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

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. +is 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.


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. e 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][10][11]). erefore, 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][15][16][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][22][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 pushback 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. us, 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 S 2M with M ∈ Z + in [26].
en, 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.
us, a nonstationary quasiinterpolatory 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. e rest of this paper is organized as follows: in Section 2, 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.

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. Let l 0 (Z) denote the linear space of real sequences with finite support. Given a sequence of initial control points q 0 � q 0 j , j ∈ Z ∈ l 0 (Z), we consider the binary nonstationary subdivision scheme where S a k is the k-level subdivision operator mapping l 0 (Z) to l 0 (Z), the sequence a k � a k i , i ∈ Z ∈ l 0 (Z) is the k-level mask with finite support, and we denote this subdivision scheme by S a k k≥0 . e k-level symbol corresponding to the mask a k is a k (z) � i∈Z a k i z i with the subsymbols a k 0 (z) � i∈Z a k 2i z 2i and a k 1 (z) � i∈Z a k 2i+1 z 2i+1 satisfying a k (z) � a k 0 (z) + a k 1 (z). Following [28], now we give the definition of convergence of the nonstationary subdivision.
Definition 1 (see [28]). e subdivision S a k k≥0 is termed uniformly convergent if, for an initial control sequence q 0 , there exists a continuous function where K is any compact set in R and f q 0 is nontrivial for at least one initial data sequence. e subdivision S a k k≥0 is termed C l convergent if the limit function f q 0 has continuous derivatives up to order l with l ∈ Z + . e following definitions and results are needed to investigate the convergence and smoothness of nonstationary subdivision schemes.
Definition 2 (see [29]). A nonstationary subdivision scheme S a k k≥0 is said to be asymptotic similar to the stationary subdivision scheme S a if the masks a k k≥0 and a { } have the same support U (i.e., a k i � a i � 0 for i ∉ U) and satisfy lim Definition 3 (see [30]). Let D n be the n-th order differential operator. A binary nonstationary subdivision scheme S a k k≥0 is said to satisfy approximate sum rules of order Theorem 1 (see [30]). Assume that the nonstationary subdivision scheme S a k k≥0 satisfies approximate sum rules of order r + 1, r ∈ N 0 and is asymptotic similar to a C r -convergent stationary scheme S a . en, the nonstationary scheme S a k k≥0 is also C r 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. us, we now review the definition of exponential polynomial spaces and related results as follows: Definition 4 (see [31]). Let T ∈ Z + , and let γ � c 0 , c 1 , . . . , c T } with c T ≠ 0 be a finite set of real or imaginary numbers. e space of exponential polynomials V T,γ is defined as e exponential polynomial space V T,γ can be characterized by the following lemma: Lemma 1 (see [31]). Let c(z) � T j�0 c j z j and denote by (θ i , τ i ) i�0,...,N the set of zeros with multiplicity, satisfying en, e following two theorems, which can be deduced from Proposition 4.2 and eorem 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 symbols
Theorem 3. With z k i � e θ i /2 k+1 , i � 1, . . . , N, a binary nonstationary subdivision scheme associated with symbols a k (z) k≥0 reproduces V T,γ if it generates V T,γ and there exists a shift parameter p such that for each k≥0

Nonstationary Quasi-Interpolatory Schemes Reproducing Exponential Polynomials
is 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).
Given n, l ∈ Z + , let the set of zeros with multiplicity in Lemma 1 be given as where t ∈ R + ∪ ι[0, π) and ι 2 � − 1. In other words, we assume c(z) in Lemma 1 has N � 2l + 1 pairwise distinct zeros, and its total number of zeros is T � 2 l j�0 τ j � 2n + 2. en, the exponential polynomial space in Lemma 1, which is actually a 2n + 2 dimensional space, can now be rewritten as Note that compared with the space in ([24] Section 6), this new one contains more exponentials like x r e ±tx , r ≥ 2. Now, we try to construct the desired nonstationary subdivision schemes, which are, in fact, quasi-interpolatory ones reproducing exponential polynomials in EP ΓΛ n . In order to describe the construction clearly, we first derive the schemes reproducing EP ΓΛ 0 and EP ΓΛ 1 .
en, by taking a generalization, we obtain the general scheme reproducing EP ΓΛ n .

e Cubic Exponential B-Spline Scheme and the Nonstationary Quasi-Interpolatory Scheme
Reproducing EP ΓΛ 0 . We start from the cubic exponential B-spline scheme. As it is known, for k≥0, the cubic exponential Bspline scheme generates the refined data sequence P k+1 i i∈Z from the coarser data sequence p k i i∈Z through the refinement rules e cubic exponential B-spline scheme (1) is known to generate the space EP ΓΛ 1 . Based on this scheme, by making a suitable modification to its refinement rules [24], a nonstationary combined subdivision scheme can be obtained as where α k 0 and β k 0 are constants depending on k, ) are the two points provided by the cubic exponential B-spline scheme (1). e nonstationary combined scheme (3) can be seen as obtained by moving the points P k+1 2i and P k+1 2i+1 to new positions according to the displacements , 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 β k 0 � 0, the combined scheme (3) can be rewritten as By writing down the symbols of the scheme (4), from eorem 3, it can be seen that the scheme (4), if convergent, reproduces EP ΓΛ 0 . Besides, when α k 0 � 0, the scheme (4) reduces to the cubic exponential B-spline scheme (1), and when α k 0 � 1/(4(1 + v k+1 )), it becomes the D-D 2-point scheme. In this way, the scheme (4) is a nonstationary scheme reproducing EP ΓΛ 0 , which combines the D-D 2point scheme and the cubic exponential B-spline scheme (1).

e Nonstationary Quasi-Interpolatory Scheme
Reproducing EP ΓΛ 1 . Now, we try to obtain the quasi-interpolatory subdivision reproducing EP ΓΛ 1 , which reproduces more exponential polynomials than the scheme (4).
In fact, we modify both rules of the scheme (4) by adding the term − β k 0 (Δp k i + ΔP k i+1 ) to the second rule of (4) and adding additionally the term α k 1 (ΔP k i− 1 + ΔP k i+1 ) to its first rule to obtain the following new scheme: e scheme (5) can be seen as obtained by moving the points P k+1 2i and P k+1 2i+1 , generated by the cubic exponential Bspline scheme (1), according to the displacements α k , respectively. In this way, the corresponding k-level symbol can be written as where For the scheme (5), from eorem 3, by solving the linear system, We can find its solution containing a free parameter, which we denote by ω k . Here, we set ω k ≔ α k 1 . en, with the parameters in this solution, the scheme (5), if convergent, reproduces EP ΓΛ 1 . We denote this new scheme reproducing EP ΓΛ 1 by S a k ΓΛ 1 ,ω k k≥0 , and the corresponding k-level symbol can be written as In addition, eorem 2 implies that the scheme S a k ΓΛ 1 ,ω k k≥0 also generates the space 4 Mathematical Problems in Engineering when . Note that, when ω k � 0, the scheme S a k ΓΛ 1 ,ω k k≥0 reduces to the nonstationary interpolatory 4-point scheme [25], and when , it actually becomes the exponential pseudospline scheme generating EP ΓΛ 2 (22) and reproducing EP ΓΛ 1 [33]. Here, we denote by e k ΓΛ 1 ,1 (z) and e k ΓΛ 1 ,2 (z) the k-level symbols of the nonstationary interpolatory 4-point scheme [25] and the aforementioned exponential pseudospline scheme, which can be written as (1 + z) 4 8 respectively. Now, let θ k � ((32v k+1 (1 + v k+1 ) 2 )/(v k+1 + 2))ω k , and then the k-level symbol a k ΓΛ 1 ,ω k (z) can be rewritten as In this way, we obtain a new nonstationary scheme reproducing EP ΓΛ 1 , which combines the nonstationary interpolatory 4-point scheme as shown in [25] and the noninterpolatory one with the k-level symbol e k ΓΛ 1 ,2 (z). Note that the k-level symbol in (24) is actually a convex combination of the symbols e k ΓΛ 1 ,1 (z) and e k ΓΛ 1 ,2 (z) for 0 ≤ θ k ≤ 1. Figure 1 shows the basic limit functions of this new scheme reproducing EP ΓΛ 1 with v 0 � cos(π/4) and ω k � − 0.015, 0, 0.015, 0.03. Figure 2 shows the curves generated by this new scheme reproducing EP ΓΛ 1 with different values of ω k and v 0 . In particular, when v 0 � cos(π/4) (solid line in the left column), 1 (solid line in the middle column), and cosh(3/5) (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 ω k on the shape of the limit functions of this new scheme.

Nonstationary Quasi-Interpolatory Schemes
Reproducing EP ΓΛ n . As it can be seen from Section 3.1, the two newly derived schemes reproducing EP ΓΛ 0 and EP ΓΛ 1 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 EP ΓΛ n .
Specifically speaking, we use more terms like α k 1 (ΔP k i− 1 + ΔP k i+1 ) and β k 0 (Δp k i + ΔP k i+1 ) to modify the first and second rules of the scheme (4) to get with the k-level symbol where In other words, the scheme (10) is obtained by moving the points P k+1 2i and P k+1 2i+1 , generated by the cubic exponential B-spline scheme (1), to new positions according to the displacements α k 0 Δp k i + Δ k 2i,n and − Δ k 2i+1,n , respectively. In particular, when n � 0, we have Δ k 2i,0 � Δ k 2i+1,0 � 0 and the scheme (10) reduces to the scheme (4). Besides, when α k 1 � · · · � α k n � 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 reproducing EP ΓΛ n can be obtained. In fact, we have the following result. Proof. Since the case of n � 1 has been investigated in Section 3.1.2, here we only need to investigate the case of n ≥ 2. From eorem 3, the scheme (10) reproduces EP ΓΛ n , if for p � 0, 1, . . . , 2τ 0 − 1 and q � 0, 1, . . . , τ j − 1, j � 1, . . . , l, the corresponding k-level symbol in (27) satisfies e two subsymbols a k 0 (z) and a k 1 (z) satisying a k (z) � a k 0 (z) + a k 1 (z) satisfy respectively. us, for r ∈ N 0 , D r a k 0 (z) is an even function if r is an even number and odd function if r is an odd number, while D r a k 1 (z) is an even function if r is an odd number and odd function if r is an even number. erefore, for r ∈ N 0 , the subsymbols a k 0 (z) and a k where t k+1 is defined as in (14). In this way, from (29), for p � 0, 1, . . . , 2τ 0 − 1, we have is leads to the result that D p a k 0 (1) � D p a k 1 (1) � δ p,0 . e 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: Now, let us find the desired α k 0 , . . . , α k n , β k 0 , . . . , β k n− 1 by solving the linear systems in (33). For the first linear system in (33), we make a substitution as follows: en, the first linear system in (33) can be rewritten as Here, A k is the (2n + 2) × (n + 2) matrix   Mathematical Problems in Engineering . Note that with the substitution in (34), a k 0 (z) satisfies Besides, due to the symmetry of a k 0 (z) (symmetric about z 0 ), if q ∈ Z + , we have us, by a process of row operation (Gaussian elimination), it can be seen that the rank of A k and (A k | g) is τ 0 + l j�1 τ j � n + 1. Since the unknown h k is a (n + 2) × 1 vector, the linear system (35) can be solved containing a free parameter.
Since h k has been found, let us now fix the desired parameters α k 0 , . . . , α k n from h k . In fact, through the substitution in (34), we obtain a linear system en, by the row operation (Gaussian elimination), this linear system is uniquely solvable. In this way, the desired parameters α k 0 , . . . , α k n are found containing a free parameter. Similarly, through the second linear system in (33), the parameters β k 0 , . . . , β k n− 1 can also be uniquely determined. en, with such chosen parameters, the scheme (10), if convergent, reproduces EP ΓΛ n .
□ Remark 1. We denote by ω k the free parameter obtained in the proof of eorem 4. For this free parameter, we set ω k � (− 1) n+1 α k n . en, if ω k is chosen to be 0, the first linear system in (33) can be uniquely solved and the solution can be found with α k 0 � 1/(4(1 + v k+1 )), α k 1 � · · · � α k n � 0. is solution forces the subsymbol a k 0 (z) to be a k 0 (z) � 1, which implies that the obtained scheme in eorem 4 is an interpolatory scheme. Besides, with eorem 2, it can be shown that, with another suitable choice of ω k , this obtained scheme in eorem 4 also generates EP ΓΛ n+1 with EP ΓΛ n ⊂ EP ΓΛ n+1 .
In this way, given the set ΓΛ n with n ≥ 0, we can derive the desired nonstationary scheme reproducing EP ΓΛ n , which, in fact, is a quasi-interpolatory scheme combining the interpolatory and noninterpolatory schemes with the same reproduction property. We denote by S a k ΓΛ n ,ω k k≥0 this newly obtained scheme and by a k ΓΛ n ,ω k (z) the corresponding k-level symbol. In particular, when n � 0, S a k ΓΛ 0 ,ω k k≥0 refers to the equivalent form of the scheme (4) obtained using the substitution in Remark 2. From the proof of eorem 4, given a nonstationary scheme with the k-level symbol h k (z), which is supported on [− 2n − 2, 2n + 2] and reproduces EP ΓΛ n , we can find the parameters α k 0 , . . . , α k n , β k 0 , . . . , β k n− 1 such that a k (z) in (27) satifies (29) (a k (z) actually becomes h k (z)), and thus the corresponding nonstationary scheme reproduces EP ΓΛ n . is means that the nonstationary scheme with the k-level symbol h k (z) falls into the family of the nonstationary schemes in this paper, and these new schemes Mathematical Problems in Engineering in this paper can actually unify the schemes which are supported on [− 2n − 2, 2n + 2] and reproduce EP ΓΛ n . 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.

Analysis of Smoothness and Approximation Order
is section is devoted to the analysis of the convergence, smoothness, and approximation power of the new scheme S a k ΓΛ n ,ω k k≥0 . Before that, we need to derive the corresponding asymptotical similar scheme first.
In order to derive the asymptotical similar scheme of the scheme S a k ΓΛ n ,ω k k≥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 EP ΓΛ 0 and EP ΓΛ 1 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 S a k ΓΛ n ,ω k k≥0 reduces to the interpolatory scheme reproducing EP ΓΛ n when ω k � 0 and becomes a noninterpolatory one generating EP ΓΛ n+1 and reproducing EP ΓΛ n with ΓΛ n ⊂ ΓΛ n+1 when ω k is chosen to be another suitable value. For these two schemes, let e k ΓΛ n ,1 � e k 1,− 2n− 1 , 0, . . . , 0, e k denote their k-level masks, and by e k ΓΛ n ,1 (z) and e k ΓΛ n ,2 (z) 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). en, 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 a k ΓΛ n ,ω k (z) is actually a convex combination of e k ΓΛ n ,1 (z) and e k ΓΛ n ,2 (z). In other words, we have the following result.

Lemma 2.
Given the set ΓΛ n , the k-level symbol a k ΓΛ n ,ω k (z) of the new scheme S a k ΓΛ n ,ω k k≥0 can be written as a k ΓΛ n ,ω k (z) � 1 − θ k e k ΓΛ n ,1 (z) + θ k e k ΓΛ n ,2 (z), where θ k is a free parameter depending on k.
Remark 3. Since the exponential pseudospline schemes in [33] are derived using polynomial correction, eorem 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 quasiinterpolatory subdivision schemes are also parameter-dependent schemes (see also [27]). Now, we let θ k � ω k /|e k 2,2n+2 | � ω k /|e k 2,− 2n− 2 | in Lemma 2. As a result, we choose ω k � |e k 2,2n+2 | in the new scheme S a k ΓΛ n ,ω k k≥0 so that it also generates EP ΓΛ n+1 . Now, we make a mild condition on the free parameter ω k . We assume that the limit of ω k as k tends to infinity exists and we denote its limit by ω, that is to say we assume lim k⟶∞ ω k ≔ ω ∈ R exists. en, together with Lemma 2, we can show that the asymptotical similar scheme of S a k ΓΛ n ,ω k k≥0 is the scheme S 2M with M � n + 1 in [26]. is is shown in the following result.

Proposition 1.
Assume lim k⟶∞ ω k ≔ ω ∈ R. en, the asymptotical similar scheme of the new nonstationary scheme S a k ΓΛ n ,ω k k≥0 reproducing EP ΓΛ n is the stationary scheme S 2M with M � n + 1.
(42) erefore, a n,ω (z) is just the symbol of the scheme S 2M with M � n + 1 in [26] (see [35]). us, the scheme S 2M with 8 Mathematical Problems in Engineering M � n + 1 in [26] is the asymptotical similar scheme of S a k ΓΛ n ,ω k k≥0 .

□
Based on Proposition 1, now we investigate the C r convergence of the scheme S a k ΓΛ n ,ω k k≥0 . In fact, we have the following result.
Theorem 5. Given the set ΓΛ n , for the new scheme S a k ΓΛ n ,ω k k≥0 , it is C r convergent with r < 2n + 4, if the free parameter ω k is such that the asymptotical similar counterpart is C r convergent.

□
Now, let us discuss the approximation order of the scheme S a k ΓΛ n ,ω k k≥0 . Let f be a function satisfying ‖D j f‖ L ∞ (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 , j ∈ Z for some κ ∈ Z + . en, for the scheme S a k ΓΛ n ,ω k k≥0 , according to [36], eorem 21 (see also [37]), we have where f ∞ denotes the limit of the scheme S a k ΓΛ n ,ω k k≥0 obtained from f κ and c f is a positive constant only dependent on f.
is implies that this newly obtained scheme S a k ΓΛ n ,ω k k≥0 has the same approximation order as its stationary counterpart (see [26] for the approximation order of the corresponding stationary counterpart).

Examples and Comparison
is 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.
us, now we first compare the new nonstationary quasi-interpolatory schemes with the interpolatory ones reproducing the same exponential polynomials. Given the set ΓΛ n , recall that the new scheme S a k ΓΛ n ,ω k k≥0 reproduces EP ΓΛ n and is supported on [− 2n − 2, 2n + 2]. From Remark 1, when ω k � 0, this scheme becomes the interpolatory subdivision with the same reproduction property, which is actually supported on [− 2n − 1, 2n + 1]. Besides, eorem 6 implies that this newly obtained scheme has the same smoothness order as its stationary counterpart. us, we can choose ω k away from 0 to increase its smoothness order at the expense of slightly larger support. is is shown by Table 1, which gives the smoothness orders of this new nonstationary scheme between the cases ω k � 0 and ω k ≠ 0 (maximal smoothness order).
at the scheme S a k ΓΛ n ,ω k k≥0 can reach a higher smoothness order by choosing ω k away from 0 is also reflected by Table 2. Table 2 gives the comparison among the cubic exponential B-spline scheme (1), the scheme S a k ΓΛ 1 ,0 k≥0 (i.e., the nonstationary interpolatory 4-point scheme in [25]), and the scheme S a k ΓΛ 1 ,ω k k≥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 S a k D 1 e k ΓΛ 3 ,2 (1) � 0, and θ k � ω k /|c k | with c k being the coefficient of the leading term of e k ΓΛ 3 ,2 (z). Note that the scheme with the k-level symbol e k ΓΛ 3 ,2 (z) also generates EP ΓΛ 4 � 1, x, { x 2 , x 3 , e ±tx , e ±2tx , e ±3tx } with ΓΛ 4 � (0, 4), (± t, 1), (± 2t, 1), { (± 3t, 1)}. From eorem 5, it can be seen that this nonstationary scheme is C r convergent if ω k is such that the asymptotical similar counterpart is C r convergent with r � 0, 1, . . . , 5. Figures 3 and 4 show some epitrochoid curves and hypotrochoid curves reproduced by the scheme S a k ΓΛ 3 ,ω k k≥0 reproducing the space in (48) with v 0 � cos(π/4) and ω k � 0.002. Table 3 lists the schemes S a k ΓΛ n ,ω k k≥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 S a k ΓΛ n ,ω k k≥0 with the change of n.

Conclusion
is 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. ese 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. erefore, 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]). us, 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].

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.