smoothness order at the expense of slightly larger support. This is shown by Table 1, which gives the smoothness orders of this new nonstationary scheme between the cases ωk=0 and ωk≠0 (maximal smoothness order).Table 1
Comparison on the smoothness orders of the schemeSaΓΛn,ωkkk≥0 reproducing EPΓΛn with different n between the cases ωk=0 and ωk≠0 (maximal smoothness order).
n
=
0
n
=
1
n
=
2
n
=
3
n
=
4
n
=
5
n
=
6
ω
k
=
0
0
1
2
3
4
4
5
ω
k
≠
0
2
3
4
5
6
6
7That the schemeSaΓΛn,ωkkk≥0 can reach a higher smoothness order by choosing ωk away from 0 is also reflected by Table 2. Table 2 gives the comparison among the cubic exponential B-spline scheme (1), the scheme SaΓΛ1,0kk≥0 (i.e., the nonstationary interpolatory 4-point scheme in [25]), and the scheme SaΓΛ1,ωkkk≥0 on the support, the space they reproduce, the approximation order, and the smoothness order. From Table 2, it can be seen that with only a slight increase of the support, the scheme SaΓΛ1,ωkkk≥0 can reach the highest smoothness order among these three schemes and higher approximation order than the cubic exponential B-spline scheme (1). Besides, it also implies that the new schemes in this paper have much higher approximation order than the exponential B-spline schemes due to the reproduction property of these two families of schemes.Table 2
Comparison between the schemeSaΓΛ1,ωkkk≥0 in cases ωk=0 and ωk≠0 and the cubic exponential B-spline scheme (1) on the support, the space they reproduce, the approximation order, and the smoothness order.
Subdivision
Support
Space reproduced
Approximation order
Smoothness order
S
a
ΓΛ
1
,
ω
k
k
k
≥
0
−
4,4
EP
Γ
Λ
1
4
3
Scheme (1)
−
2,2
EP
Γ
Λ
0
2
2
S
a
1,0
k
k
≥
0
−
3,3
EP
Γ
Λ
1
4
1From Lemma 2 and Theorem 4, the new schemeSaΓΛn,ωkkk≥0 also contains a noninterpolatory scheme besides the interpolatory one with the same reproduction property. Compared with this noninterpolatory scheme, the scheme SaΓΛn,ωkkk≥0 has at least the same smoothness order. Besides, due to the existence of the parameter ωk, the scheme SaΓΛn,ωkkk≥0 differs with different values of ωk but with the same reproduction property. Also, the parameter ωk makes the scheme SaΓΛn,ωkkk≥0 have better flexibility than the exponential pseudosplines with the same reproduction property in curve design, which is reflected by Figures 1 and 2 obtained by the scheme in the following example 2.Now, we present some concrete examples to illustrate the performance of this new family of nonstationary schemes.Example 1.
Whenn=0, from Remark 2, the new nonstationary scheme SaΓΛ0,ωkkk≥0 reproducing EPΓΛ0 is actually the scheme (4). From Theorem 1, it is Cr convergent if ωk is such that the corresponding stationary scheme is Cr convergent with r=0,1,2.Example 2.
Whenn=1, the new scheme SaΓΛ1,ωkkk≥0 is just the scheme with the k-level symbol in (21). Theorem 1 implies that this new scheme is Cr convergent if ωk is such that the asymptotical similar scheme is Cr convergent with r=0,1,2,3.Example 3.
Whenn=2, let ΓΛ2=0,2,±t,1,±2t,1. Then, we have(44)EPΓΛ2=1,x,e±tx,e±2tx.
From Theorem 4, by suitably choosingα0k,α1k,α2k and β0k,β1k, we can derive the new nonstationary quasi-interpolatory scheme SaΓΛ2,ωkkk≥0 reproducing the space EPΓΛ2 in (44), and the corresponding k-level symbol can be written as(45)aΓΛ2,ωkkz=1−θkeΓΛ2,1kz+θkeΓΛ2,2kz.
Here,eΓΛ2,1kz is the k-level symbol of the interpolatory scheme reproducing the space in (44) ([12], Section 4.2):(46)eΓΛ2,2kz=1+z48z2+2vk+1z+12vk+1+1z2+4vk+12−2z+14vk+12ν1k1+z4+ν2kz+z3+1−2ν1k−2ν2kz2z−6,with(47)ν1k=4vk+14+10vk+13+4vk+12+vk+1+18vk+12vk+1+122vk+1−12vk+12−1,ν2k=−8vk+16+24vk+15+14vk+14+4vk+13+3vk+12+2vk+1+14vk+122vk+1−12vk+12−11+vk+12being the k-level symbol of the scheme generating EPΓΛ3≔1,x,x2,x3,e±tx,e±2tx with ΓΛ3=0,4,±t,1,±2t,1 and reproducing the space in (44) and θk=512vk+1+13vk+142vk+1−12vk+12−1/4vk+14+10vk+13+4vk+12+vk+1+1ωk.
From Theorem 5, this new scheme isCr convergent if the parameter ωk is such that the asymptotical similar counterpart is Cr convergent with r=0,1,2,3,4.Example 4.
Whenn=3, let ΓΛ3≔0,2,±t,1,±2t,1,±3t,1, and then we have(48)EPΓΛ3=1,x,e±tx,e±2tx,e±3tx.
From Theorem 4, with a suitable choice of the parametersα0k,…,α3k and β0k,β1k,β2k, we can derive the scheme SaΓΛ3,ωkkk≥0 reproducing the space in (48) with the k-level symbol(49)aΓΛ3,ωkkz=1−θkeΓΛ3,1kz+θkeΓΛ3,2kz.
Here,eΓΛ3,1kz is the k-level symbol of the scheme (32)-(33) in [24]:(50)eΓΛ3,2kz=1+z48z2+2vk+1z+12+2vk+1z2+4vk+12−2z+14vk+12z2+2vk+14vk+12−3z+12+2vk+14vk+12−3qkzz−8,where qkz=rk1+z6+skz+z5+tkz2+z4+1−2rk−2sk−2tkz3 with rk, sk, and tk satisfying(51)eΓΛ3,2k1=eΓΛ3,2ke±tk+1=eΓΛ3,2ke±2tk+1=eΓΛ3,2ke±3tk+1=2,D1eΓΛ3,2k1=0,and θk=ωk/γk with γk being the coefficient of the leading term of eΓΛ3,2kz. Note that the scheme with the k-level symbol eΓΛ3,2kz also generates EPΓΛ4=1,x,x2,x3,e±tx,e±2tx,e±3tx with ΓΛ4=0,4,±t,1,±2t,1,±3t,1.
From Theorem 5, it can be seen that this nonstationary scheme isCr convergent if ωk is such that the asymptotical similar counterpart is Cr convergent with r=0,1,…,5.
Figures3 and 4 show some epitrochoid curves and hypotrochoid curves reproduced by the scheme SaΓΛ3,ωkkk≥0 reproducing the space in (48) with v0=cosπ/4 and ωk=0.002.
Table3 lists the schemes SaΓΛn,ωkkk≥0 with n=0,1,2,3 in the aforementioned examples on the support, smoothness order (smooth. order), dimension of the exponential polynomial space generated (dim. space. generated), dimension of the exponential polynomial space reproduced (dim. space. reproduced), and the approximation order (approxi. order). From Table 3, we can see the change of these properties of the scheme SaΓΛn,ωkkk≥0 with the change of n.Figure 3
Epitrochoid curves reproduced by the schemeSaΓΛ3,ωkkk≥0 reproducing the space in (48) with ωk=0.002 and v0=cosπ/4.Figure 4
Hypotrochoid curves reproduced by the schemeSaΓΛ3,ωkkk≥0 reproducing the space in (48) with ωk=0.002 and v0=cosπ/4.Table 3
The schemes in the above 4 examples and the different properties.
Subdivision
Support
Smooth. order
Dim. space. generated
Dim. space. reproduced
Approxi. order
S
a
ΓΛ
0
,
ω
k
k
k
≥
0
−
2,2
2
4
2
2
S
a
ΓΛ
1
,
ω
k
k
k
≥
0
−
4,4
3
6
4
4
S
a
ΓΛ
2
,
ω
k
k
k
≥
0
−
6,6
4
8
6
6
S
a
ΓΛ
3
,
ω
k
k
k
≥
0
−
8,8
5
10
8
8


## 6. Conclusion

This paper presents a new family of nonstationary subdivision schemes by suitably using the so-called push-back operation. Each one of these new schemes is a nonstationary quasi-interpolatory subdivision and reproduces a certain exponential polynomial space. These new schemes unify and extend the existing interpolatory schemes with the same reproduction property and the noninterpolatory ones like the cubic exponential B-spline scheme. We show that they have higher smoothness orders than the interpolatory ones with the same reproduction property and better accuracy than the exponential B-spline schemes. Note that this new family of nonstationary schemes are actually constant reproducing nonstationary schemes and their symbols are odd symmetric. Therefore, future works may focus on the investigation of the generalization of the schemes in this paper, which can reproduce even more exponential polynomials. Besides, the basic limit functions of these new schemes satisfy nonstationary refinable equations, which are the key ingredients to derive the nonstationary multiresolution analysis (see also [38]). Thus, future works can also focus on the construction of nonstationary (biorthogonal) wavelets using these new schemes. Similar works include the generalized Daubechies wavelets in [4] and the biorthogonal wavelets in [3].



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