Analytic functions in shift-invariant spaces and analytic limits of level dependent subdivision

The structure of exponential subspaces of finitely generated shift-invariant spaces is well understood and the role of such subspaces for the approximation power of refinable function vectors and related multi-wavelets is well studied. In this paper, in the univariate setting, we characterize all analytic subspaces of finitely generated shift-invariant spaces and provide explicit descriptions of elements of such subspaces. Consequently, we depict the analytic functions generated by level dependent (non-stationary) subdivision schemes with masks of unbounded support. And we confirm the belief that the exponential polynomials are indeed the only analytic functions generated by such subdivision schemes with finitely supported masks.


Introduction
Shift-invariant spaces have been well studied in the literature in various contexts. On one hand, such spaces arise naturally in the signal processing, where they model signals generated by integer shifts of some basic signals [1,39,41]. On the other hand, approximation properties of shift-invariant spaces have been put to good use in the study of representation systems, affine synthesis and related issues, see e.g. [22,40] and references therein. One of the most popular and well studied applications is the theory of wavelets and multiwavelets. Deep analysis of structural and approximation properties of wavelet generated spaces can be found in [2,3,4,16,25,26]. A closely related subject is subdivision schemes for numerical approximation and for generating curves and surfaces. The set of limit functions of a subdivision scheme is a shift-invariant space.
There is a vast number of results on finitely generated shift-invariant spaces. In this paper, we focus on the case of compactly supported generators and address the problem of classifying all analytic functions in the corresponding shift-invariant space. Note that the generators themselves are not analytic, since they are compactly supported. From the practical point of view our results can be interesting for both signal processing (for characterizations of all analytic signals generated by shifts of finitely many signals) and for wavelet theory (e.g. in the context of wavelet expansions of the solutions of differential equations). Our original motivation for studying the structure of analytic subspaces of shift-inariant spaces comes from subdivision schemes and refinability.
Intrinsic connection between refinable functions and recursive algorithms called subdivision schemes is well known [6,7,9,17,19,30,33,43]. Indeed, convergent subdivision schemes generate refinable functions or sequences of jointly refinable functions. In return, existence of solutions of refinement equations or systems of refinement equations together with linear independence guarantee the convergence of the underlying subdivision schemes.
Approximation and structural properties of the corresponding shift-invariant spaces depend on the structure of their polynomial or exponential polynomial subspaces, see e.g. [5,12,21,28,37] and references therein. In most cases the solutions of refinement equations or the corresponding subdivision limits are not known analytically. Therefore, the description of their approximation/generation properties is usually done in terms of the finite sequences of coefficients of the refinement equations or, equivalently, in terms of the corresponding trigonometric polynomials or, equivalently, in terms of the Strang-Fix conditions on the Fourier transforms of the refinable functions. Such characterizations are well known in both level independent (stationary) and level dependent (non-stationary) settings, see e.g. [6,7,8,13,19,24]. Moreover, polynomial generation is necessary for linear independence or the convergence of subdivision in the stationary setting [7]. In the non-stationary setting, similar results involving exponential polynomial generation are not true. Indeed, there exist convergent non-stationary schemes that do not generate any exponential polynomials [14]. Naturally, a question arises if there exist other classes of analytic functions rather than exponential polynomials whose generation is necessary for the convergence of non-stationary subdivision. Furthermore, existence of such classes of functions would enrich our understanding of the palette of shapes generated by subdivision.
In this paper, we deal with the univariate setting and answer three related open questions: What is the exact structure of all analytic subspaces of the shift-invariant spaces generated by finitely many compactly supported (not necessarily refinable) functions?
How does the number of generators effect the analytic subspaces of the corresponding shift-invariant space?
How big are such analytic subspaces generated by level dependent subdivision schemes with finite masks?
The answer to the first question also provides a characterization of all analytic subspaces generated by vector subdivision schemes. Polynomial subspaces generated by vector subdivision schemes and approximation properties of the related multi-wavelets were thoroughly studied in the stationary case e.g. in [5,11,27,29,31,32].
The answer to the second question sheds the light onto approximation properties of the non-stationary subdivision scheme with masks of unbounded support e.g. by Rvachev [38] or of the related constructions in [10,23].
This paper is organized as follows: In section 2, we show first that the analytic subspace of the shiftinvariant space generated by compactly supported (not necessarily refinable) functions φ 1 , . . . , φ n consists of the analytic functions for some pairwise distinct modulo 2πi complex numbers λ j , some d j ∈ N 0 and some 1-periodic analytic functions π j,k : R → C. Moreover, see section 3, in the case of a single generator φ = φ 1 , if the integer shifts of φ are linearly independent, then the structure of H is characterized by the exponential decay of the sequences derived from the Fourier transform (analytically extended to C) of φ. Furthermore, see section 4, if additionally the Fourier transform of the generator φ = φ 1 satisfies the generalized refinability propertŷ with some trigonometric polynomials a j , then the 1-periodic analytic functions π j,k are trigonometric polynomials. If all trigonometric polynomials a j are the same, then all π j,k are constant. In section 5, we first recall the basic facts about subdivision and then interpret the results from section 2 accordingly. In particular, we confirm the following beliefs: every analytic limit of a stationary subdivision scheme is a polynomial; every analytic limit of a non-stationary subdivision scheme is a exponential polynomial. In section 6, we study the structure of the piecewise analytic basic limit functions of non-stationary subdivision.

Analytic functions in finitely generated shift invariant spaces
There is a multitude of results in the literature about the properties of shift-invariant spaces generated by Φ = {φ 1 , . . . , φ n } with each φ j of compact support, see [37] and references therein. We denote by V Φ the corresponding shift-invariant space with ℓ(Z) being the space of sequences over C. Such spaces for n = 1 arise in the context of stationary and non-stationary subdivision, while for n > 1 in the context of vector subdivision schemes and multi-wavelets. Our goal is to expose the classes of analytic functions that belong to V Φ . Under analytic functions [15] we mean functions infinitely differentiable on R and having a power series expansion around each point in R. Our characterizations make use of the following exponential spaces.
Definition 2.1. Let Λ ⊆ C and P k the space of polynomials of degree less than or equal to k. The space with the integer k(λ) being the multiplicity of λ ∈ Λ.
In subsection 2.1, we characterize the set of all analytic functions that belong to V Φ without any additional assumptions on the generators in Φ. In section 3, we study the special case of n = 1 under the assumption that the integer shifts of φ = φ 1 are linearly independent. In section 4, we additionally assume the generalized refinability of φ = φ 1 .

Case n ≥ 1
In this subsection, we study the structure of the subspace H of analytic functions in the shift-invariant space V Φ . The main result of this section, Theorem 2.8, states that the expected contribution of the exponential spaces U to the structure of H has to be unexpectedly augmented by contributions of certain 1-periodic analytic functions. The proof of Theorem 2.8 is based on auxiliary results from subsection 2.1.1 and on Theorem 2.7. The next Example shows that the presence of 1-periodic analytic functions in the subspace H is very natural.
Example 2.2. Let π be 1-periodic analytic function. For a B-spline ψ, define Then, due to the partition of unity property of ψ and periodicity of π, we have i.e. π ∈ V φ .

Auxiliary results
To study the structure of the subspace H, we make use of the following well known difference operator.
In Lemma 2.4, we recall the action of the powers of the difference operator ∇ λ on the products of the form π(t)p(t)e λt with a 1-periodic function π and an algebraic polynomial p. Such products are shown in the sequel to be the building blocks of the elements of H. Clearly, powers of ∇ λ annihilate π(t)p(t)e λt . On the contrary, ∇ λ does not affect the structure of such products, if we replace the term e λt by e µt with µ λ. We present the proof of this straightforward fact to illustrate that, even in the presence of the 1-periodic function π, the action of the difference operator ∇ λ is inherited from its action on the exponential spaces U.
Our next result, Lemma 2.6, states that any 1-periodic function, appearing in the representation of f ∈ H, is analytic. It also exposes the finer structure of H. We first illustrate the idea of the proof of Lemma 2.6 on the following example.
Lemma 2.6. Let H ⊆ V Φ be a subspace of all analytic functions. If there exist s ∈ N, λ j ∈ C (pairwise distinct modulo 2πi) and d j ∈ N 0 , j = 1, . . . , s, such that p ℓ,k (t)π ℓ,k (t) belongs to H and all its coefficients c k,m are non-zero, since c k,m are products of the factors e λ j , j = 1, . . . , d, or e λ j −λ ℓ − 1, j ℓ, and of the binomial coefficients in the expansions of (t + 1) k , k = 0, . . . , d ℓ . Therefore, deg(p ℓ,k ) = k, k = 0, . . . , d ℓ . Note also that the leading term of f for k = d ℓ is of the form c d ℓ ,d ℓ π ℓ,d ℓ (t) t d ℓ e λ ℓ t . Thus, by Lemma 2.4, applying ∇ d ℓ λ ℓ to f leaves us with d ℓ ! c d ℓ ,d ℓ π ℓ,d ℓ (t) e λ ℓ t ∈ H. The analyticity of π ℓ,d ℓ (t) e λ ℓ t and e λ ℓ t implies that the function π ℓ,d ℓ is analytic. Next we apply to f the operator ∇ d ℓ −1 Its analyticity and the analyticity of π ℓ,d ℓ and e λ ℓ t imply that π ℓ,d ℓ −1 is analytic. Continuing iteratively yields the claim.

Structure of H
We show first that every f ∈ H is indeed of the form (1) with 1-periodic analytic functions π j,k . Moreover, Proof. Due to the compact supports of φ j , j = 1, . . . , n, there are only finitely many functions φ j (· − ℓ), j = 1, . . . , n, ℓ ∈ Z, whose supports intersect with the interval [0, 1]. This finite number we denote by where some of the coefficients a ℓ ∈ C are non-zero. If, furthermore, f ∈ H, then, due to the analyticity of this linear combination, we have The identity (3) implies that, for every τ ∈ [0, 1], the sequence of numbers { f (τ + ℓ) : ℓ ∈ Z} satisfies the linear difference equation with constant coefficients a 0 , . . . , a N . Let α j ∈ C, j = 1, . . . , s, s ≤ N, be the roots of the characteristic polynomial corresponding to (3) and µ j ∈ N be the multiplicity of α j . Then, the solution of (3), for τ ∈ [0, 1], has the form Extend b j,k to a 1-periodic function over R. Then substitution t = τ + ℓ and the binomial identity lead to The functions π j,k : can be extended to R to be 1-periodic. By Lemma 2.6 (i), due to f ∈ H, all the functions π j,k are analytic.
Finally, in Theorem 2.8, we observe that the intrinsic building blocks of H depend on the invariant spaces of the shift operators A d j with d j , j = 1, . . . , s, in (2). The operator A d j acts on the space i.e. on the direct sum of the spaces P k of algebraic polynomials of degrees at most k, by of size k+1, k = 0, . . . , d j (B k maps the vector of coefficients of p k to the coefficients of p k (·+1)). More precisely, B k is a lower triangular matrix with ones on the main diagonal and the ℓ-th column of B k contains (starting with the main diagonal element) the binomial coefficients of the expansions of (t + 1) k+1−ℓ , ℓ = 1, . . . , k + 1, respectively. (4) such that H is a linear span of spaces L 1 , . . . , L s with Moreover, every subspace of H has the same form (5) with the same λ j and π j,k but with some subspaces N ′ j ⊆ N j , j = 1, . . . , s. Proof. For f in H, by Lemma 2.6 (ii), we get that . . , s and describe the transformation f → g j in Lemma 2.6 (ii). Moreover, we observe that the shift operator g(·) → g(· + 1) leaves the functions π j,k unchanged and maps the vector p j,0 , . . . , the linear span of all integer shifts of g. Hence, it contains all functions e λ j t d j k=0p j,k π j,k with p j,0 , . . . ,p j,d j from the minimal invariant subspace of the operator A d j that contains the vector p j,0 , . . . , p j,d j . The invertibility of

. , s, one chooses an arbitrary invariant subspace N j of the corresponding blockdiagonal matrices A d j . This defines the functional space (5). The direct sum of those s spaces is the space of all analytic functions in a shift-invariant space V Φ . It is interesting to note that the matrices A d j defined in (4) may have a very rich variety of invariant spaces. It would be interesting to obtain the description of such invariant spaces at least for A d j with small number of diagonal-blocks.
In the example below we consider the simplest case of two diagonal blocks of sizes 1 and 2 and show that already in this case there are four possible spaces N 1 .
The following example shows that the structure of the invariant subspaces of the matrices A d j in (4) is highly nontrivial.
Example 2.10. By Theorem 2.8, every subspace H of analytic functions generated by the integer shifts of a finite set of compactly supported functions is a direct sum of spaces L j of the form (5). Consider the simplest case s = 1, i.e., H = L 1 , d 1 = 3 and the 3 × 3 matrix M 3 has two blocks, of sizes one and two. For the sake of simplicity in what follows we denote λ 1 = λ, M 3 = M and N 1 = N. Let λ ∈ C, (a, b, c) ∈ R 3 and π 1,0 , π 2,2 : R → C be 1-periodic analytic functions. We classify all invariant subspaces N ⊆ M = P 0 ⊕ P 1 of the linear operator A : M → M which, by (4), has the matrix representation By Theorem 2.8 there is a one-to-one correspondence between these invariant subspaces N of A and the subspaces L 1 = L in (5).
1). The first subspace of the matrix A is N (1) = (0, b, c) (b, c) ∈ R 2 and the corresponding subspace of the analytic functions is given by 3). Moreover, for every vector (a 0 , Surprisingly, these are not all invariant subspaces of A. There is one more family of invariant subspaces. 4). For every (u, v) ∈ R 2 \ {0}, the matrix A has a two-dimensional invariant subspace Thus, there are four possible choices for the corresponding space L 1 . Note that even in this simple example the classification of invariant subspaces of the matrix A is nontrivial.

Analytic functions in single generated shift-invariant spaces
In this section, we characterize the structure of the subspace H of analytic functions in a singly generated shift-invariant space This characterization, stated in Theorem 3.1, relates the structure of H to the exponential decay of the sequences derived from the Fourier transform (analytically extended to C) of a compactly supported distribution φ.
Due to Theorem 2.8, we consider analytic functions . . , d}, we set the corresponding 1-periodic analytic function ω j to be identically zero on R. We assume that ω d (t) 0.
(ii) Conversely, if the sequences in (6) decay exponentially for some λ ∈ C and d ∈ N 0 , then the space V φ contains the (d + 1)-dimensional subspace of analytic functions where ω k are 1-periodic analytic functions given by We first prove (ii). Assume that the sequences in (6) decay exponentially. Then, by Payley-Wiener theorem, the 1-periodic functions in (9) are analytic and, by the Poisson summation formula, we have for k = 0, . . . , d and t ∈ R. Hence, for k = 0, . . . , d, the functions are analytic and belong to H λ ⊆ V φ . The proof of (i) is by induction on d. In the case d = 0, the polynomial p 0 is constant, w.l.g p 0 (t) ≡ 1. Then The periodicity of ω implies that which is equivalent to the identity Due to the linear independence of the integer shifts of ψ, we obtain a ℓ+1 − a ℓ = 0 for all ℓ ∈ Z. Or, equivalently, w.l.g. a ℓ = 1 for all ℓ ∈ Z. Therefore, by the Poisson summation formula, we obtain Due to the analyticity of ω, the above identity holds if and only if there exists a constant C > 0 and q ∈ (0, 1) such that φ − iλ 2π + ℓ ≤ C q |ℓ| for all ℓ ∈ Z.
The linear independence of the integer shifts of ψ implies that a ℓ+1 − a ℓ = p(ℓ) for all ℓ ∈ Z. Theory of difference equations ensures that every solution of this difference equation is given by a ℓ =p(ℓ), ℓ ∈ Z, for some polynomialp of degree d. We writep(ℓ) = α ℓ d + q(ℓ), α ∈ R \ {0} and deg (q) ≤ d − 1, and have and, hence, the function is analytic due to the analyticity of g and, by the inductive assumption, analyticity of ℓ∈Z q(ℓ) ψ(· − ℓ). Consequently and due to the inductive assumption, the function is analytic as well. Therefore, by the Poisson summation formula and the analyticity implies that the sequence φ (d) − iλ 2π + ℓ : ℓ ∈ Z decays exponentially. Thus, we have shown that (6) is satisfied for k = 0, . . . , d.
Corollary 3.2. The set of analytic functions spanned by the shifts of a compactly supported function φ is a linear span of spaces H λ in (7) over all λ ∈ C such that the sequences in (6) decay exponentially.

Single generated shift-invariant spaces with generalized refinability
Additional assumption on the generalized refinability of φ, i.e. the property for some trigonometric polynomials replaces the requirement in Theorem 3.1 on the exponential decay of sequences in (6) by a requirement that only finitely many of the sequence elements are non-zero (i.e. the corresponding 1-periodic analytic functions in (9) are trigonometric polynomials).
The main result of this section finalizes our knowledge about the structure of H. If, for some λ ∈ C and d ∈ N 0 , d ≤ N, the analytic function e λ t d k=0 contain (all together) at most N non-zero elements.
The In the proof of Proposition 4.3 we use the idea of the method of counting of zeros elaborated in [35]. The essence of the method is the following: if the infinite product of trigonometric polynomials has too many zeros on a segment [0, r], then one of the polynomials must have more than N zeros which leads to the contradiction. However, for proving Proposition 4.3, this idea should be significantly modified since here we have to count not zeros but in a sense "almost zeros" of polynomials. That is why we begin with Lemma 4.2, which states that the maximum norm of a trigonometric polynomial of degree N is small, if its point evaluations at arbitrary (well-separated) N + 1 pairwise distinct points in [0, 1) are small. This result generalizes the well known fact that an algebraic polynomial of degree N is identically zero, if it vanishes at N + 1 points.
where the chord length |z − z k | ≤ 2 for |z| = 1. On the other hand, the length of an arbitrary chord of a unit circle is at least the length of the shortest arc defined by this chord multiplied by 2 π (this estimate is achieved for diameters). Therefore, |z m − z k | ≥ 2 π · 2π · |y m − y k | = 4 |y m − y k |. Thus, Taking maximum over y ∈ [0, 1), we arrive at the desired estimate (14). Now we are ready to prove Theorem 4.1. Proof. Let d = 0 and assume that there are at least N + 1 non-zero elements in the corresponding sequence in (6) with α = − iλ 2π , λ ∈ C. 1.
Step: W.l.g. φ(α) 0. Then by (12), for every ε > 0 there exists r α ∈ N such that for all r ≥ r α thus, this product is bounded away from zero uniformly for all r ≥ r α . The fact that a j (0) = 1, j ∈ N, implies that there exists R ∈ N such that for all i.e. the above product is also uniformly bounded away from zero. Next, we split the infinite product appearing in the definition of φ(α + ℓ) into three products accordingly to the properties in (15)- (16). For ℓ = 2 r , r ≥ r α , due to the 1-periodicity of the trigonometric polynomials a j , we have Due to (15)- (16), the exponential decay of the sequence in (6) implies that r+R j=r+1 a j (2 − j α + 2 − j+r ) ≤ Cq 2 r , q ∈ (0, 1).
Hence, at least one of the factors in (17) (has an almost zero) is in the absolute value smaller than or equal to Cq 2 r /R . Repeating the argument with ℓ = 2 r+n , n ∈ N, we conclude that J trigonometric polynomials a r+1 , . . . a r+1+J , r ≥ r α , J >> R, have at least J − R almost zeros. The possible almost zeros for each a k , k ∈ {r + 1, . . . , r + 1 + J} are at the distinct complex points in (17) 2

3.
Step: We use Lemma 4.2 to get a contradiction to the fact that a k (0) = 1. First note that all the points (we set have, due to α = − i λ 2π , λ ∈ C, the same imaginary part Moreover, these points w ℓ are separated by the distance of at least 2 −k for k > R. Indeed, let n, s ∈ {1, . . . , R}, n s, and ℓ,l ∈ L ∪ {0}, ℓ l . Then, for k > R, due to |α ℓ − αl| = |ℓ −l| being an integer bigger than or equal to 1, we have Secondly, for Note that lim k→∞ a k −ã k ∞ = 0 and that the minimal distance between the real points (which are real parts of w ℓ 's) is given by 2 −k , due to all w ℓ 's having the same imaginary part. Also note that the almost zeros w ℓ of a k are closely related to the almost zeros ofã k by a k (w ℓ ) =ã k (y ℓ ), ℓ ∈ L ∪ {0}. Therefore, by Lemma 4.2, we get On the other hand, a k (0) = 1 and lim k→∞ a k −ã k ∞ = 0 lead to a contradiction.
Next we provide the inductive step that completes the proof of Theorem 4.1. Proof. The base of the induction follows from Proposition 4.3. We assume that, for k = 0, . . . , d − 1, the sequences in (6) with α = − iλ 2π , λ ∈ C, have in total finitely many non-zero elements. This implies the existence of r 0 such that for all r > r 0 we hateφ The inductive step we prove by contradiction assuming that there are at least N + 1 non zero elements in the sequences in (6) for k = 0, . . . , d. 1.Step By the argument in Proposition 4.3 1.

2.
Step For the same α, the additional information about the exponential decay of the other sequences in (6) for k = 1, . . . , d supplies another d(J − R) almost zeros. Indeed, due toφ(α) 0, there exist ρ ∈ (0, 1) and a constant C 0 > 0 such that Furthermore, for ε > 0 there exists r α,t ∈ N such that for all r ≥ r α,t thus, this product decays slower that t d uniformly for all r ≥ r α,t . Making use of the Taylor expansion ofφ at α + 2 r + t, r ≥ max{r 0 , r α,t }, we obtain with the constant C 2 > 0 depending on the constant C > 0 that governs the exponential decay in (6) and on the error term in the Taylor expansion. For ℓ = 2 r , r ≥ max{r 0 , r α,t }, due to the 1-periodicity of the trigonometric polynomials a j , we have Due to (20), (21) and similar argument to (16), the decay in (22) implies that Hence, at least one of the factors in (17) (has an almost zero) is in the absolute value smaller than C 2 q 2 r /R . Repeating the argument with ℓ = 2 r+n , n ∈ N, we conclude that J trigonometric polynomials a r+1 , . . . a r+1+J , r ≥ max{r 0 , r α , r α,t }, J >> R, have at least 2(J − R) almost zeros. The possible almost zeros for each a k , k ∈ {r + 1, . . . , r + 1 + J} are at the distinct complex points in (23) 3.
Step with the corresponding t j in 3.
Step of Proposition 4.3, by the pigeonhole principle, there exists a k , k ∈ {r + 1, . . . , r + 1 + J} with N + 1 distinct almost zeros of the form in (18) and in (24). The claim follows by the argument similar to the one in 3.
Step of Proposition 4.3 with the minimal distance of 2 −k−s d between the almost zeros in (18) and in (24).
A consequence of Theorem 4.1 states that the analytic subspaces of the shift-invariate space V φ , in the case all the trigonometric polynomials a j = a, j ∈ N, are the same, consist only of polynomials. Proof. Let α = − λi 2π . By Theorem 4.1, there are only finitely many ℓ ∈ Z such that φ(α + ℓ) 0. Case λ 0 is impossible. There exists at least one ℓ ∈ Z (w.l.g ℓ = 0) such that φ(α + ℓ) 0. Otherwise, if φ(α + ℓ) = 0, ℓ ∈ Z, then (9) implies that ω 0 (t) ≡ 0 and, by (8), the integer shifts of φ are linearly dependent. Choose R ∈ N such that for all r ≥ R we have φ(α + 2 r ) = 0. Ensuring these conditions we arrive at the contradiction to the fact that the trigonometric polynomial a has degree N. Indeed, by the 1-periodicity of a, we have which implies that at least one of the factors a(2 − j α + 2 − j+r ) = 0 for some j ∈ N, j ≥ r + 1. None of such factors, however, occur again for different r. Thus, to ensure that φ(α + 2 r ) = 0, r ∈ N, we are forced to choose a with infinitely many different zeros.
Remark 4.6. Note that the case λ 0 is possible in the setting of generalized refinability as the infinitely many zeros in (25) can be redistributed among the trigonometric polynomials a j (y), j ∈ N.

Generation properties of level dependent subdivision
In this section, we discuss the analytic limits of level dependent subdivision schemes. In subsection 5.1, we link the results from section 4 with generation properties of such subdivision schemes which are iterative algorithms mapping seqiuences c j = {c j,k : k ∈ Z} from ℓ(Z) into ℓ(Z). The linear subdivision operators S a j depend on the finite sequences (masks) a j = {a j,k : k ∈ Z} of real numbers and are defined by If the level dependent scheme associated with the sequence {a j : j ∈ N} of masks converges, then all its limits belong to V φ with

Analytic limits
We say that a level dependent scheme associated with the mask sequence {a j : j ∈ N} generates U, if the subdivision limit lim j→∞ S a j . . . S a 1 c 1 belongs to U for every starting sequence c 1 in (27) sampled from a function in U. The generation properties of subdivision schemes are well understood and are characterized in terms of socalled zero conditions or generalized zero conditions, see e.g. [7,20,24], on the mask symbols The zero conditions determine uniquely if the subdivision limit belongs to the exponential function space U in Definition 2.1 or not.
Note that the requirement in (13) boils down to the generalized zero conditions (or, equivalently, to the generalized Strang-Fix conditions [26,42]) on the trigonometric polynomials a j (or subdivision symbols a [ j] ). We first illustrate this fact on the following example.
Example 5.1. It is well known [20] that the generation of two exponential polynomials e λt and t e λt , λ ∈ C, by a level dependent subdivision scheme is equivalent to the requirement that the corresponding symbols satisfy the generalized zero conditions (for λ = 0, zero conditions at −1) In this case, generalized refinability (12) together with a standard assumptions (for λ = 0, conditions at 1) imply that the only non-zero elements of the sequences in (6) for k = 0, 1 are In other words, by Theorem 3.1, the 1-periodic analytic functions ω 0 and ω 1 are constant. Indeed, a straightforward computation yields and Furthermore, for β ∈ Z \ {0}, let j ′ ∈ N be the number of 2's in the prime number decomposition of |β|. Define j = j ′ + 1. Then we have Therefore, by (28), we arrive at the generalized Strang-Fix conditions Similarly, if (30)- (32) are satisfied, then, by Theorem 3.1, H λ is spanned by e λt and t e λt , which is equivalent to (28).
The main result of subsection 4, Theorem 4.1, essentially states that if a function belongs to the subspace H ⊆ V φ of analytic functions, then it must satisfy the generalized zero conditions. Thus, completing the quest for exhibiting all possible analytic functions generated by level dependent subdivision. We restate Theorem 4.1.
Theorem 5.2. Every analytic limit of a level dependent subdivision scheme is an exponential polynomial.
In the special level independent (stationary) case, i.e. a j = a for all j ∈ N. The Corollary 4.5 can be restated as follows.
Corollary 5.3. Every analytic limit of a level independent subdivision scheme is a polynomial.

Piecewise analytic basic limit functions
In this section, we discuss the piecewise analytic limits of level dependent subdivision schemes. We provide sufficient conditions that guarantee that restrictions of piecewise analytic φ to certain subsets of R belong to the exponential function space U in Definition 2.1.
Note that the generalized refinability in (12) implies the existence of the sequence of the basic limit functions which are jointly refinable In this subsection, we restrict our attention to the piecewise analytic basic limit functions that have diminished smoothness at the same integer points. We follow the presentation in [34] and extend it to the level dependent setting, see Lemma 6.2 and Theorem 6.3. In Theorem 6.9, we state sufficient conditions that guarantee that restrictions of piecewise analytic φ 1 to certain intervals of R belong to the exponential function space U.
Remark 6.1. The assumption in Lemma 6.2 that the discontinuous derivatives of φ 1 and φ 2 have the same integer break points cannot be guaranteed by the joint refinability of φ 1 and φ 2 . In the level independent case, this assumption on the break points is obsolete, see [34,Lemma 1]. On the other hand, our assumption that the first derivatives of φ 1 and φ 2 have the same break points is supported by the properties of exponential B-splines. (ii) for all j = 0, . . . , N lim x→ j − φ 2 (x) and lim x→ j + φ 2 (x) exist and are finite, Proof. Part (i): Assume that x j ∈ {x 0 , . . . , x N }, is the smallest non-integer point satisfying the assumptions of this Lemma. Let x = x j + h, h ∈ (−ε, ε), ε > 0. The mutual refinability of φ 1 and φ 2 implies that or, equivalently, due to a 1,0 0, Note that the points . , x j − N are all smaller than x j and are non-integer. Thus, they do not coincide with the knots x 0 , . . . , x j−1 . Therefore, for all small enough ε > 0, the right-hand side and, consequently, the left-hand sides of the identity (33) are both continuously differentiable. Thus, φ 2 ∈ C 1 (x j − ε, x j + ε), which contradicts the assumption that x j is a knot of φ (1) 2 . Therefore, all points x 0 , . . . , x N are integers. Assume next that x 0 0, then the identity (33) with x j = x 0 implies that x 0 in not a knot. Part (ii): Similar to the stationary case, first study φ 2 and, then consider φ 1 . Part (iii): Assume φ (1) 2 is continuous at 0. Choose the smallest j ∈ {1, . . . , N} such that φ (1) 2 is discontinuous at j. From (33) with x j = j, for h going to either 0 + or 0 − , we obtain that Therefore, φ (1) 2 is discontinuous at 0 and, due to supp φ 2 = [0, N], The mutual refinability of φ 1 and φ 2 implies that Due to the uniform convergence of integrals of uniformly convergent function sequences, Theorem 6.3 can be also formulated for functions whose global smoothness is one less than their local smoothness on the intervals ( j, j + 1). Note that due to Lemma 6.2, the end points of the smoothness intervals are integers.  (iv) converge uniformly on R to φ.
Then φ is a linear spline with integer knots.
Remark 6.4. Condition (iii) of Theorem 6.3 follows either from asymptotic equivalence [18] of a non-stationary scheme {S a j : j ∈ N} to a stationary scheme S a with the basic limit function φ; or from asymptotic similarity, i.e. lim j→∞ a j = a, and generation of one exponential polynomial [14].
Proof. For k ∈ N such that 2 k > N + 1, define the jumps and define g j = S a j . . . S a 1 δ.