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Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials

  • Zeze Zhang , Hongchan Zheng EMAIL logo and Lulu Pan
Published/Copyright: August 28, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we propose a family of non-stationary combined ternary ( 2 m + 3 ) -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach 2 m + 3 approximation order. Moreover, we discuss its smoothness and show that it can produce C 2 m + 2 limit curves. Several numerical examples are given to show the performance of the schemes.

Keywords: non-stationary combined ternary scheme; exponential polynomial generation/reproduction; approximation order; smoothness
MSC 2010: 65D17

1 Introduction

Subdivision schemes are an efficient tool to design smooth curves and surfaces from a given initial polyline/polyhedral mesh. Over the past two decades, subdivision schemes have shown their usefulness in several application contexts ranging from computer aided geometric design and signal/image processing to computer graphics and animation. Recently, subdivision schemes have become of interest also in biomedical imaging applications (see, e.g., [1,2,3]) and isogeometric analysis (IgA), a modern computational approach that integrates finite element analysis into conventional CAD systems (see, e.g., [4,5, 6,7]). Generally, given a set of initial values on a coarse grid, a subdivision scheme is a set of rules that recursively define sets of values on finer grids. If the rules are the same at all refinement levels, then the scheme is stationary, otherwise it is non-stationary.

One of the important capabilities of non-stationary schemes (which stationary schemes do not have) is the reproduction of exponential polynomials. Such schemes may be useful for the processing of families of oscillatory signals that are well approximated by combinations of exponential polynomials, such as speech signals, and narrowband signals in general. Hence, there have been continuous works on non-stationary subdivision schemes generating exponential polynomials. Romani [8] converted three exponential B-spline schemes into interpolatory schemes without changing the generation property. Conti et al. [9] transformed the non-stationary approximating schemes into interpolatory ones with the same generation property. For other references on this method, see, e.g., [10,11,12, 13,14,15]. All of the aforementioned works convert a known approximating subdivision scheme into a new interpolatory subdivision scheme by a suitable polynomial correction of the symbol of the approximating one. Another way to construct interpolatory schemes is to derive them from approximating ones using the push-back operation presented in [16]. With this method, Lin et al. [17] derived interpolatory surface subdivision from the approximating subdivision. Zhang and Wang [18] gave a semi-stationary subdivision scheme by this operation. Novara and Romani [19] used the push-back operation and constructed a combined ternary 4-point scheme which can unify quite a number of existing approximating and interpolatory schemes. However, previous works are restricted to the stationary schemes which can only generate/reproduce algebraic polynomials. Zheng and Zhang [20] first generalized the push-back operation to the non-stationary case and constructed a non-stationary combined subdivision scheme. As ternary subdivision schemes compare favorably with their binary analogues because of generating limit functions with the same (or higher) smoothness but smaller support (see, e.g., [21]), Zhang et al. [22] generalized the push-back operation to the non-stationary ternary case and presented a non-stationary combined ternary 5-point subdivision scheme. In this paper, to generate and reproduce more general exponential polynomials, we propose a family of non-stationary combined ternary ( 2 m + 3 ) -point subdivision schemes based on the generalized ternary scheme of order 4 proposed in [22]. For such a scheme, we investigate its properties, including the support, exponential polynomial generation/reproduction, approximation order and smoothness. Some examples are given to illustrate the feature of the proposed scheme.

The rest of the paper is organized as follows: Section 2 recalls some basic knowledge about subdivision schemes. Section 3 is devoted to the construction of the new family of non-stationary combined ternary ( 2 m + 3 ) -point subdivision schemes, which reproduces high-order exponential polynomials. The support, exponential polynomial generation/reproduction, approximation order and smoothness are investigated in Section 4. Section 5 concludes the paper with a short summary and further research work.

2 Background

In this section, we review a few basic definitions and results about ternary subdivision scheme which form the basis of the rest of this paper.

This paper is mainly concerned with non-stationary ternary subdivision schemes formalized as follows. Given a sequence of initial control points P 0 = { P i 0 ∈ R : i ∈ Z } , the control points P k = { P j k : j ∈ Z } are iteratively refined with the rule

( P k + 1 ) i = ( S a k P k ) i ≔ ∑ j a i − 3 j k P j k , k ∈ N ,

where S a k is the k -level subdivision operator mapping ℓ 0 ( Z ) to ℓ 0 ( Z ) , and ℓ 0 ( Z ) denotes the linear space of real sequences with finite support. The set of coefficients a k = { a i k : i ∈ Z } in the aforementioned equations is termed as the mask of the subdivision scheme at level k and the Laurent polynomial a k ( z ) = ∑ i ∈ Z a i k z i , z ∈ C ⧹ { 0 } is called the k -level symbol of the non-stationary ternary scheme. A non-stationary ternary subdivision scheme is denoted as { S a k } k ∈ N , and when all subdivision operators are the same, the subdivision scheme is simply denoted as { S a } . We denote the subsymbols of the subdivision scheme { S a k } k ∈ N by a h k ( z ) = ∑ i ∈ Z a 3 i + h k z 3 i + h , h = 0 , 1 , 2 , and remark that the j th derivative of a subsymbol is

D j a h k ( z ) = ∑ i ∈ Z ρ j , h ( i ) a 3 i + h k z 3 i + h − j , ρ j , h ( x ) = ∏ n = 0 j − 1 ( 3 x + h − n ) ,

where D j denotes the j th order differential operator.

In the following, we recall some knowledge about the generation/reproduction of exponential polynomials.

Definition 2.1

[8] Let T ∈ N and χ = { χ 0 , χ 1 , … , χ T } with χ T ≠ 0 a finite set of real or imaginary numbers. The T -dimensional space of exponential polynomials V T , χ is defined by

V T , χ ≔ f : R → C , f ∈ C T ( R ) : ∑ j = 0 T χ j D j f = 0 .

Lemma 2.2

[8] Let χ ( z ) = ∑ j = 0 T χ j z j and denote by { ( θ i , τ i ) } i = 0 , … , N the set of zeros with multiplicity, satisfying

D r χ ( θ i ) = 0 , r = 0 , … , τ i − 1 , i = 0 , … , N .

Then T = ∑ i = 0 N τ i , V T , χ ≔ span { x r e θ i x , r = 0 , … , τ i − 1 , i = 0 , … , N } .

Definition 2.3

[23] Let { a k ( z ) } k ∈ N be a sequence of subdivision symbols. The subdivision scheme associated with the sequence of symbols { a k ( z ) } k ∈ N is said to be V T , χ -generating, if it is convergent and for f ∈ V T , χ there exists an initial sequence f ( 0 ) uniformly sampled from f ˜ ∈ V T , χ such that

lim k → ∞ S a n + k S a n + k − 1 ⋯ S a n f ( 0 ) = f , ∀ n ≥ 0 .

Definition 2.4

[23] Let { a k ( z ) } k ∈ N be a sequence of subdivision symbols. The subdivision scheme associated with the sequence of symbols { a k ( z ) } k ∈ N is said to be V T , χ -reproducing, if it is convergent and for f ∈ V T , χ and an initial sequence f ( 0 ) uniformly sampled from f , it holds

lim k → ∞ S a n + k S a n + k − 1 ⋯ S a n f ( 0 ) = f , ∀ n ≥ 0 .

The following results give conditions on a k ( z ) for { S a k } k ∈ N to generate/reproduce V T , χ .

Theorem 2.5

[24] A non-stationary ternary subdivision scheme associated with symbols { a k ( z ) } k ∈ N generates V T , χ , if the following conditions are satisfied

D r a k ( μ ) = 0 , r = 0 , … , τ i − 1 ,

for all μ ∈ U k ≔ { ε e − θ i 3 − k − 1 : ε ∈ { e 2 π ι / 3 , e 4 π ι / 3 } , ι 2 = − 1 , i = 1 , … , N } .

Theorem 2.6

[24] Let z i k ≔ e − θ i / 3 k + 1 , i = 1 , … , N . A non-stationary ternary subdivision scheme associated with symbols { a k ( z ) } k ∈ N reproduces V T , χ if it generates V T , χ and there exists a shift parameter p such that for each k ∈ N

D r a k ( z i k ) = 3 ( z i k ) 2 p − r q r ( 2 p ) , q r ( p ) = ∏ j = 0 r − 1 ( p − j ) , r = 1 , … , τ i − 1 , 1 , r = 0 .

3 Family of non-stationary combined ternary ( 2 m + 3 ) -point subdivision schemes

In this section, our goal is to construct a family of non-stationary combined ternary ( 2 m + 3 ) -point subdivision schemes reproducing exponential polynomials, on the basis of the generalized ternary subdivision scheme of order 4 in [22].

For the reader’s convenience, refinement rules of the generalized ternary subdivision scheme of order 4 are recalled here:

P ¯ 3 i − 1 k + 1 P ¯ 3 i k + 1 P ¯ 3 i + 1 k + 1 = 4 ( v k ) 2 + 4 v k + 2 3 ( 1 + 2 v k ) 2 8 ( v k ) 2 + 8 v k 3 ( 1 + 2 v k ) 2 1 3 ( 1 + 2 v k ) 2 2 v k + 2 3 ( 1 + 2 v k ) 2 12 ( v k ) 2 + 8 v k − 1 3 ( 1 + 2 v k ) 2 2 v k + 2 3 ( 1 + 2 v k ) 2 1 3 ( 1 + 2 v k ) 2 8 ( v k ) 2 + 8 v k 3 ( 1 + 2 v k ) 2 4 ( v k ) 2 + 4 v k + 2 3 ( 1 + 2 v k ) 2 P i − 1 k P i k P i + 1 k ,

where

(1) v k = 1 2 e t 3 k + 1 + e − t 3 k + 1 , t ∈ { 0 , s , ι s ∣ s > 0 } , k ∈ N .

From Proposition 2 of [25], we know v k + 1 and v k indeed satisfy the following iteration v k + 1 = 1 2 Re ( ( v k + ( v k ) 2 − 1 ) 1 / 3 + ( v k + ( v k ) 2 − 1 ) − 1 / 3 ) .

We construct the non-stationary combined ternary ( 2 m + 3 )-point scheme from the generalized ternary subdivision scheme of order 4, by adding variable parameters ξ k , α i k ( i = − m , … , m , m ∈ N ) which depend on the subdivision level k ∈ N . The subdivision rules are obtained as

(2) P 3 i − 1 k + 1 = P ¯ 3 i − 1 k + 1 + ∑ j = − m m α − j k Δ P i − j k , P 3 i k + 1 = P ¯ 3 i k + 1 + ξ k Δ P i k , P 3 i + 1 k + 1 = P ¯ 3 i + 1 k + 1 + ∑ j = − m m α j k Δ P i − j k ,

where Δ P i − j k = − ( P i − j − 1 k − 2 P i − j k + P i − j + 1 k ) , and the corresponding k -level symbol can be written as

(3) a k ( z ) = a 0 k ( z ) + a 1 k ( z ) + a 2 k ( z ) ,

where

(4) a 0 k ( z ) = 2 v k + 2 3 ( 1 + 2 v k ) 2 − ξ k ( z 3 + z − 3 ) + 12 ( v k ) 2 + 8 v k − 1 3 ( 1 + 2 v k ) 2 + 2 ξ k , a 1 k ( z ) = − α m k z 3 m + 4 + ( 2 α m k − α m − 1 k ) z 3 m + 1 + ∑ j = − m + 1 m − 1 ( − α − j + 1 k + 2 α − j k − α − j − 1 k ) z 1 − 3 j + ( 2 α − m k − α − m + 1 k ) z − 3 m + 1 − α − m k z − 3 m − 2 + 1 3 ( 1 + 2 v k ) 2 z 4 + 8 ( v k ) 2 + 8 v k 3 ( 1 + 2 v k ) 2 z + 4 ( v k ) 2 + 4 v k + 3 3 ( 1 + 2 v k ) 2 z − 2 , a 2 k ( z ) = − α − m k z 3 m + 2 + ( 2 α − m k − α − m + 1 k ) z 3 m − 1 + ∑ j = − m + 1 m − 1 ( − α j − 1 k + 2 α j k − α j + 1 k ) z − 1 − 3 j + ( 2 α m k − − α m − 1 k ) z − 1 − 3 m − α m k z − 3 m − 4 + 4 ( v k ) 2 + 4 v k + 3 3 ( 1 + 2 v k ) 2 z 2 + 8 ( v k ) 2 + 8 v k 3 ( 1 + 2 v k ) 2 z − 1 + 1 3 ( 1 + 2 v k ) 2 z − 4 .

Suppose the refinement rules in (2) do not depend on the level k , i.e., ξ k ≡ ξ , α i k ≡ α i and v k ≡ 1 ( t = 0 in (1)), then the scheme (2) reduces to the stationary one

(5) P 3 i − 1 k + 1 = 10 27 P i − 1 k + 16 27 P i k + 1 27 P i + 1 k + ∑ j = − m m α − j Δ P i − j k , P 3 i k + 1 = 4 27 P i − 1 k + 19 27 P i k + 4 27 P i + 1 k + ξ Δ P i k , P 3 i + 1 k + 1 = 1 27 P i − 1 k + 16 27 P i k + 10 27 P i + 1 k + ∑ j = − m m α j Δ P i − j k .

Remark 1

The new scheme (2) is a combined scheme, in the sense that it reduces to an approximating scheme when ξ k ≠ 2 ( v k + 1 ) 3 ( 1 + 2 v k ) 2 and an interpolatory scheme when ξ k = 2 ( v k + 1 ) 3 ( 1 + 2 v k ) 2 . Note that, when m = 1 , it turns into the non-stationary combined ternary 5-point subdivision scheme in [22]. In particular, if ( α − 1 k , α 0 k , α 1 k , ξ k ) = − ω k + 2 v k + 2 3 ( 2 v k − 1 ) ( 1 + 2 v k ) 3 , 2 v k ω k ( 4 ( v k ) 2 − 3 ) + 4 v k ( 1 + v k ) 3 ( 1 + 2 v k ) 3 ( 2 v k − 1 ) , − ω k , 2 v k + 2 3 ( 1 + 2 v k ) 3 , where ω k ∈ R is the free parameter, it actually becomes the non-stationary ternary 5-point interpolatory C 2 subdivision scheme in [26].

4 Properties of the non-stationary combined ternary ( 2 m + 3 ) -point subdivision scheme

In this section, we discuss the properties of the proposed non-stationary combined ternary subdivision scheme (2), including the support, exponential polynomial generation/reproduction, approximation order and smoothness.

4.1 Support

The support of a subdivision scheme represents how far one vertex affects its neighboring points whose size directly influences local support property of the subdivision curve. In this subsection, we study the support of the proposed scheme (2).

Theorem 4.1

Let ϕ n ≔ lim k → ∞ S a n + k S a n + k − 1 … S a n δ 0 , n ≥ 0 be the family of basic limit functions generated by the non-stationary combined ternary subdivision scheme (2), where δ 0 is the delta-sequence, i.e., δ 0 = { δ α , 0 } α ∈ Z . Then the scheme has support width 3 m + 4 , i.e., the basic limit functions vanish outside the interval − 3 m + 4 2 , 3 m + 4 2 .

Proof

Assume the supports of the k -level masks associated with a non-stationary subdivision scheme are [ l ( k ) , r ( k ) ] , k ∈ N . In fact, the supports of the limit basic functions ϕ n are proved to be included in [ L n , R n ] = ∑ k = n ∞ 3 n − k − 1 l ( k ) , ∑ k = n ∞ 3 n − k − 1 r ( k ) (see, e.g., [22,27]).

From the k -level subsymbols of the scheme (2) in (4), we have l ( k ) = − ( 3 m + 4 ) , r ( k ) = 3 m + 4 , for k ∈ N . Thus, for n ≥ 0 , the left endpoint

L n = ∑ k = n ∞ 3 n − k − 1 l ( k ) = − ( 3 m + 4 ) × 3 n − 1 ∑ k = n ∞ 3 − k = − 3 m + 4 2 ,

while the right endpoint

R n = ∑ k = n ∞ 3 n − k − 1 r ( k ) = ( 3 m + 4 ) × 3 n − 1 ∑ k = n ∞ 3 − k = 3 m + 4 2 .

Hence, the supports of the limit functions are − 3 m + 4 2 , 3 m + 4 2 , which completes the proof.□

For m = 1 , when

( α − 1 k , α 0 k , α 1 k , ξ k ) = − 4 v k − 3 27 ( v k + 1 ) ( 2 v k + 1 ) 2 , − 4 v k ( v k + 1 ) 27 ( 1 + 2 v k ) 2 , 1 54 ( v k + 1 ) ( 2 v k + 1 ) 2 , − ( 8 v k + 7 ) ( 2 v k − 1 ) 54 ( v k + 1 ) ( 2 v k + 1 ) ,

the non-stationary combined ternary ( 2 m + 3 )-point scheme (2) turns into an approximating 5-point subdivision scheme

(6) P 3 i − 1 k + 1 = 4 v k + 3 27 ( v k + 1 ) ( 2 v k + 1 ) 2 P i − 2 k + 40 ( v k ) 3 + 80 ( v k ) 2 + 50 v k + 12 27 ( v k + 1 ) ( 1 + 2 v k ) 2 P i − 1 k + 128 ( v k ) 3 + 256 ( v k ) 2 + 136 v k + 5 54 ( 1 + v k ) ( 1 + 2 v k ) 2 P i k + 4 ( v k ) 3 + 8 ( v k ) 2 + 13 ( v k ) + 10 27 ( v k + 1 ) ( 1 + 2 v k ) 2 P i + 1 k − 1 54 ( v k + 1 ) ( 2 v k + 1 ) 2 P i + 2 k , P 3 i k + 1 = 32 ( v k ) 3 + 64 ( v k ) 2 + 64 v k + 29 54 ( 1 + v k ) ( 1 + 2 v k ) 2 P i − 1 k + 76 ( v k ) 3 + 152 ( v k ) 2 + 71 v k − 2 27 ( 1 + v k ) ( 1 + 2 v k ) 2 P i k + 32 ( v k ) 3 + 64 ( v k ) 2 + 64 v k + 29 54 ( 1 + v k ) ( 1 + 2 v k ) 2 P i + 1 k , P 3 i + 1 k + 1 = − 1 54 ( v k + 1 ) ( 2 v k + 1 ) 2 P i − 2 k + 4 ( v k ) 3 + 8 ( v k ) 2 + 13 ( v k ) + 10 27 ( v k + 1 ) ( 1 + 2 v k ) 2 P i − 1 k + 128 ( v k ) 3 + 256 ( v k ) 2 + 136 v k + 5 54 ( 1 + v k ) ( 1 + 2 v k ) 2 P i k + 40 ( v k ) 3 + 80 ( v k ) 2 + 50 v k + 12 27 ( v k + 1 ) ( 1 + 2 v k ) 2 P i + 1 k + 4 v k + 3 27 ( v k + 1 ) ( 2 v k + 1 ) 2 P i + 2 k .

The basic limit functions of the approximating scheme (6) are shown in Figure 1 with different parameters v 0 = 0.01 (solid line), 1 (dashed line), 100 (dotted line), respectively.

Figure 1 The basis limit functions of the approximating 5-point scheme (6).
Figure 1

The basis limit functions of the approximating 5-point scheme (6).

4.2 Exponential polynomial generation/reproduction property

The goal of this subsection is to study the exponential polynomial generation/reproduction property of the scheme (2). We start by defining the exponential polynomial spaces E P Ω ≔ span { x r e θ x : 0 ≤ r < τ , ( θ , τ ) ∈ Ω } for any finite Ω ⊂ C × N 0 , with the convention that, for τ = 0 , no function is added to the generators of E P Ω .

Now, given

Γ = ( γ 0 , γ 1 , … , γ ℓ ) ∈ { 1 , 2 } × N × N 0 ℓ − 1 , ℓ ∈ N ,

one can define

m ≔ ∑ s = 0 ℓ γ s − 1 , m ′ ≔ m − 1 , m ∈ N ,

then, fixed t ∈ R + ⋃ ι [ 0 , π ) ,

(7) Γ Λ t ≔ { ( 0 , 2 γ 0 + 1 ) } ∪ { ( ± s t , γ s ) } s = 1 ℓ , Γ Λ t ′ ≔ { ( 0 , 2 γ 0 ) } ∪ { ( ± s t , γ s ) } s = 1 ℓ .

The exponential polynomial spaces E P Γ Λ t and E P Γ Λ t ′ are actually 2 m + 3 and 2 m + 4 dimensional spaces.

The following theorem shows that the proposed scheme can reproduce high-order exponential polynomials with suitable choices of parameters ξ k , α − m k , … , α m k .

Theorem 4.2

Given the sets Γ Λ t and Γ Λ t ′ , t ∈ R + ⋃ ι [ 0 , π ) defined by (7), under suitable choices of the parameters ξ k , α − m k , … , α m k , if the scheme (2) is convergent, then it can reproduce E P Γ Λ t and generate E P Γ Λ t ′ . Moreover, the reproduction of exponential polynomial e ± t x implies that the scheme (2) reduces to an interpolatory one.

Proof

In view of Theorem 2.6, the scheme (2) can reproduce E P Γ Λ t , if for p = 0 , 1 , … , 2 γ 0 and q = 0 , 1 , … , γ s , s = 1 , … , ℓ , the corresponding k -level symbol satisfies

(8) D p a k ( ε ) = 0 , D p a k ( 1 ) = 3 δ p , 0 , ε ∈ { e 2 π ι / 3 , e 4 π ι / 3 } , D q a k ( ε e ± s t k + 1 ) = 0 , D q a k ( e ± s t k + 1 ) = 3 δ q , 0 , t k + 1 = t / 3 k + 1 .

Note that

D p a k ( ε ) = ∑ h = 0 2 D p a h k ( ε ) = ∑ h = 0 2 ε h − p ∑ i ∈ Z ρ p , h ( i ) a 3 i + h k ε 3 i = ∑ h = 0 2 ε h − p D p a h k ( 1 ) ,

because ε 3 i = 1 for all i ∈ Z . Then the conditions D p a k ( ε ) = 0 and D p a k ( 1 ) = 3 δ p , 0 can be written as the linear system

ε − p D p a 0 k ( 1 ) + ε 1 − p D p a 1 k ( 1 ) + ε 2 − p D p a 2 k ( 1 ) = 0 , D p a 0 k ( 1 ) + D p a 1 k ( 1 ) + D p a 2 k ( 1 ) = 3 δ p , 0 ,

where ξ k and α i k ( i = − m , … , m ) are the unknowns of the system. This leads to the result that D p a 0 k ( 1 ) = D p a 1 k ( 1 ) = D p a 2 k ( 1 ) = δ p , 0 . The other two equations in (8) can be dealt with in a similar way. Thus, fixed p and q , (8) is actually equivalent to the following three linear systems:

(9) D p a 0 k ( 1 ) = δ p , 0 , D q a 0 k ( e ± s t k + 1 ) = δ q , 0 , D p a 1 k ( 1 ) = δ p , 0 , D q a 1 k ( e ± s t k + 1 ) = δ q , 0 , D p a 2 k ( 1 ) = δ p , 0 , D q a 2 k ( e ± s t k + 1 ) = δ q , 0 .

Because a 1 k ( z ) = a 2 k ( z − 1 ) , we have D p a 1 k ( e j t k + 1 ) = D p a 2 k ( e − j t k + 1 ) , which implies the latter two linear systems in (9) are equivalent.

Now, let us find the desired ξ k , α − m k , … , α m k by solving the linear systems in (9). For the first system in (9), we can easily get the result ξ k = 2 v k + 2 3 ( 1 + v k ) 2 . For the second linear system in (9), we make a substitution as follows:

(10) t 0 k = − α 1 k + 2 α 0 k − α − 1 k , t m k = 2 α m k − α m − 1 k , t m + 1 k = − α m k , t 2 m + 1 k = 2 α − m k − α − m + 1 k , t j k = − α j + 1 k + 2 α j k − α j − 1 k , t m + j + 1 k = − α − j + 1 k + 2 α − j k − α − j − 1 k , j = 1 , … , m − 1 , t 2 m + 2 k = − α − m k , t 2 m + 3 k = 4 ( v k ) 2 + 4 v k + 2 ( 1 + 2 v k ) 2 , t 2 m + 4 k = 8 ( v k ) 2 + 8 v k ( 1 + 2 v k ) 2 , t 2 m + 5 k = 1 3 ( 1 + 2 v k ) 2 .

Then, the second linear system (9) can be written as

(11) A k t k = g ,

where A k = ( A 0 k , A 1 k , A 1 k ¯ , … , A ℓ k , A ℓ k ¯ ) T is a ( 2 m + 3 ) × ( 2 m + 6 ) matrix with

A 0 k = D 0 ( z ) ( 1 ) 0 ⋯ 0 D 0 ( z 4 ) ( 1 ) D 1 ( z 4 ) ( 1 ) ⋯ D 2 γ 0 ( z 4 ) ( 1 ) ⋮ ⋮ ⋮ ⋮ D 0 ( z 3 m + 4 ) ( 1 ) D 1 ( z 3 m + 4 ) ( 1 ) ⋯ D 2 γ 0 ( z 3 m + 4 ) ( 1 ) D 0 ( z − 2 ) ( 1 ) D 1 ( z − 2 ) ( 1 ) ⋯ D 2 γ 0 ( z − 2 ) ( 1 ) D 0 ( z − 5 ) ( 1 ) D 1 ( z − 5 ) ( 1 ) ⋯ D 2 γ 0 ( z − 5 ) ( 1 ) ⋮ ⋮ ⋮ ⋮ D 0 ( z − 3 m − 2 ) ( 1 ) D 1 ( z − 3 m − 2 ) ( 1 ) ⋯ D 2 γ 0 ( z − 3 m − 2 ) ( 1 ) D 0 ( z − 2 ) ( 1 ) D 1 ( z − 2 ) ( 1 ) ⋯ D 2 γ 0 ( z − 2 ) ( 1 ) D 0 ( z ) ( 1 ) 0 ⋯ 0 D 0 ( z 4 ) ( 1 ) D 1 ( z 4 ) ( 1 ) ⋯ D 2 γ 0 ( z 4 ) ( 1 ) ,

A s k = D 0 ( z ) ( e s t k + 1 ) 0 ⋯ 0 D 0 ( z 4 ) ( e s t k + 1 ) D 1 ( z 4 ) ( e s t k + 1 ) ⋯ D 2 γ 0 ( z 4 ) ( e s t k + 1 ) ⋮ ⋮ ⋮ ⋮ D 0 ( z 3 m + 4 ) ( e s t k + 1 ) D 1 ( z 3 m + 4 ) ( e s t k + 1 ) ⋯ D 2 γ 0 ( z 3 m + 4 ) ( e s t k + 1 ) D 0 ( z − 2 ) ( e s t k + 1 ) D 1 ( z − 2 ) ( e s t k + 1 ) ⋯ D 2 γ 0 ( z − 2 ) ( e s t k + 1 ) D 0 ( z − 5 ) ( e s t k + 1 ) D 1 ( z − 5 ) ( e s t k + 1 ) ⋯ D 2 γ 0 ( z − 5 ) ( e s t k + 1 ) ⋮ ⋮ ⋮ ⋮ D 0 ( z − 3 m − 2 ) ( e s t k + 1 ) D 1 ( z − 3 m − 2 ) ( e s t k + 1 ) ⋯ D 2 γ 0 ( z − 3 m − 2 ) ( e s t k + 1 ) D 0 ( z − 2 ) ( e s t k + 1 ) D 1 ( z − 2 ) ( e s t k + 1 ) ⋯ D 2 γ 0 ( z − 2 ) ( e s t k + 1 ) D 0 ( z ) ( e s t k + 1 ) 0 ⋯ 0 D 0 ( z 4 ) ( e s t k + 1 ) D 1 ( z 4 ) ( e s t k + 1 ) ⋯ D 2 γ 0 ( z 4 ) ( e s t k + 1 ) , s = 1 , … , ℓ ,

A s k ¯ = D 0 ( z ) ( e − s t k + 1 ) 0 ⋯ 0 D 0 ( z 4 ) ( e − s t k + 1 ) D 1 ( z 4 ) ( e − s t k + 1 ) ⋯ D 2 γ 0 ( z 4 ) ( e − s t k + 1 ) ⋮ ⋮ ⋮ ⋮ D 0 ( z 3 m + 4 ) ( e − s t k + 1 ) D 1 ( z 3 m + 4 ) ( e − s t k + 1 ) ⋯ D 2 γ 0 ( z 3 m + 4 ) ( e − s t k + 1 ) D 0 ( z − 2 ) ( e − s t k + 1 ) D 1 ( z − 2 ) ( e − s t k + 1 ) ⋯ D 2 γ 0 ( z − 2 ) ( e − s t k + 1 ) D 0 ( z − 5 ) ( e − s t k + 1 ) D 1 ( z − 5 ) ( e − s t k + 1 ) ⋯ D 2 γ 0 ( z − 5 ) ( e − s t k + 1 ) ⋮ ⋮ ⋮ ⋮ D 0 ( z − 3 m − 2 ) ( e − s t k + 1 ) D 1 ( z − 3 m − 2 ) ( e − s t k + 1 ) ⋯ D 2 γ 0 ( z − 3 m − 2 ) ( e − s t k + 1 ) D 0 ( z − 2 ) ( e − s t k + 1 ) D 1 ( z − 2 ) ( e − s t k + 1 ) ⋯ D 2 γ 0 ( z − 2 ) ( e − s t k + 1 ) D 0 ( z ) ( e − s t k + 1 ) 0 ⋯ 0 D 0 ( z 4 ) ( e − s t k + 1 ) D 1 ( z 4 ) ( e − s t k + 1 ) ⋯ D 2 γ 0 ( z 4 ) ( e − s t k + 1 ) ,

t k = ( t 0 k , t 1 k , … , t 2 m + 5 k ) T and g is the ( 2 m + 3 ) × 1 vector, g = ( 1 , 0 , … , 0 ︸ 2 γ 0 , 1 , 0 , … , 0 ︸ γ 1 − 1 , 1 , 0 , … , 0 ︸ γ 1 − 1 , … , 1 , 0 , … , 0 ︸ γ ℓ − 1 , 1 , 0 , … , 0 ︸ γ ℓ − 1 ) T . Thus, by Gaussian elimination, it can be seen that the rank of A k and ( A k ∣ g ) is 2 γ 0 + 1 + 2 ∑ s = 1 ℓ γ s = 2 m + 3 . Since the unknown t k is a ( 2 m + 6 ) × 1 vector which has 2 m + 3 unknown elements t 0 k , t 1 k , … , t 2 m + 2 k , then the linear system (11) is uniquely solvable. Hence, with such chosen parameter, the scheme (2) reproduces E P Γ Λ t .

Similarly, using Theorem 2.5 we have the result for the generation of E P Γ Λ t ′ . By Theorem 2.5, the scheme (2) can generate E P Γ Λ t ′ , if for p 1 = 0 , 1 , … , 2 γ 0 − 1 and q 1 = 0 , 1 , … , γ s , s = 1 , … , ℓ , the corresponding k -level symbol satisfies

(12) D p 1 a k ( ε ) = 0 , D q 1 a k ( ε e ± s t k + 1 ) = 0 .

Let

h 0 k = − α 1 k + 2 α 0 k − α − 1 k , h m k = 2 α m k − α m − 1 k , h m + 1 k = − α m k , h 2 m + 1 k = 2 α − m k − α − m + 1 k , h j k = − α j + 1 k + 2 α j k − α j − 1 k , h m + j + 1 k = − α − j + 1 k + 2 α − j k − α − j − 1 k , j = 1 , … , m − 1 , h 2 m + 2 k = − α − m k , h 2 m + 3 k = 1 3 ( 1 + 2 v k ) 2 , h 2 m + 4 k = 2 v k + 2 3 ( 1 + 2 v k ) 2 − ξ k , h 2 m + 5 k = 4 ( v k ) 2 + 4 v k + 2 ( 1 + 2 v k ) 2 , h 2 m + 6 k = 8 ( v k ) 2 + 8 v k ( 1 + 2 v k ) 2 , h 2 m + 7 k = 12 ( v k ) 2 + 8 v k − 1 ( 1 + 2 v k ) 2 + 2 ξ k ,

then it is not difficult to observe that equation (12) is equivalent to

(13) B k h k = G ,

where B k = ( B 0 k , B 1 k , B 1 k ¯ , … , B ℓ k , B ℓ k ¯ ) T is a ( 2 m + 4 ) × ( 2 m + 8 ) matrix with

B 0 k = D 0 ( z + z − 1 ) ( ε ) D 1 ( z + z − 1 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z + z − 1 ) ( ε ) D 0 ( z 2 + z − 2 ) ( ε ) D 1 ( z 2 + z − 2 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z 2 + z − 2 ) ( ε ) D 0 ( z 5 + z − 5 ) ( ε ) D 1 ( z 5 + z − 5 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z 5 + z − 5 ) ( ε ) ⋮ ⋮ ⋮ ⋮ D 0 ( z 3 m + 2 + z − 3 m − 2 ) ( ε ) D 1 ( z 3 m + 2 + z − 3 m − 2 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z 3 m + 2 + z − 3 m − 2 ) ( ε ) D 0 ( z 4 + z − 4 ) ( ε ) D 1 ( z 4 + z − 4 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z 4 + z − 4 ) ( ε ) D 0 ( z 7 + z − 7 ) ( ε ) D 1 ( z 7 + z − 7 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z 7 + z − 7 ) ( ε ) ⋮ ⋮ ⋮ ⋮ D 0 ( z 3 m + 4 + z − 3 m − 4 ) ( ε ) D 1 ( z 3 m + 4 + z − 3 m − 4 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z 3 m + 4 + z − 3 m − 4 ) ( ε ) D 0 ( z 4 + z − 4 ) ( ε ) D 1 ( z 4 + z − 4 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z 4 + z − 4 ) ( ε ) D 0 ( z 3 + z − 3 ) ( ε ) D 1 ( z 3 + z − 3 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z 3 + z − 3 ) ( ε ) ⋮ ⋮ ⋮ ⋮ D 0 ( z + z − 1 ) ( ε ) D 1 ( z + z − 1 ) ( ε ) ⋯ D 2 γ 0 − 1 ( z + z − 1 ) ( ε ) D 0 ( 1 ) 0 ⋯ 0 ,

B s k = D 0 ( z + z − 1 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z + z − 1 ) ( ε e s t k + 1 ) D 0 ( z 2 + z − 2 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 2 + z − 2 ) ( ε e s t k + 1 ) D 0 ( z 5 + z − 5 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 5 + z − 5 ) ( ε e s t k + 1 ) ⋮ ⋮ ⋮ D 0 ( z 3 m + 2 + z − 3 m − 2 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 3 m + 2 + z − 3 m − 2 ) ( ε e s t k + 1 ) D 0 ( z 4 + z − 4 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 4 + z − 4 ) ( ε e s t k + 1 ) D 0 ( z 7 + z − 7 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 7 + z − 7 ) ( ε e s t k + 1 ) ⋮ ⋮ ⋮ D 0 ( z 3 m + 4 + z − 3 m − 4 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 3 m + 4 + z − 3 m − 4 ) ( ε e s t k + 1 ) D 0 ( z 4 + z − 4 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 4 + z − 4 ) ( ε e s t k + 1 ) D 0 ( z 3 + z − 3 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 3 + z − 3 ) ( ε e s t k + 1 ) ⋮ ⋮ ⋮ D 0 ( z + z − 1 ) ( ε e s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z + z − 1 ) ( ε e s t k + 1 ) D 0 ( 1 ) ⋯ 0 , s = 1 , … , ℓ ,

B s k ¯ = D 0 ( z 3 m + 2 + z − 3 m − 2 ) ( ε e − s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 3 m + 2 + z − 3 m − 2 ) ( ε e − s t k + 1 ) D 0 ( z 4 + z − 4 ) ( ε e − s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 4 + z − 4 ) ( ε e − s t k + 1 ) D 0 ( z 7 + z − 7 ) ( ε e − s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 7 + z − 7 ) ( ε e − s t k + 1 ) ⋮ ⋮ ⋮ D 0 ( z 3 m + 4 + z − 3 m − 4 ) ( ε e − s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 3 m + 4 + z − 3 m − 4 ) ( ε e − s t k + 1 ) D 0 ( z 4 + z − 4 ) ( ε e − s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 4 + z − 4 ) ( ε e − s t k + 1 ) D 0 ( z 3 + z − 3 ) ( ε e − s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z 3 + z − 3 ) ( ε e − s t k + 1 ) ⋮ ⋮ ⋮ D 0 ( z + z − 1 ) ( ε e − s t k + 1 ) ⋯ D 2 γ 0 − 1 ( z + z − 1 ) ( ε e − s t k + 1 ) D 0 ( 1 ) ⋯ 0 ,

h k = ( h 0 k , h 1 k , … , h 2 m + 7 k ) T and G is the ( 2 m + 4 ) × 1 vector, G = ( 1 , 0 , … , 0 ︸ 2 γ 0 − 1 , 1 , 0 , … , 0 ︸ γ 1 − 1 , 1 , 0 , … , 0 ︸ γ 1 − 1 , … , 1 , 0 , … , 0 ︸ γ ℓ − 1 , 1 , 0 , … , 0 ︸ γ ℓ − 1 ) T . By Gaussian elimination, we conclude that the rank of B k and ( B k ∣ G ) is 2 m + 4 . Since the ( 2 m + 8 ) × 1 vector h k has 2 m + 5 unknown variables, the linear system (13) can be solved containing a free parameter. Since the scheme (2) is assumed to be convergent, we have h 0 k + h 1 k + ⋯ + h 2 m + 7 k = 1 , which implies parameters ξ k , α − m k , … , α m k to be unique. Hence, the scheme (2) can generate E P Γ Λ t ′ .

To conclude, we show that the scheme (2) turns into an interpolatory scheme, if the scheme reproduces e ± t x . In view of Theorem 2.6, it suffices to show that

a k ( e ± t k + 1 ) = 3 , a k ( ε e ± t k + 1 ) = 0 ,

if the scheme (2) reproduces e ± t x . Then a direct calculation gives the result ξ k = 2 v k + 2 3 ( 1 + v k ) 2 , which implies that the scheme (2) reduces to an interpolatory one.□

As an example, we take m = 1 in (2) and obtain the non-stationary combined ternary 5-point subdivision scheme in [22]

(14) P 3 i − 1 k + 1 = P ¯ 3 i − 1 k + 1 + α − 1 k Δ P i − 1 k + α 0 k Δ P i k + α 1 k Δ P i + 1 k , P 3 i k + 1 = P ¯ 3 i k + 1 + ξ k Δ P i k , P 3 i + 1 k + 1 = P ¯ 3 i + 1 k + 1 + α 1 k Δ P i − 1 k + α 0 k Δ P i k + α − 1 k Δ P i + 1 k .

The scheme (14) can reproduce E P Γ Λ t 1 ≔ { 1 , x , x 2 , e ± t x } , when

(15) α − 1 k = 8 ( v k ) 2 + 12 v k + 1 9 ( 2 v k − 1 ) ( 2 v k + 1 ) 4 , α 0 k = 2 v k ( 8 ( v k ) 3 + 16 ( v k ) 2 + 18 v k + 9 ) 9 ( 2 v k + 1 ) 4 , α 1 k = − 4 ( v k ) 2 + 6 v k + 5 9 ( 2 v k − 1 ) ( 2 v k + 1 ) 4 , ξ k = 2 ( v k + 1 ) 3 ( 2 v k + 1 ) 2 .

As a consequence, the conic sections such as circles, parabolas and hyperbolas can be exactly reproduced by the scheme (14), but not the curves like the cardioid. To reproduce more exponential polynomials, we take m = 2 in (2). The scheme (2) becomes the non-stationary combined ternary 7-point scheme

(16) P 3 i − 1 k + 1 = P ¯ 3 i − 1 k + 1 + α − 2 k Δ P i − 2 k + α − 1 k Δ P i − 1 k + α 0 k Δ P i k + α 1 k Δ P i + 1 k + α 2 k Δ P i + 2 k , P 3 i k + 1 = P ¯ 3 i k + 1 + ξ k Δ P i k , P 3 i + 1 k + 1 = P ¯ 3 i + 1 k + 1 + α 2 k Δ P i − 2 k + α 1 k Δ P i − 1 k + α 0 k Δ P i k + α − 1 k Δ P i + 1 k + α − 2 k Δ P i + 2 k .

Set

(17) α − 2 k = − 128 ( v k ) 6 + 128 ( v k ) 5 − 144 ( v k ) 4 − 56 ( v k ) 3 + 64 ( v k ) 2 − 12 v k − 3 9 ( 2 v k − 1 ) 4 ( 1 + 2 v k ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) ( 4 ( v k ) 2 − 3 ) , α − 1 k = 1024 ( v k ) 9 + 1536 ( v k ) 8 − 1152 ( v k ) 7 − 1664 ( v k ) 6 + 672 ( v k ) 5 + 576 ( v k ) 4 − 160 ( v k ) 3 − 36 ( v k ) 2 + 2 9 ( 2 v k − 1 ) 3 ( 1 + 2 v k ) 6 ( 8 ( v k ) 3 − 6 v k + 2 ) , α 0 k = 1024 ( v k ) 10 + 1024 ( v k ) 9 − 1152 ( v k ) 7 − 768 ( v k ) 6 + 672 ( v k ) 5 + 304 ( v k ) 4 − 200 ( v k ) 3 − 12 ( v k ) 2 + 4 v k + 1 9 ( 1 + 2 v k ) 6 ( 2 v k − 1 ) 4 , α 1 k = − 512 ( v k ) 9 + 768 ( v k ) 8 − 832 ( v k ) 6 − 384 ( v k ) 5 + 432 ( v k ) 4 + 160 ( v k ) 3 − 108 ( v k ) 2 + 4 9 ( 2 v k − 1 ) 3 ( 1 + 2 v k ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) , α 2 k = 64 ( v k ) 6 + 64 ( v k ) 5 − 64 ( v k ) 3 − 4 ( v k ) 2 + 30 v k − 6 9 ( 2 v k − 1 ) 4 ( 1 + 2 v k ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) ( 4 ( v k ) 2 − 3 ) , ξ k = 2 ( v k + 1 ) 3 ( 1 + 2 v k ) 2 ,

the scheme (16) reproduces the exponential polynomial space E P Γ Λ t 2 , 1 ≔ { 1 , x , x 2 , e ± t x , e ± 2 t x } which contains the exponential polynomial space E P Γ Λ t 1 . Similarly, with a suitable choice of ( α − 2 k , α − 1 k , α 0 k , α 1 k , α 2 k , ξ k ) , the scheme (16) can reproduce the exponential polynomial space E P Γ Λ t 2 , 2 ≔ { 1 , x , x 2 , e ± t x , e ± 3 t x } , when

α − 2 k = − 2 ( 2 v k ) 14 + 6 ( 2 v k ) 13 − 21 ( 2 v k ) 12 − 60 ( 2 v k ) 11 + 99 ( 2 v k ) 10 + 231 ( 2 v k ) 9 − 268 ( 2 v k ) 8 − 402 ( 2 v k ) 7 + 414 ( 2 v k ) 6 9 ( 2 v k − 1 ) 3 ( 2 v k + 1 ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) ( 4 ( v k ) 2 − 3 ) − 258 ( 2 v k ) 5 − 297 ( 2 v k ) 4 + 12 ( 2 v k ) 3 + 32 ( 2 v k ) 2 + 3 9 ( 2 v k − 1 ) 3 ( 2 v k + 1 ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) ( 4 ( v k ) 2 − 3 ) , α − 1 k = 2 ( 2 v k ) 23 + 6 ( 2 v k ) 22 − 39 ( 2 v k ) 21 − 104 ( 2 v k ) 20 + 342 ( 2 v k ) 19 + 933 ( 2 v k ) 18 − 1784 ( 2 v k ) 17 − 4275 ( 2 v k ) 16 9 ( 2 v k + 1 ) 6 ( 2 v k − 1 ) 3 ( 4 ( v k ) 2 − 3 ) ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) + 6129 ( 2 v k ) 15 + 11904 ( 2 v k ) 14 − 2871 2 ( 2 v k ) 13 − 20367 ( 2 v k ) 12 + 22699 ( 2 v k ) 11 + 20415 ( 2 v k ) 10 − 23025 ( 2 v k ) 9 9 ( 2 v k + 1 ) 6 ( 2 v k − 1 ) 3 ( 4 ( v k ) 2 − 3 ) ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) + − 10374 ( 2 v k ) 8 + 13536 ( 2 v k ) 7 + 1641 ( 2 v k ) 6 − 3970 ( 2 v k ) 5 + 198 ( 2 v k ) 4 + 654 ( 2 v k ) 3 − 156 ( 2 v k ) 2 − 72 v k + 18 9 ( 2 v k + 1 ) 6 ( 2 v k − 1 ) 3 ( 4 ( v k ) 2 − 3 ) ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) , α 0 k = ( 2 v k ) 19 + 4 ( 2 v k ) 18 − 5 ( 2 v k ) 17 − 44 ( 2 v k ) 16 − 35 ( 2 v k ) 15 + 184 ( 2 v k ) 14 + 381 ( 2 v k ) 13 − 348 ( 2 v k ) 12 − 1443 ( 2 v k ) 11 9 ( 8 ( v k ) 3 − 6 v k + 1 ) 3 ( 2 v k − 1 ) 2 ( 2 v k + 1 ) 6 ( 4 ( v k ) 2 − 3 ) + 255 ( 2 v k ) 10 + 2901 ( 2 v k ) 9 − 48 ( 2 v k ) 8 − 3268 ( 2 v k ) 7 + 230 ( 2 v k ) 6 + 1886 ( 2 v k ) 5 − 385 ( 2 v k ) 4 − 403 ( 2 v k ) 3 9 ( 8 ( v k ) 3 − 6 v k + 1 ) 3 ( 2 v k − 1 ) 2 ( 2 v k + 1 ) 6 ( 4 ( v k ) 2 − 3 ) + 122 ( 2 v k ) 2 + 42 v k − 15 9 ( 8 ( v k ) 3 − 6 v k + 1 ) 3 ( 2 v k − 1 ) 2 ( 2 v k + 1 ) 6 ( 4 ( v k ) 2 − 3 ) , α 1 k = − ( 2 v k ) 23 + 3 ( 2 v k ) 22 − 15 ( 2 v k ) 21 − 57 ( 2 v k ) 20 + 81 ( 2 v k ) 19 + 480 ( 2 v k ) 18 − 118 ( 2 v k ) 17 − 2367 ( 2 v k ) 16 − 648 ( 2 v k ) 15 9 ( 2 v k − 1 ) 3 ( 2 v k + 1 ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) ( 4 ( v k ) 2 − 3 ) − 7584 ( 2 v k ) 14 + 3564 ( 2 v k ) 13 − 16434 ( 2 v k ) 12 − 7465 ( 2 v k ) 11 + 23982 ( 2 v k ) 10 + 7140 ( 2 v k ) 9 − 22329 ( 2 v k ) 8 9 ( 2 v k − 1 ) 3 ( 2 v k + 1 ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) ( 4 ( v k ) 2 − 3 ) − − 1584 ( 2 v k ) 7 + 11607 ( 2 v k ) 6 − 1805 ( 2 v k ) 5 − 2466 ( 2 v k ) 4 + 750 ( 2 v k ) 3 + 150 ( 2 v k ) 2 − 144 v k 9 ( 2 v k − 1 ) 3 ( 2 v k + 1 ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) ( 4 ( v k ) 2 − 3 ) , α 2 k = ( 2 v k ) 14 + 3 ( 2 v k ) 13 − 6 ( 2 v k ) 12 − 30 ( 2 v k ) 11 + 129 ( 2 v k ) 9 + 73 ( 2 v k ) 8 − 309 ( 2 v k ) 7 − 171 ( 2 v k ) 6 + 426 ( 2 v k ) 5 9 ( 2 v k − 1 ) 3 ( 2 v k + 1 ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) ( 4 ( v k ) 2 − 3 ) + 81 ( 2 v k ) 4 − 273 ( 2 v k ) 3 + 79 ( 2 v k ) 2 + 36 v k − 12 9 ( 2 v k − 1 ) 3 ( 2 v k + 1 ) 6 ( 8 ( v k ) 3 − 6 v k + 1 ) 4 ( 8 ( v k ) 3 − 6 v k − 1 ) ( 4 ( v k ) 2 − 3 ) , ξ k = 2 ( v k + 1 ) 3 ( 1 + 2 v k ) 2 .

Figure 2 shows the limit curves generated by the subdivision scheme (16) with parameters in (17) and v 0 = cos π 4 , 1 , 10 , respectively. Figure 3 illustrates visual comparison of the limit curves generated by the schemes in [22,26] and the scheme (16) from the same control polygons with v 0 = cos π 4 . From Figure 3, we can see that only the scheme (16) with parameters in (17) can exactly reproduce the cardioid and pascal’s limacon. Figure 4 displays the reproduction of the exponential polynomial space E P Γ Λ t 2 , 2 by the subdivision scheme (16) with parameters in (18).

Figure 2 Limit curves generated by the subdivision scheme (16) from the same control polygon with parameters in (17) and v 0 = cos π 4 {v}^{0}=\cos \frac{\pi }{4} (solid line), 1 (dashed line), 10 (dash dotted line), respectively.
Figure 2

Limit curves generated by the subdivision scheme (16) from the same control polygon with parameters in (17) and v 0 = cos π 4 (solid line), 1 (dashed line), 10 (dash dotted line), respectively.

Figure 3 Comparison between the limit curves obtained by the scheme in [26] with ω k = 1 32 ( 8 ( v k ) 3 − 2 v k + 2 ) {\omega }^{k}=\frac{1}{32\left(8{\left({v}^{k})}^{3}-2{v}^{k}+2)} (left), the scheme in [22] with parameters in (15)(center) and the scheme (16) with parameters in (17)(right) from the same control polygons. The corresponding initial parameter is chosen as v 0 = cos π 4 {v}^{0}=\cos \left(\frac{\pi }{4}\right) .
Figure 3

Comparison between the limit curves obtained by the scheme in [26] with ω k = 1 32 ( 8 ( v k ) 3 − 2 v k + 2 ) (left), the scheme in [22] with parameters in (15)(center) and the scheme (16) with parameters in (17)(right) from the same control polygons. The corresponding initial parameter is chosen as v 0 = cos π 4 .

Figure 4 Reproduction of the epicycloid, epitrochoid and astroid by the subdivision scheme (16) with parameters in (18) and v 0 = cos π 4 {v}^{0}=\cos \frac{\pi }{4} .
Figure 4

Reproduction of the epicycloid, epitrochoid and astroid by the subdivision scheme (16) with parameters in (18) and v 0 = cos π 4 .

4.3 Approximation order

This subsection is devoted to the analysis of approximation order of the proposed scheme (2). For this purpose, first we recall the following definitions.

Definition 4.3

[28] Suppose that { S a k } k ∈ N be a non-stationary ternary scheme and f 0 = { f i 0 : i ∈ Z } is sampled from an underlying function f with density 3 − k 0 for some k 0 ∈ N . If there exists the largest exponent d > 0 such that

‖ S a k ∞ f 0 − f ‖ L ∞ ( R ) ≤ C 3 − k 0 d

with a constant C > 0 independent of k 0 , then the exponent d is called the approximation order of the subdivision scheme.

Definition 4.4

[29] A non-stationary subdivision scheme { S a k } k ∈ N is said to be asymptotically similar to a stationary subdivision scheme S a if the masks { a k } k ∈ N and { a } have the same support D (i.e., a i k = a i = 0 for i ∉ D ) and satisfy

lim k → ∞ a α k = a α , α ∈ D .

The study of the approximation order for non-stationary binary schemes was carried out in [30], and we extend it here to the case of non-stationary ternary case. Theorem 4.5 can appear trivial after reading the proof of Theorem 21 in [30].

Theorem 4.5

Assume that a non-stationary ternary scheme { S a k } k ∈ N is V T -reproducing where V T ≔ span { φ 0 ( x ) , … , φ T − 1 ( x ) } and is asymptotically similar to a convergent stationary ternary scheme { S a } . Assume further that the initial data are of the form f d ≔ { f i d = f ( 3 − d i ) , i ∈ Z } for some fixed d ∈ N and for some function f ∈ W ∞ γ ( R ) where γ ∈ N , γ ≤ T . If the Wronskian matrix W V γ ( 0 ) of V γ ⊆ V T is invertible, where the T × T Wronskian matrix W V T ( x ) is defined by

(19) W V T ( x ) ≔ 1 β ! d β φ α ( x ) d x β α , β = 0 , … , T − 1 ,

then

‖ g f d − f ‖ L ∞ ( R ) ≤ C f 3 − γ d , d ∈ N ,

with a constant C f > 0 depending only on f , where g f d is the limit of the subdivision scheme obtained from the initial data f d .

In the following corollary, we give the approximation order of the non-stationary combined ternary subdivision scheme (2) by Theorem 4.5.

Corollary 4.6

If the stationary scheme (5) is convergent, then the non-stationary ternary combined subdivision scheme (2) reproducing the space E P Γ Λ t has approximation order 2 m + 3 .

Proof

From Definition 4.4, we know that the non-stationary ternary scheme (2) is asymptotically similar to the ternary stationary scheme (5). Set E P Γ Λ t ≔ V 2 m + 3 = { φ 0 ( x ) , … , φ 2 m + 2 ( x ) } . Since the non-stationary ternary scheme (2) reproduces the exponential space E P Γ Λ t , from (19) we get the Wronskian matrix of V 2 m + 3 as follows:

(20) W V 2 m + 3 ( x ) = ( C 0 ( x ) , C 1 ( x ) , C 1 ( x ) ¯ , … , C ℓ ( x ) , C ℓ ( x ) ¯ ) T

with

C 0 ( x ) = 1 x ⋯ x 2 γ 0 0 1 ⋯ D 1 ( x 2 γ 0 ) ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ D 2 m + 2 ( x 2 γ 0 ) , C s ( x ) = e s t x x e s t x ⋯ x γ s − 1 e s t x D 1 ( e s t x ) D 1 ( x e s t x ) ⋯ D 1 ( x γ s − 1 e s t x ) ⋮ ⋮ ⋮ ⋮ D 2 m + 2 ( e s t x ) D 2 m + 2 ( x e s t x ) ⋯ D 2 m + 2 ( x γ s − 1 e s t x ) , s = 1 , … , ℓ , C s ( x ) ¯ = e − s t x x e − s t x ⋯ x γ s − 1 e − s t x D 1 ( e − s t x ) D 1 ( x e − s t x ) ⋯ D 1 ( x γ s − 1 e − s t x ) ⋮ ⋮ ⋮ ⋮ D 2 m + 2 ( e − s t x ) D 2 m + 2 ( x e − s t x ) ⋯ D 2 m + 2 ( x γ s − 1 e − s t x ) .

From (20), we get W T γ ( 0 ) , γ ≤ 2 m + 3 is invertible.

Moreover, the non-stationary ternary combined subdivision scheme (2) is asymptotically similar to the convergent stationary scheme (5), hence the non-stationary ternary combined subdivision scheme (2) has approximation order 2 m + 3 from Theorem 4.5.□

4.4 Smoothness

In this subsection, we discuss the smoothness of the non-stationary ternary scheme (2) and get that the scheme (2) reproducing the exponential space E P Γ Λ t can reach C r convergence where r ≤ 2 m + 2 , when the stationary counterpart (5) of the scheme (2) is C r convergent. Before investigating the smoothness of the scheme (2), we recall the following results.

Theorem 4.7

[21] Let S be a stationary ternary subdivision scheme with the symbol a ( 0 ) ( z ) and its j th order difference scheme S j ( j = 1 , … , n + 1 ) exist with symbol a ( j ) ( z ) = 3 z 1 + z + z 2 j a ( 0 ) ( z ) . Suppose the symbol a ( l ) ( z ) satisfies

∑ i ∈ Z a 3 i ( l ) = ∑ i ∈ Z a 3 i + 1 ( l ) = ∑ i ∈ Z a 3 i + 2 ( l ) , l = 0 , … , n .

If there exists an L ≥ 1 such that 1 3 S n + 1 L ∞ < 1 , then the scheme S is C n convergent, where

1 3 S n + 1 L ∞ = max ∑ j ∣ b i − 3 L j L ∣ : 0 ≤ i < 3 L , b L ( z ) = ∏ l = 0 L − 1 b ( z 3 l ) , b ( z ) = 1 3 a ( n + 1 ) ( z ) .

Definition 4.8

[31] A non-stationary ternary subdivision scheme { S a k } k ∈ N is said to satisfy approximate sum rules of order N , N ∈ N , if, given

μ k ≔ ∣ a k ( 1 ) − 3 ∣ , δ k ≔ max η = 0 , … , N − 1 ∣ 3 − k η D η a k ( ε ) ∣ , ε ∈ { e 2 π ι / 3 , e 4 π ι / 3 } ,

we have

∑ k ∈ N μ k < ∞ , ∑ k ∈ N 3 k ( N − 1 ) δ k < ∞ .

In the non-stationary case, approximate sum rules of order N and asymptotical similarity to a stationary C N − 1 subdivision scheme are sufficient conditions for C N − 1 convergence of non-stationary schemes (see [31]). By Corollary 1 of [31], we have the following result.

Theorem 4.9

Assume that the non-stationary ternary subdivision scheme { S a k } k ∈ N is said to satisfy approximate sum rules of order N , N ∈ N and is asymptotic similar to a C N − 1 convergent stationary scheme S a . Then, the non-stationary ternary scheme { S a k } k ∈ N is C N − 1 convergent.

In the following corollary, we study the approximate sum rules of the scheme (2).

Corollary 4.10

If the non-stationary ternary subdivision scheme (2) reproduces the exponential space E P Γ Λ t , then it satisfies approximate sum rules of order 2 m + 3 .

The proof of Corollary 4.10 will be given in Appendix, which is in analogue to the proof of Theorem 10 in [30]. And as a direct consequence of Theorem 4.9 and Corollary 4.10, we have the following result.

Theorem 4.11

For the non-stationary ternary subdivision scheme (2) reproducing the exponential space E P Γ Λ t , it is C r convergent with r ≤ 2 m + 2 , if the asymptotical similar counterpart (5) of the scheme (2) is C r convergent.

5 Conclusion

In this paper, we construct a family of non-stationary combined ternary ( 2 m + 3 ) -point subdivision schemes which can reproduce high-order exponential polynomials. The construction of the family is based on the generalized ternary scheme of order 4 introduced in [22] by adding variable parameters. The proposed schemes are able to generate/reproduce various exponential polynomial spaces E P Γ Λ t ′ / E P Γ Λ t and have approximation order of 2 m + 3 . Moreover, they are C 2 m + 2 convergent for a suitable choice of the parameters. In the future, we may focus on the analysis of other property of the scheme, such as the shape-preserving property, and the construction of non-stationary combined ternary even-point schemes which have high-order exponential polynomial generation/reproduction.

Acknowledgment

This study was supported by the National Science Foundation for Young Scientists of China (Grant No. 61803129).

  1. Conflict of interest: Authors state no conflict of interest.

Appendix Proof of Corollary 4.10

Proof

Let E P Γ Λ t ≔ V 2 m + 3 = { φ 0 ( x ) , … , φ 2 m + 2 ( x ) } and define the functions

(A1) ℱ j , β ( x ) ≔ ∑ n = 0 2 m + 2 m j , β , n k φ n ( x ) , j = 0 , 1 , 2 , β = 0 , 1 , … , 2 m + 2 ,

where the coefficient vector m j , β k = ( m j , β , n k ) n = 0 , 1 , … , 2 m + 2 is the unique solution of the Hermite interpolation problem

(A2) D λ ℱ j , β ( j ⋅ 3 − k − 1 ) = δ β , λ ( − 1 ) λ λ ! , λ = 0 , 1 , … , 2 m + 2 , j = 0 , 1 , 2 ,

where δ β , λ is the Kronecker delta symbol. Indeed, by (A1), (A2) is equivalent to

(A3) W V 2 m + 3 ( j ⋅ 3 − k − 1 ) ( m j , β k ) T = c β T , j = 0 , 1 , 2

with c β ≔ δ β , λ ( − 1 ) λ λ ! ( λ = 0 , … , 2 m + 2 ) . From Corollary 4.6, we get that W V 2 m + 3 ( 0 ) is invertible. Furthermore, there exists a neighborhood δ of zero such that W V 2 m + 3 ( x ) is invertible for all x ∈ δ . Hence, the linear system (A3) clearly has a unique solution.

Next, we define

(A4) Q j , β ≔ ∑ i ∈ Z a j − 3 i k ( ( j ⋅ 3 − 1 − i ) 3 − k ) β − ℱ j , β ( j ⋅ 3 − k − 1 ) , j = 0 , 1 , 2 .

By (A2), we get that

(A5) ℱ 0 , β ( 0 ) = ℱ 1 , β ( 3 − k − 1 ) = ℱ 2 , β ( 2 ⋅ 3 − k − 1 ) = δ β , 0 .

Consequently, ℱ 0 , 0 ( 0 ) = ℱ 1 , 0 ( 3 − k − 1 ) = ℱ 2 , 0 ( 2 ⋅ 3 − k − 1 ) = 1 , and

(A6) ∑ i ∈ Z a i k − 3 = Q 0 , 0 + Q 1 , 0 + Q 2 , 0 .

Moreover, due to (A5), we have

(A7) 3 − β ( k + 1 ) ∑ i ∈ Z ε i i β a i k = 3 − β ( k + 1 ) ∑ i ∈ Z ε − 3 i ( − 3 i ) β a − 3 i k + ∑ i ∈ Z ε 1 − 3 i ( 1 − 3 i ) β a 1 − 3 i k + ∑ i ∈ Z ε 2 − 3 i ( 2 − 3 i ) β a 2 − 3 i k = ∑ i ∈ Z ( − i ⋅ 3 − k ) β a − 3 i k + ε ∑ i ∈ Z ( ( 3 − 1 − i ) 3 − k ) β a 1 − 3 i k + ε 2 ∑ i ∈ Z ( ( 2 ⋅ 3 − 1 − i ) 3 − k ) β a 2 − 3 i k = Q 0 , β + ε Q 1 , β + ε 2 Q 2 , β .

We only estimate Q 0 , β , Q 1 , β and Q 2 , β can be estimated analogously. Since ℱ 0 , β is a linear combination of exponential polynomials in V 2 m + 3 and the non-stationary ternary scheme (2) reproduces V 2 m + 3 , we obtain the identity ℱ 0 , β ( 0 ) = ∑ i ∈ Z a − 3 i k ℱ 0 , β ( i 3 − k ) . Plugging this into (A4) for j = 0 leads to

(A8) Q 0 , β ≔ ∑ i ∈ Z a − 3 i k [ ( − i 3 − k ) β − ℱ 0 , β ( i 3 − k ) ] .

Now, we exploit the arguments of Taylor expansion for ℱ 0 , β . Precisely, let T ℱ 0 , β 2 m + 3 be the degree ( 2 m + 2 ) Taylor polynomial of the function ℱ 0 , β around zero, that is,

T ℱ 0 , β 2 m + 3 ( x ) ≔ ∑ λ = 0 2 m + 2 x λ λ ! D λ ℱ 0 , β ( 0 ) .

Then we replace ℱ 0 , β in Q 0 , β by its Taylor polynomial T ℱ 0 , β 2 m + 3 plus the remainder term ℛ ℱ 0 , β 2 m + 3 , such that we have the form ℱ 0 , β ( i ⋅ 3 − k ) = T ℱ 0 , β 2 m + 3 ( i ⋅ 3 − k ) + ℛ ℱ 0 , β 2 m + 3 ( i ⋅ 3 − k ) . Thus, from the Hermite interpolation conditions in (A2), we have T ℱ 0 , β 2 m + 3 ( i ⋅ 3 − k ) = ( − i ⋅ 3 k ) β , which implies that

(A9) Q 0 , β = − ∑ i ∈ Z a − 3 i k ℛ ℱ 0 , β 2 m + 3 ( i ⋅ 3 − k ) .

Moreover, by (A3), ( m 0 , β k ) T = W V 2 m + 3 ( 0 ) − 1 c β T such that for a given β , and the coefficient vector m 0 , β k for ℱ 0 , β in (A1) can be bounded independently of k ∈ N , which implies that the α th derivative of ℱ 0 , β for each α = 0 , 1 , … , 2 m + 2 is uniformly bound around 0. Accordingly, ∣ ℛ ℱ 0 , β 2 m + 3 ( i 3 − k ) ∣ = O ( 3 − k ( 2 m + 3 ) ) and together with (A8) we have ∣ Q 0 , β ∣ = O ( 3 − k ( m + 3 ) ) as k → ∞ . Similarly, ∣ Q 1 , β ∣ = O ( 3 − k ( m + 3 ) ) and ∣ Q 2 , β ∣ = O ( 3 − k ( m + 3 ) ) as k → ∞ . Therefore, combining (A6) and (A7), we arrive at

∑ i ∈ Z a i k − 3 = O ( 3 − k ( 2 m + 3 ) ) , ∑ i ∈ Z ε i i β a i k = O ( 3 − k ( 2 m + 3 − β ) ) ,

which leads to

∑ k = 0 ∞ ∣ a k ( 1 ) − 3 ∣ < ∞ , ∑ k = 0 ∞ 3 k ( 2 m + 2 ) δ k < ∞ ,

for δ k ≔ max β = 0 , … , 2 m + 2 3 − k β ∣ D β a k ( ε ) ∣ . Hence, the non-stationary ternary subdivision scheme (2) satisfies approximate sum rules of order 2 m + 3 .□

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Received: 2020-07-21
Revised: 2021-04-02
Accepted: 2021-06-03
Published Online: 2021-08-28

© 2021 Zeze Zhang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2021-0058
Keywords for this article
non-stationary combined ternary scheme; exponential polynomial generation/reproduction; approximation order; smoothness
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  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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