Dimension preserving approximation

This article introduces the novel notion of dimension preserving approximation for continuous functions defined on [0, 1] and initiates the study of it. Restrictions and extensions of continuous functions in regards to fractal dimensions are also investigated.


Introduction
We start our discussion with a function g ∈ C[0, 1] := {f : [0, 1] → R :f is continuous on [0,1]} with dim G g > 1. Here and in the following, we denote the graph of a function g by G g . In the present informal discussion, we use dim to denote a fractal dimension. For the existence of such functions g, see, for instance, [24].
The function f : [0, 1] → R defined by f (x) := x 0 g(t)dt will have the following properties: If we approximate f by Bernstein polynomials B n (f ) of order n, then B n (f ) converges uniformly to f and (B n (f )) converges uniformly to f = g. (We refer the interested reader to [13] for Bernstein polynomials and their properties.) Note that (B n (f )) is again a polynomial and, thus, the fractal dimension of (B n (f )) is equal to one. The above conveys that the approximation of a function by Bernstein polynomials preserves the function class but not the dimension of its derivative. The current article targets to study approximation aspects with respect to fractal dimensions of a function and its derivative.
The structure of this paper is as follows. After a brief introduction to fractal dimensions in Sect. 2, the novel concept of dimension preserving approximation is introduced in Sect. 3 and some of its properties are discussed. Section 4 deals with the restriction and extension of continuous functions in regards to fractal dimensions.

Hausdorff dimension, box dimension, and packing dimension
In this section, we introduce those fractal dimensions that are relevant for the present paper. These are the Hausdorff dimension, the box dimension, and the packing dimension defined for nonempty subsets of a separable metric space (X, d X ). For more details about these fractal dimensions and for proofs, we refer the interested reader to, for instance, [11,12,19].
To this end, let (X, d X ) be a separable metric space. For a non-empty subset U of X, the diameter of U is defined as Let F be a subset of X and s a non-negative real number. The s-dimensional Hausdorff measure of F is defined by where the infimum is taken over all countable covers {U i } i∈N of F by sets respectively. If the above two expressions are equal, their common value is called the box dimension of F : Let us introduce a few notions that will lead to the definition of packing dimension. Let s ≥ 0 and δ > 0. We denote by It is observed that P s δ (F ) decreases with δ. This further implies that the limit exists. As P s δ is only a pre-measure, one defines Thus, the packing measure P s of F is the infimum of the packing pre-measures P s 0 of countable covers of F .

Definition 2.3.
Let F ⊂ X and s ≥ 0. The packing dimension of F is defined by dim P F := inf{s : P s (F ) = 0} = sup{s : P s (F ) = ∞}.
It is known that the following inequalities hold between these types of fractal dimensions [11]: Although there are several other notions of fractal dimension, this article will deal only with those that were introduced above.

Dimension preserving approximation
In this section, we present some results relating to the invariance of fractal dimensions under certain maps. In what follows, let (X, d X ) be a separable metric space, and (Y, d Y ) be a separable normed linear space. We equip the space X × Y with a metric d defined by The Lipschitz constant of a map f : X → Y is given by .
The following result is a generalization of Theorem 1 in [20].

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Therefore, T f is a bi-Lipschitz map. Since the Hausdorff dimension and packing dimension are Lipschitz invariant (see, for instance, [12]), we have that Since the upper and lower box dimensions and the box dimension (if it exists) are also Lipschitz invariant (cf. [12]), the previous lemma also holds for these fractal dimensions.
It is well-known that the set of Lipschitz functions [0, 1] → R, which we denote by Lip[0, 1], is a dense subset of C[0, 1] when the latter is endowed with the supremum norm · ∞ . We use this fact to prove the following theorem. 1] which converges to f uniformly. Now let g ∈ Lip[0, 1] be arbitrary but fixed and fix an h ∈ S β . We define a sequence (f k ) by With the convergence of (f k ) to g, a basic real analysis result completes the proof.
It is known that the box dimension, Hausdorff dimension and packing dimension are also Lipschitz invariant and therefore the above theorem is also valid for these dimensions.
For the next result, we require the following definition.
be a set-valued map between two metric spaces.
, establishing the proof.
Remark 3.6. The set-valued map D is not closed. If we choose a sequence of polynomials (p n ) converging to a Weierstrass-type nowhere differentiable function f with Hausdorff dimension > 1 (for examples of such functions, see, e.g., [20]) then (1, p n ) ∈ G D and (1, Therefore, we deduce that G D is not closed.
The following result is well-known in analysis but repeated for the sake of completeness.
Note that if f is a continuously differentiable function, then dim(G f ) = 1. However, we cannot say anything about the dimension of its derivative. For example, take a Weierstrass-type nowhere differentiable continuous function g : [0, 1] → R as in, for instance [24], Moreover, we emphasize the fact that functions f defined by an integral formula are always absolutely continuous. Hence, for such functions f we have dim G f = 1.
Then there exists a sequence of continuously differentiable functions (f n ) satisfying dim G f n = β, and (f n ) converges uniformly to f . Proof. From Theorem 3.3 we obtain a sequence of continuous functions (g n ) with dim G gn = β, which converges uniformly to f . Define a function f n : . Then, f n = g n and (f n ) converges to f . Moreover, one verifies that the sequence f n (0) converges to zero. In view of Theorem 3.7, the sequence (f n ) converges uniformly to f with the required condition dim G f n = β.
Remark 3.9. The above theorem can be extended as follows. Suppose f is a k−times continuously differentiable function with dim G f (k) = β for some 1 ≤ β ≤ 2. Then there exists a sequence of k−times continuously differentiable functions (f n ) satisfying dim G f (k) n = β, which converges uniformly to f . The next theorem deals with both dimension preserving and shape preserving approximation of a continuous function. Proof. The proof uses arguments similar to those given in Theorems 3.3 and 3.8, and is omitted.

Construction of dimension preserving approximants
Hutchinson constructed parametrized curves in [16] and Barnsley [5] used iterated function systems (IFSs) to define a class of functions called fractal interpolation functions (FIFs). A FIF is a continuous function whose graph is the (attractor) invariant set of a suitably chosen IFS. For the benefit of the reader, we briefly revisit the construction of a fractal interpolation function. For material about IFSs and FIFs, we refer the interested reader to, e.g., [6,19].
To this end, let (X, d X ) be a complete metric space and let f : X → X. The map f is said to be a contraction (on X) if Lip(f ) < 1.
and consider the IFS W = (I × R, w i : i ∈ J). The function f in the above theorem whose graph is the attractor of an IFS is termed a fractal interpolation function. Main features of FIFs are that their graphs are self-referential in the sense of (3.1) and that they usually have non-integral box or Hausdorff dimension.
For a special choice of mappings F i , namely, F i (x, y) := c i x + d i + α i y, where the coefficients c i and d i are determined by the conditions (3.2), and the α i ∈ (−1, 1) are free parameters, the resulting FIF is called affine.
Estimates for the Hausdorff dimension of an affine FIF were presented in [5] and also in [10]. The box dimension of classes of affine FIFs was computed in [7,8,14] and for FIFs generated by bilinear maps in [9]. In [15], a formula for the box dimension of FIFs R n → R m was derived.
In [21,22,26], the idea of fractal interpolation was explored further leading to a class of fractal functions associated with a given (classical) function f ∈ C(I) as follows. (See also, [19] for a similar approach.) Let Δ := (x 0 , x 1 , . . . , x N ) be a partition of I := [0, 1] such that, without loss of generality, 0 = x 0 < x 1 < · · · < x N = 1. For i ∈ J, let L i : I → I i be affine (see above) and F i : I × R → R be given by where b = f is any continuous function satisfying and α := (α 1 , α 2 , . . . , α N ) ∈ (−1, 1) N . The corresponding FIF, denoted by f α Δ,b , is called an α-fractal function. In [22], it is noted that α-fractal functions satisfy the self-referential equation The following result is a special case of Theorem 3 in [8]   where D is the unique positive solution of denotes the graph of f α Δ,b . Note 3.15. We define the second modulus of smoothness with step-weight func- where the second supremum is taken over those values of x for which every argument belongs to the domain [0, 1]. In [25] the following estimate was proved: for some constant C > 0. Here, B n : C(I) → Π n denotes the n-th order Bernstein operator and Π n the space of polynomials of degree ≤ n.
Now we are ready to prove the next result. Further assume, without loss of generality, that the sampling points in x i , f (x i ) : i = 0, 1, 2 corresponding to f are not collinear. Let (p n ) n∈N be the sequence of Bernstein polynomials p n = B n (f ) that converges uniformly to f . For each fixed n ∈ N, construct the α-fractal function (p n ) α Δ,Bn(pn) corresponding to p n by choosing the parameter function b (see above) as B n (p n ). In the light of Eq. (3.3) and Note 3.15, a simple and straightforward calculation produces We therefore conclude that the sequence (p n ) α Δ,Bn(pn) converges uniformly to f . For each fixed n ∈ N, the functions p n and B n (p n ) are Lipschitz continuous and the set of data points {(x i , p n (x i )) : i = 0, 1, 2} is not collinear. Hence, with the help of Theorem 3.14, the box dimensions of the graphs of (p n ) α Δ,Bn(pn) , which depend only on the partition and the scaling vector, are all the same and are equal to β. Remark 3.17. The previous theorem also determines the order of approximation by fractal functions. More precisely, for a given function f ∈ C(I) and β ∈ (1, 2) we have the following estimate where (f n ) is a sequence of fractal functions as in the above theorem and the constant C depends only on β. The above order of approximation is not claimed to be the optimal. Note that though there are many other approximation polynomials, so called Bernstein-type polynomials, we have used only the Bernstein polynomials in the previous theorem. The reader is encouraged to consult [13] for a more detailed study on order of convergence by Bernsteintype polynomials.
Next, we approximate a given function by a sequence of fractal functions having the same Hausdorff dimension. For this purpose, we need to quote the following result that can be found in [4] and is based on work presented in [3].
Let f be an affine FIF associated with the above data set and denote by G f its graph.
Note that Theorem 3.18 implies that under the condition (3.4) the box dimension of G f equals its Hausdorff dimension. Proof. We consider a sequence of data sets , for some i = j, in order to verify condition (3.4). For each n ≥ 2, we define a data set˜ n bỹ where y 0 = f (x 0 ) + 1 n , y i = f (x i ) for i = 1, 2, . . . , n. Finally, we obtain a sequence (g n ) of fractal interpolation functions generated by the data set n and the aforementioned scale vector α converging to f and satisfying the desired condition.

Restrictions and extensions of continuous functions
In this section, we focus on some restrictions and extensions of continuous functions in regards to fractal dimensions. For this purpose, we need to state some known results. In [17] the above result was also established for the packing dimension.
The same result holds for the packing dimension.
Proof. For β ≤ dim G f . Using Lemma 4.3 we have a compact subset K 1 of G f such that dim K 1 = β. We now show that there exists a compact set By the compactness of K 1 there exists a convergent subsequence of (x n , f(x n )) . Denote this convergent subsequence again by (x n ) and let (x, f (x)) ∈ K 1 be its limit. Hence, (x n ) converges to x and x ∈ K 2 completing the proof. The next theorem is a modification of Proposition 2.3 appeared in [18]. For the convenience of the reader, we include the proof. The result also holds for the packing dimension.
Proof. Let X be a proper compact subset of [0, 1] and f : X → R a continuous function. Now we consider the following possibilities: (1) 0, 1 ∈ X.
We write [0, 1]\X for each of the four cases above as follows: By the finite stability of the Hausdorff dimension (cf. [12]), we claim that it is enough to deal with the first case. Applying Lemma 4.5 for each intervals [a i , b i ], we extend the function f as follows: Hence we obtain the result for the Hausdorff dimension. Since the packing dimension is also countably stable, the result for dim P follows immediately.

Summary
In this article we investigated a new notion of constrained approximation through fractal dimensions. Further, we constructed dimension preserving approximants to a prescribed function. In the last part of the article, we introduced and investigated the restrictions to and extensions of continuous functions in terms of fractal dimensions.