Non-stationary Fractal Interpolation

We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps $\{\mathcal{F}_k\}_{k\in \mathbb{N}}$ where each $\mathcal{F}_k$ maps $\mathcal{H}(X)\to \mathcal{H}(X)$ and arises from an iterated function system. Employing the recently developed theory of non-stationary versions of fixed points [11] and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior, and extend fractal interpolation to this new, more flexible setting.


Introduction
Contractive operators on complete function spaces play an important role in the theory of differential and integral equations and are fundamental for the development of iterative solvers. One class of contractive operators is defined on the graphs of functions using a special type of iterated function system (IFS). The fixed point of such an IFS is the graph of a function that exhibits fractal characteristics. There is a vast literature on IFSs and fractal functions including, for instance, [2,13,14].
Up to now, the construction of contractive operators on sets or functions uses primarily sequences of iterates of one operator. Recently, motivated by non-stationary subdivision algorithms, a more general class of sequences consisting of different contractive operators was introduced in [11] and their limit properties studied. These ideas were then extended in [6] to sequences of different contractive operators mapping between different spaces. Using different contractive operators provides one with the ability to construct limit attractors that have different shapes or features at different scales.
This article uses the aforementioned new ideas to introduce the novel concept of non-stationary IFS and non-stationary fractal interpolation. These new ideas widen the applicability of fractal functions and fractal interpolation as they now include scale and location dependent features.
The outline of this paper is as follows. After providing some necessary preliminaries in Section 2, some results from [11] are presented in Section 3. In Section 4, (stationary) fractal interpolation and the associated (stationary) IFSs are reviewed. Non-stationary fractal functions are constructed in Section 5 and non-stationary fractal interpolation is introduced in Section 6. The final Section 7 defines non-stationary fractal functions on the Bochner-Lebesgue L p -spaces with 0 < p ≤ ∞.

Preliminaries
Let (X, d) be a complete metric space. For a map f : X → X, we define the Lipschitz constant associated with f by d(x, y) .
A map f is said to be Lipschitz if Lip(f ) < +∞ and a contraction on X if Lip(f ) < 1.
Definition 2.1. Let (X, d) be a complete metric space and F := {f 1 , . . . , f n } a finite family of contractions on X. Then the pair (X, F) is called a contractive iterated function system (IFS) on X.
(a) As we deal exclusively with contractive IFSs in this article, we drop the adjective "contractive" in the following. (b) In order to avoid trivialities, we henceforth assume that the number of maps in an IFS is an integer greater than 1.
With an IFS (X, F) and its point maps f ∈ F, we can associate a setvalued mapping, also denoted by F, as follows. Let (H(X), h) be the hyperspace of all nonempty compact subsets of X endowed with the Hausdorff metric h(S 1 , S 2 ) := max{d(S 1 , S 2 ), d(S 2 , S 1 )}, where d(S 1 , S 2 ) := sup Define the mapping F : H(X) → H(X) by [2,9] (2.1) It is known that for contractive mappings f ∈ F, the set-valued map F defined by (2.1) is a contractive Lipschitz map on H(X) with Lipschitz constant Lip(F) = max{Lip(f i ) : i ∈ N n }. Here, we set N n := {1, . . . , n}. Moreover, the completeness of (X, d) implies the completeness of (H(X), h). The next definition is motivated by the validity of the Banach Fixed Point Theorem in the above setting.
Definition 2.2. The unique fixed point A ∈ H(X) of the contractive setvalued map F is called the attractor of the IFS (X, F).
Note that since A satisfies the self-referential equation the attractor is in general a fractal set. It follows directly from the proof of the Banach Fixed Point Theorem that the attractor A is obtained as the limit (in the Hausdorff metric) of the iterative process A k := F(A k−1 ), k ∈ N: for an arbitrary A 0 ∈ H(X). Here, F k denotes the k-fold composition of F with itself. We refer to the element A k ∈ H(X) as the k-th level approximant of A or as a pre-fractal of rank k [13].

Systems of Function Systems (SFS)
In [11], a generalization of IFSs was presented. The idea for this generalization comes from the theory of subdivision schemes. Instead of using only one set-valued map F to obtain an iterative process {A n } n∈N with initial A 0 ∈ H(X), a sequence of function systems consisting of different families F is considered.
To this end, let (X, d) be a complete metric space and let {T k } k∈N be a sequence of transformations T k : X → X.
A criterion for obtaining an invariant domain for a sequence {T k } k∈N of transformations on X is given below. Proof. For the proof, we refer the interested reader to [11].
where F k = {f i,k : i ∈ N n k } is a family of contractions constituting an IFS on a complete metric space (X, d). Setting s i,k := Lip(f i,k ), we obtain that Lip(F k ) = max{s i,k : i ∈ N n k } < 1.
The following definitions are taken from [11,Section 4].
are called the forward and backward trajectories of A 0 , respectively.
For our current setting, it was shown in [11,Corollary 4.2] that if (i) n := n k , for all k ∈ N; (ii) there exists a common nonempty compact invariant set I ⊆ X for the maps then the forward trajectory {Φ k (A 0 )} converges for an arbitrary A 0 ⊆ I to the unique attractor of (X, F).
It was observed in [11] that the limits of forward trajectories do not lead to new classes of fractals. On the other hand, backward trajectories converge under rather mild conditions, even when forward trajectories do not converge to a (contractive) IFS, and generate new types of fractal sets.
As the convergence of backward trajectories is important for this article, we summarize the result in the next theorem whose proof the reader can find in [11].
Then the backward trajectories {Ψ k (A 0 )} converge for any initial A 0 ⊆ I to a unique attractor A ⊆ I . (a) In [11,Proposition 3.11], it is required that the invariant set I be compact. However, it suffices to only require that I is closed as (X, d) is complete. (See the proof of Proposition 3.11 in [11].) (b) The conditions for convergence of the forward and backward trajectories are more general in [11]. For our purposes and setting, the above criteria are however sufficient.
(c) Fractals generated by backwards trajectories allow for more flexibility in their shapes. By a proper choice of IFSs, one can construct fractals exhibiting different local behavior. (Cf. [11].) This is due to the fact that in the sequence the global shape of the attractor is determined by the initial maps F 1 • F 2 . . ., whereas the local shape is given by the final maps [5] was also undertaken in [11,Section 4.1], showing that SFSs have weaker prerequisites than Vvariable fractals.

Fractal Interpolation
Before introducing the new concept of non-stationary fractal interpolation, we need to briefly recall the rudimentaries of (stationary) fractal interpolation and (stationary) fractal functions. This is the purpose of the current section.

Stationary Fractal Interpolation. Suppose we are given a finite family {l
Recall that the set B(X, Y ) := {g : X → Y : g is bounded} when endowed with the metric becomes a complete metric space. For i ∈ N n , let F i : X × Y → Y be a mapping which is uniformly contractive in the second variable, i.e., there exists a c ∈ [0, 1) so that for all y 1 , y 2 ∈ Y Define an operator T : where χ M denotes the characteristic function of a set M . Such operators are referred to as Read-Bajractarević (RB) operators. The operator T is well-defined and since g is bounded and each F i contractive in the second variable, T g ∈ B(X, Y ). Equivalently, (4.5) can also be written in the form To achieve notational simplicity, we set Therefore, by the Banach Fixed Point Theorem, T has a unique fixed point f * in B(X, Y ). This unique fixed point is called the bounded fractal function (generated by T ) and it satisfies the self-referential equation or, equivalently, The fixed point f * ∈ B(X, Y ) is obtained as the limit of the sequence of mappings Next, we would like to consider a special choice for the mappings F i . To this end, we require the concept of an F -space. We recall that a metric d : Now assume that Y is an F -space, i.e., a topological vector space whose topology is induced by a complete translation-invariant metric d, and in addition that this metric is homogeneous. This setting allows us to consider mappings F i of the form where q i ∈ B(X, Y ) and S i : X → R is a function.
As the metric d Y is homogeneous, the mappings (4.11) satisfy condition (4.4) provided that the functions S i are bounded on X with bounds in [0, 1). For then Here, · ∞ denotes the supremum norm and s := max{ S i ∞ : i ∈ N n }. Henceforth, we will assume that all functions S i are bounded above by s ∈ [0, 1).
With the choice (4.11), the RB operator T becomes an affine operator on B(X, Y ) of the form Next, we exhibit the relation between the graph G(f * ) of the fixed point f * of the operator T given by (4.5) and the attractor of an associated contractive IFS.
To this end, consider the complete metric space X × Y and define map- Assume that the mappings F i in addition to being uniformly contractive in the second variable are also uniformly Lipschitz continuous in the first variable, i.e., that there exists a constant L > 0 so that for all y ∈ Y , Denote by a := max{a i : i ∈ N n } the largest of the contractivity constants of the l i and let θ := 1−a 2L . Then the mapping d θ : The next theorem is a special case of a result presented in [4]. Equation (4.15) can be represented by the following commutative diagram On the other hand, suppose that F = (X × Y, w 1 , w 2 , . . . , w n ) is an IFS whose mappings w i are of the form (4.14) where the functions l i are contractive injections satisfying (4.1) and (4.2), and the mappings F i are uniformly Lipschitz continuous in the first variable and uniformly contractive in the second variable. Then we can associate with the IFS F an RB operator T F of the form (4.5). The attractor A F of F is then the graph G(f ) of the fixed point f of T F . (This was the original approach in [3] to define a fractal interpolation function on a compact interval in R.) The commutativity of the diagram (4.16) then holds with F T replaced by F and T replaced by T F .
We now specialize even further and choose arbitrary f, b ∈ B(X, Y ) and set (4.17) q Then the RB operator T becomes , i ∈ N n . and, under the assumption that s < 1 its unique fixed point f * ∈ B(X, Y ) satisfies the self-referential equation (4.19) f In the case of univariate fractal interpolation on the real line with X := [a, b], −∞ < a < b < +∞, the base function b can be chosen to be the affine function whose graph connects the points (a, f (a)) and (b, f (b)).
If we consider the complete metric space of continuous functions (C(X, R), d) instead of (B(X, R), d), define x 0 := a, x n := b, and x i := l i (b), i ∈ N n , and impose the join-up conditions the fixed point f * will be a continuous function whose graph interpolates the set {(x j , f (x j )) : j = 0, 1, . . . , n}. Such functions are usually referred to as fractal interpolation functions [3,9]. As the RB operator is the same at each level of recursion (4.10), we refer to this as stationary fractal interpolation.

non-stationary Fractal Functions
Here, we introduce non-stationary versions of the concepts of fractal functions as presented in the previous section.
To this end, consider a doubly-indexed family of injective contractions {l i k ,k : i k ∈ N n k , k ∈ N} from X → X generating a partition of X for each k ∈ N in the sense of (4.1) and (4.2).
It is straight-forward to verify that each RB operator T k is a contraction on B(X, Y ) with Lipschitz constant Proposition 5.1. Let {T k } k∈N be a sequence of RB operators of the form (5.1) on (B(X, Y ), d). Suppose that the elements of {q i k ,k : i k ∈ N n k , k ∈ N} satisfy Now let x ∈ X. Then there exists an i k ∈ N n k with x ∈ l i k ,k (X). Thus, for any f ∈ B(X, Y ), , 0), which shows, after taking the sup over x ∈ X, that d(T k f, 0) ≤ s d(f, 0)+M . Proposition (3.1) now yields the statement.
Considering the backward trajectories {Ψ k } k∈N of the sequence {T k } k∈N of RB operators defined above and using Theorem (3.1), we obtain the next result. A fixed point f * generated by a sequence {T k } of different RB operators will be called a non-stationary fractal function (of class B(X, Y )).  1], and where τ denotes the Takagi function [17] and that T k 2 → q, where q(x) = 4x(1 − x). Consider the alternating sequence {T i } i∈N of RB operators given by Two images of this hybrid attractor of the backward trajectory Ψ k starting with f 0 ≡ 0 are shown in Figure 1. For k ∈ N, let {l i k ,k : i k ∈ N n k , k ∈ N} be family of injections from [0, 1] → [0, 1] generating a partition of [0, 1] in the sense of (4.1) and (4.2). Assume w.l.o.g. that l 1,k (0) = 0 and l n k ,k (1) = 1 and define x i k −1,k := l i k ,k (0), x i k ,k := l i k ,k (1), i k ∈ N n k where x 0,k := 0 and x n k ,k := 1. By relabelling -if necessary -we may assume that 0 = x 0,k < · · · < x i k −1,k < x i k ,k < · · · x n k ,k = 1.
Let f ∈ C[0, 1] be arbitrary. Define a metric subspace of C[0, 1] by and note that C * [0, 1] becomes a complete linear metric space when endowed with the metric induced by the sup-norm on continuous functions. Additionally, let b ∈ C * [0, 1] be the unique affine function whose graph connects the points (0, f (0)) and (1, f (1)): Note that we have continuity of T k g at the points x i k ,k ∈ [0, 1]: For, Therefore, T k g ∈ C * [0, 1] and T k g interpolates P k in the sense that where s is given by (5.3).
Proof. Using the form (4.17) for the functions q i k ,k , we obtain from (5.4) the estimate q i k ,k ∞ ≤ f ∞ + s b ∞ , which by Proposition 5.1 yields the result.
In connection with Theorem 5.1, the above arguments prove the next result. We refer to the fixed point f * ∈ C * [0, 1] as a continuous non-stationary fractal interpolation function.
To illustrate the above results, we refer to Remark 3.1(c) and present the following example.
Two images of the hybrid attractor of the backward trajectory Ψ k starting with the function f 0 (x) = x, x ∈ [0, 1], are shown below in Figure 2. Remark 6.2. Theorem 4.1 holds in the case of non-stationary fractal functions as well. For k ∈ N, a non-stationary IFS is associated with T k by setting The conditions imposed on S i k ,k and the form of the second component allows the immediate transfer of the proof of Theorem 4.1. Hence, even in the nonstationary case, one may choose the geometry (IFS) or the analytic (RB operator) approach when defining non-stationary fractal functions.

non-stationary Fractal Functions in Bochner-Lebesgue Spaces
In this section, we construct non-stationary fractal functions in the Bochner-Lebesgue spaces L p with 0 < p ≤ ∞. To this end, assume that X is a closed subspace of a Banach space X and that X := (X, Σ, µ) is a measure space. Further suppose that (Y, · Y ) is a Banach space.
Recall that the Bochner-Lebesgue space L p (X, Y), 1 ≤ p ≤ ∞, consists of all Bochner measurable functions f : X → Y such that For 0 < p < 1, the spaces L p (X, Y) are defined using a metric instead of a norm to obtain completeness. More precisely, for 0 < p < 1, define Then (L p (X, Y), d p ) becomes an F -space. (Note that the inequality (a+b) p ≤ a p + b p holds for all a, b ≥ 0.) For more details, we refer to [1,16].
In order to work in both cases simultaneously, we define ρ p : with the usual modification for p = ∞. We use the notation and terminology of Section 5 and assume that (A1) {q i k ,k : i k ∈ N n k , k ∈ N} ⊂ L p (X, Y); (A2) {S i k ,k : i k ∈ N n k , k ∈ N} ⊂ L p (X, R); (A3) {l i k ,k : i k ∈ N n k , k ∈ N} is a family of µ-measurable diffeomorphisms X → X generating for each k ∈ N a partition of X in the sense of (4.1) and (4.2). If we define for each k ∈ N an RB operator T k on L p (X, Y) of the form (5.1), whose maps satisfy assumptions (A1), (A2), and (A3), then a straightforward computation shows that T k has the following Lipschitz constants on L p (X, Y): where L i k ,k denotes the Lipschitz constant of Dl −1 i k ,k and D the Fréchet derivative on X. Now set (7.1) Imposing the condition  Proof. Only (3.4) needs to be established. This, however, carries over directly from the proof of Theorem 5.1 with γ p instead of s.
The attractor f * : X → Y whose existence is guaranteed by Theorem 7.1 is called a non-stationary fractal function of class L p (X, Y ).