Super Fractal Interpolation Functions 1 Introduction

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.19(2015) No.1,pp.20-29

Super Fractal Interpolation Functions G. P. Kapoor1 , Srijanani Anurag Prasad2 ∗ 2

1 Department of Mathematics and Statistics,Indian Institute of Technology Kanpur, India Department of Mathematics, Indian Institute of Science Education and Research Bhopal, India.

(Received 3 February 2014 , accepted 26 November 2014)

Abstract: In the present work, the notion of Super Fractal Interpolation Function (SFIF) is introduced for finer simulation of the objects of nature or outcomes of scientific experiments that reveal one or more structures embedded in to another. In the construction of SFIF, an IFS is chosen from a pool of several IFSs at each level of iteration leading to implementation of the desired randomness and variability in fractal interpolation of the given data. Further, an expository description of our investigations on the integral, the smoothness and determination of conditions for existence of derivatives of an SFIF is given in the present work. Keywords: Fractal; Interpolation; Super Fractals; Iteration; Attractor; Iterated Function Systems; Smoothness; Dimension

1

Introduction

Barnsley [1] introduced Fractal Interpolation Function (FIF) using the theory of Iterated Function System (IFS). Since then, a growing number of papers have been published showing relation between fractals and wavelets [2], fractal functions and Kiesswetter-like functions [3] and on fractal dimension [4, 5]. Later, Barnsley et. al. [6] extended the idea of FIF to produce more flexible interpolation functions called Hidden-variable FIF (HFIF) which were generally non-self affine. Dalla [7] found bounds on fractal dimension for the graphs of non-affine FIFs. In 1989, Barnsley and Harrington [8] constructed an IFS to show that a FIF can be integrated infinite times, giving rise to a hierarchy of smoother functions and developed results on differentiability of a FIF. Different kinds of FIFs like Hermite FIFs, Spline FIFs were constructed in [9–13] and various properties like smoothness of FIF, perturbations were discussed in [14–16]. The constructions of multivariable FIFs generated by using higher dimensional or recurrent IFSs are treated in [17–21]. Fractal Interpolation Function, constructed as attractor of a single Iterated Function System (IFS) by virtue o f selfsimilarity alone, is not rich enough to describe an object found in nature or output of a certain scientific experiment. The objects of nature generally reveal one or more structures embedded in to another. Similarly, the outcomes of several scientific experiments exhibit randomness and variation at various stages. Therefore, more than one IFSs are needed to model such objects. Barnsley [22–24] introduced the class of super fractal sets constructed by using multiple IFSs to simulate such objects. Massopust [25] constructed super fractal functions and V-variable fractal functions by joining pieces of fractal functions which are attractors of finite family of IFSs. However, for a data set arising from nature or a scientific experiment, a solution of fractal interpolation problem based on several IFSs has not been investigated so far. To fill this gap, the notion of Super Fractal Interpolation Function (SFIF) is introduced in the present work. The construction of SFIF requires the use of more than one IFS wherein, at each level of iteration, an IFS can be chosen from a pool of several IFSs. This approach is likely to ensure desired randomness and variability needed to facilitate better geometrical modeling of objects found in nature and results of certain scientific experiments. The construction of SFIF is followed in the present paper by the investigations of its smoothness, its integral and determination of conditions for existence of its derivatives. The organization of the present paper is as follows: In Section 2, for a given finite set of data, the method of construction of a Super Fractal Interpolation Function (SFIF) is developed. At each level of iteration, an IFS is chosen from a pool of IFS in our construction of SFIF. For a sample interpolation data, a computational model of SFIF, illustrating the construction method given in Section 2, is presented in Section 3. The fractal dimension and average fractal distance are computed for various SFIFs constructed in this section. Finally, in Section 4, it is found that for an SFIF passing through ∗ Corresponding

author.

E-mail address: [email protected] c Copyright⃝World Academic Press, World Academic Union IJNS.2015.02.15/848

G.P.Kapoor, S.A.Prasad: Super Fractal Interpolation Functions

21

a given interpolation data, its integral is also an SFIF, albeit for a different interpolation data. An expository description of smoothness of an SFIF and conditions for the existence of derivatives of an SFIF is also given in this section.

2

Construction of an SFIF

The notion of Super Fractal Interpolation Function (SFIF) is introduced in this section via its construction based on more than one IFS. A hyperbolic Iterated Function System (IFS), denoted by {X; ωn , n = 1, 2, . . . , N }, consists of a metric space X together with a finite set of contraction mappings ωn : X → X with contractivity factors sn , 0 ≤ sn < 1, satisfying d(ωn (x), ωn (y)) ≤ sn d(x, y), n = 1, 2, . . . , N with respect to metric d on X. The contractivity factor of the IFS is defined as s = max{sn : n = 1, 2, . . . , N }. It is known [26] that for a hyperbolic IFS {X; ωn , n = 1, 2 . . . N }, the set ∪N valued Hutchinson map W : H(X)→ H(X) defined by W (A) = n=1 ωn (A) is a contraction map with contractivity factor s. Thus, by the Banach fixed point theorem, there exists a G in H(X) such that W (G) = G which is called the attractor of IFS. For developing the notion of Super Fractal Interpolation Function (SFIF) via its construction, let x0 < x1 < . . . < xN and I = [x0 , xN ] ⊂ R. Consider the set S0 = {(xi , yi ) ∈ I × R : i = 0, 1, . . . , N } of the given interpolation data. For k = 1, . . . , M , M > 1 and n = 1, . . . , N , let the functions ωn,k : I × R → I × R be defined by ωn,k (x, y) = (Ln (x), Gn,k (x, y)) for all (x, y) ∈ I × R

(1)

where, the contractive homeomorphisms Ln : I → In = [xn−1 , xn ] are given by Ln (x) = an x + bn =

(xn − xn−1 )x + (xN xn−1 − x0 xn ) (xN − x0 )

(2)

and the functions Gn,k : I × R → R defined by Gn,k (x, y) = en,k x + γn,k y + fn,k

(3)

satisfy the join-up conditions Gn,k (x0 , y0 ) = yn−1

and Gn,k (xN , yN ) = yn .

(4)

Here, γn,k are free parameters chosen such that |γn,k | < 1 and γn,k ̸= γn,l for k ̸= l. By (4), it is observed that ωn,k are continuous functions. The Super Iterated Function System (SIFS) that is needed to construct SFIF corresponding to the set of given interpolation data S0 = {(xi , yi ) ∈ R2 : i = 0, 1 . . . , N } is now defined as the pool of IFS {{ } } R2 ; ωn,k : n = 1, . . . , N , k = 1, . . . , M (5) where, the functions ωn,k are given by (1). To introduce a SFIF associated with SIFS (5), let {Wk : H(R2 ) → H(R2 ), k = 1, . . . , M }, be a collection N ∪ of continuous functions defined by Wk (G) = ωn,k (G) where, ωn,k (G) := {ωn,k (x, y), (x, y) ∈ G}. Since, n=1 { } h(Wk (A), Wk (B)) ≤ max |γn,k | h(A, B), where h is Hausdorff metric on H(R2 ), H(R2 ); W1 , . . . , WM is a 1≤n≤N

hyperbolic IFS. Hence, by Banach fixed point theorem, there exists an attractor A ∈ H(H(R2 )). Let Λ be the code space on M natural numbers 1, 2, . . . , M . For σ = σ1 σ2 . . . σk . . . ∈ Λ, define the function ϕ : Λ → H(R2 ) by ϕ(σ) = lim Wσk ◦ Wσk−1 ◦ . . . ◦ Wσ1 (G), G ∈ H(R2 ), k→∞

(6)

where the limit is taken with respect to the Hausdorff metric. It is shown that [27] ϕ(σ) exists, belongs to A and is independent of G ∈ H(R2 ). Also, the function ϕ is onto and continuous [27]. In the construction of SFIF, for a σ = σ1 σ2 . . . ∈ Λ, let the action of SIFS (5) at the iteration level j be defined by Sj = Wσj (Sj−1 ), where S0 is the set of given interpolation data. It is easily seen that the set, Gσ ≡ ϕ(σ) =

lim Wσk ◦ . . . ◦ Wσ1 (S0 ) = lim Sk ,

k→∞

k→∞

(7)

where the limit is taken with respect to the Hausdorff metric, is the attractor of SIFS (5) for a fixed σ ∈ Λ. The following theorem shows that Gσ is the graph of a continuous function gσ .

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International Journal of Nonlinear Science, Vol.19(2015), No.1, pp.20-29

Theorem 1 Let Gσ be the attractor of SIFS (5) for σ = σ1 σ2 . . . σk . . . ∈ Λ. Then, Gσ is graph of a continuous function gσ : I → R such that gσ (xn ) = yn for all n = 0, . . . , N . Proof. Let g0 be a function whose graph is S0 . Then, the set Sk , k ≥ 1,(is graph of the function gσk , where gσk (x) = ( ( ) −1 −1 −1 Gik ,σk L−1 ik (x), gσk−1 (Lik (x)) . It is easily seen that gσk (x) = Gik ,σk Lik (x), Gik−1 ,σk−1 ., . . . Gi1 ,σ1 (Li1 ◦ . . . ◦ )) −1 −1 −1 Lik (x), g0 (Li1 ◦ . . . ◦ Lik (x))) . . . . Therefore, it follows by (7) that the set Gσ is graph of the function gσ = lim gσk .

k→∞

For proving the continuity of the function gσ , consider τ1∗ τ2∗ . . . τj∗ . . . ∈ Λ where τj∗ ̸= 1 for some j ∈ N and τi∗ = 1 for i ∈ N and i ̸= j. We first show that Gτ ∗ is graph of a continuous function gτ ∗ . If not, then Gτ ∗ = ϕ(τ ∗ ) is graph of a function gτ ∗ that is not continuous so that there exist a δ1 > 0 such that whenever u1 , u2 ∈ I and |u1 − u2 | < δ1 , |gτ ∗ (u1 ) − gτ ∗ (u2 )| > ϵ.

(8)

It is known that [1], for τ = ¯ 1 ∈ Λ, Gτ = ϕ(τ ), with ϕ defined by (6), is graph of a continuous function gτ : I → R such that gτ (xn ) = yn , n = 0, 1, . . . , N . Consequently, there exists a δ2 > 0 such that |u1 − u2 | < δ2 implies |gτ (u1 ) − gτ (u2 )| < 3ϵ . Also, since ϕ is a continuous map, there exists δ3 > 0 such that, for τ and τ ∗ satisfying |τj −τ ∗ |

dc (τ, τ ∗ ) = (M +1)j j < δ3 , the Hausdorff distance between Gτ = ϕ(τ ) and Gτ ∗ = ϕ(τ ∗ ) is less than 3ϵ which implies that max |gτ (u) − gτ ∗ (u)| < 3ϵ . Thus, for δ = min(δ1 , δ2 , δ3 ) and u1 , u2 satisfying |u1 − u2 | < δ, |gτ ∗ (u1 ) − gτ ∗ (u2 )| ≤ u∈I

|gτ ∗ (u1 )−gτ (u1 )|+|gτ (u1 )−gτ (u2 )|+|gτ (u2 )−gτ ∗ (u2 )| < ϵ, a contradiction to (8). Hence, Gτ ∗ is graph of continuous function gτ ∗ . Now, consider the sequence σn = σ1,n σ2,n . . . .... with σj,n = σj for j ≤ n and σj,n = 1 for j > n. It is easily seen that as n tends to infinity, σn tends to σ with respect to the metric dc . Using the arguments of previous paragraph inductively, it follows that Gσn = ϕ(σn ) is graph of a continuous function gσn defined on I. Let Gσ = ϕ(σ) be graph of a function gσ . By continuity of ϕ, Gσn tends to Gσ with respect to Hausdorff metric h as n → ∞, which implies that gσn tends to gσ with respect to Maximum metric as n → ∞. Hence, there exist an ϵ > 0 such that max |gσn (u) − gσ (u)| < 3ϵ . u∈I

Since gσn is continuous on I, there exits a δ > 0 such that |u1 − u2 | < δ implies |gσn (u1 ) − gσn (u2 )| < 3ϵ . Therefore, it is easily seen that |gσ (u1 ) − gσ (u2 )| ≤ |gσ (u1 ) − gσn (u1 )| + |gσn (u1 ) − gσn (u2 )| + |gσn (u2 ) − gσ (u2 )| < ϵ for |u1 − u2 | < δ which implies that the function gσ is continuous on I. This establishes that the attractor Gσ of SIFS (5) is the graph of continuous function gσ . Theorem 1 is instrumental in defining a SFIF associated with SIFS (5) as follows: Definition 1 The Super Fractal Interpolation Function (SFIF) for the given interpolation data {(xi , yi ) : i = 0, 1, . . . , N } is defined as the continuous function gσ whose graph Gσ is the attractor of SIFS (5). Remark 2 Consider the family of continuous functions G = {f : I → R such that f is continuous, f (x0 ) = y0 and f (xN ) = yN } with metric dG (f, g) = max |f (x) − g(x)|. Since G is a complete metric space, it is easily seen that, for a fixed x∈I

σ ∈ Λ, Read-Bajraktarevi´ c operator T : Λ × G → G defined as { ( ( ( −1 −1 T (σ, g)(x) = lim Gik ,σk L−1 ik (x), Gik−1 ,σk−1 Lik−1 ◦ Lik (x), Gik−2 ,σk−2 ., . . . k→∞ )} )) −1 −1 −1 −1 , Gi1 ,σ1 (Li1 ◦ . . . ◦ Lik (x), g(Li1 ◦ . . . ◦ Lik (x))) . . .

(9)

is a contraction map on G and so it has a unique fixed point in G. It is observed that, SFIF gσ satisfies gσ = T (σ, gσ ).

3

Computational Model of SFIF

Our method of construction developed in Section 2 is employed in the present section for generating various SFIFs{ for a sample interpolation data S0 }= {(0, 0), (30, 90), (60, 70), (100, 10)}. For identifying the corresponding SIFS { 2 } R ; ωn,k : n = 1, 2, 3 , k = 1, 2 , the maps ωn,k , k = 1, 2 (c.f. (1)) are obtained by computing (c.f. Table 1) the values of ai , bi (c.f. (2)) and ei,1 , fi,1 ; ei,2 and fi,2 (c.f. (4)) with γi,1 = 0.4 and γi,2 = 0.6 for i = 1, 2, 3.

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23

Table 1: Computed Values of ai , bi , ei,1 , fi,1 , ei,2 , fi,2 , i = 1, 2, 3, for sample data S0 i=1 0.3 0 0.86 0 0.84 0

a b ei,1 fi,1 ei,2 fi,2

i=2 0.3 30 -0.24 90 -0.26 90

i=3 0.4 60 -0.64 70 -0.66 70

In the construction of SFIF for a σ = σ1 σ2 . . . ∈ Λ, the set Sj = Wσj (Sj−1 ), j = 1, 2, . . . , representing the action of SIFS (5) at the iteration level j is computed. The SFIF gσ(b) for σ(b) = ¯1 (c.f. Figs. 1(a)- 1(c), blue curve) is constructed by the action of IFS {R2 ; ωn,1 , n = 1, . . . , N } at every level of iteration. Similarly, SFIF gσ(g) for σ(g) = ¯2 (c.f. Figs. 1(a)1(c), green curve) is constructed by the action of IFS {R2 ; ωn,2 , n = 1, . . . , N } at every level of iteration. The SFIF gσ(r) for σ(r) = 112 (c.f. Fig. 1(a), red curve) is constructed by the action of IFS {R2 ; ωn,1 , n = 1, . . . , N } at j th level of iteration if j is not divisible by 3 and by the action of IFS {R2 ; ωn,2 , n = 1, . . . , N } if j is divisible by 3. Likewise, SFIF gσ(r) for σ(r) = 221 (c.f. Fig. 1(b), red curve) is constructed by the action of IFS {R2 ; ωn,1 , n = 1, . . . , N } at j th level of iteration if j is divisible 3 and otherwise by the action of IFS {R2 ; ωn,2 , n = 1, . . . , N }. Finally, SFIF gσ(r) for σ(r) = 12 (c.f. Fig. 1(c), red curve) is constructed by the action of IFS {R2 ; ωn,1 , n = 1, . . . , N } at j th level of iteration if j is not divisible by 2 and by the action of IFS {R2 ; ωn,2 , n = 1, . . . , N } if j is divisible by 2.

200

200

200

180

180

180

160

160

160

140

140

140

120

120

120

100

100

100

80

80

80

60

60

60

40

40

40

20

20

0

0

10

20

30

40

50

60

70

(a) σ(r) = 112

80

90

100

0

20

0

10

20

30

40

50

60

70

80

90

100

0

0

10

20

(b) σ(r) = 221

30

40

50

60

70

80

90

100

(c) σ(r) = 12

Blue Curve (—): SF IF gσ(b) for σ(b) = ¯ 1 in(a), (b) and (c), Green Curve (—): SF IF gσ(g) for σ(g) = ¯ 2 in(a), (b) and (c), Red Curve (—): SF IF gσ(r) for σ(r) = 112 in(a), for σ(r) = 221 in (b) and for σ(r) = 12 in (c)

Figure 1: SFIFs for σ(b) = ¯1, σ(g) = ¯2 and different choices of σ(r) ¯ and gσ(g) for σ(g) = 2¯ are in fact FIFs (c.f. Figs. 1(a)- 1(c), blue and green curves ), since The SFIFs gσ(b) for σ(b) = 1 these are constructed with a single element of SIFS (5). Heuristically, in terms of their fractal dimension [1], the graphs of SFIF gσ(r) appear to fill more space in R2 than the graph of FIF gσ(b) and less space in R2 than the graph of FIF gσ(g) . In fact, the fractal dimension of graphs of FIF gσ(b) and gσ(g) are computed as 1.3069 and 1.5199 respectively whereas the fractal dimension of SFIF gσ(r) with σ(r) = 112 (c.f. Fig. 1(a), red curve) is 1.3632, the fractal dimension of SFIF gσ(r) with σ(r) = 221 (c.f. Fig. 1(b), red curve) is 1.4572 and the fractal dimension of SFIF gσ(r) with σ(r) = 12 (c.f. Fig. 1(c), red curve) is 1.4182. ( )1/2 ∫b 1 2 |f (x) − g(x)| dx Further, for FIFs gσ(b) and gσ(g) , the average fractal distance defined as dF (f, g) = (b−a) a

for the functions f and g, continuous on a closed interval [a, b], is dF (gσ(b) , gσ(g) ) = 0.297. It is observed that (i) for

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International Journal of Nonlinear Science, Vol.19(2015), No.1, pp.20-29

SFIF gσ(r) with σ(r) = 112, dF (gσ(b) , gσ(r) ) = 0.022 while dF (gσ(g) , gσ(r) ) = 0.276. So, if the data generating function is at one third average fractal distance from FIF gσ(b) , then SFIF gσ(r) is a better approximation of the data generating function, since gσ(r) is closer to gσ(b) than gσ(g) (c.f. Fig. 1(a)) i.e. dF (gσ(b) , gσ(r) ) < dF (gσ(g) , gσ(r) ). (ii) For SFIF gσ(r) with σ(r) = 221, dF (gσ(b) , gσ(r) ) = 0.228 while dF (gσ(g) , gσ(r) ) = 0.071. So, if the data generating function is at one third average fractal distance from FIF gσ(g) , then SFIF gσ(r) is a better approximation of such data generating function, since gσ(r) is closer to gσ(g) than gσ(b) (c.f. Fig. 1(b)) and (iii) for gσ(r) with σ(r) = 12, dF (gσ(b) , gσ(r) ) = 0.138 and dF (gσ(g) , gσ(r) ) = 0.162. So, if the data generating function lies in the middle of FIFs gσ(b) and gσ(g) , then SFIF gσ(r) (c.f. Fig. 1(c)) is a better approximation of such data generating function.

4

Integral and Derivative of SFIF

In this section, for an SFIF passing through a given interpolation data, its integral is shown to be also an SFIF, albeit for a different interpolation data. Further, in this section, the smoothness of SFIF is investigated in terms of its Lipschitz exponent and it is found that, in general, an SFIF may not be differentiable. This, as a natural follow up, led to determining in this section the conditions for existence of derivatives of an SFIF. In order to study the integral of an SFIF, an SIFS {{ } } R2 ; ωn,k (x, y) = (Ln (x), Gn,k (x, y)) : n = 1, . . . , N , k = 1, . . . , M , (10) associated with the data {(xi , yi ) ∈ R2 : i = 0, . . . , N } is considered, where Ln (x) = an x + bn are given by (2) and the functions Gn,k (x, y) defined by Gn,k (x, y) = γn,k y + qn,k (x),

n = 1, . . . , N.

(11)

satisfy the join up conditions given by (4). Here, γn,k are free parameters chosen such that |γn,k | < 1 and γn,k ̸= γn,l for k ̸= l and qn,k (x) are continuous functions. Condition (4) ensures that there exits a unique attractor Gσ ∈ H(R2 ) of SIFS (10). By the arguments similar to those in the proof of Theorem 1, Gσ is graph of a continuous function gσ . The following notations [8] are needed in the sequel for tidy presentation of our results:  γˆn,k = an γn,k   [ x∫N ]  N ∑   aj qj,k (t) dt   j=1 x0   yˆN,k = yˆ0 +  N ∑   aj γj,k 1− j=1 (12) x∫N n [ ]  ∑  yˆn,k = yˆ0 + aj γj,k (ˆ yN,k − yˆ0 ) + qj,k (t) dt     j=1 x0   x ∫    qˆn,k (x) = yˆn−1,k − an γn,k yˆ0 + an qn,k (t) dt  x0

where, yˆ0 is an arbitrary real number. To determine an interpolation data through which the integral of SFIF passes, let the functions qn,k (x) in (11) satisfy : N ∑ j=1

1−

x∫N

aj

x0 N ∑ j=1

N ∑

qj,k

aj γj,k

=

j=1

1−

x∫N

aj

qj,l

x0 N ∑

̸= 1 for k ̸= l, k, l = 1, . . . , M.

(13)

aj γj,l

j=1

For example, for aj = N1 , γj,k = γk and qj,k = (1 − γk )(ej x + fj ) for j = 1, . . . , N , the condition (13) is satisfied. Then, yˆi,k = yˆi,l = yˆi for i = 0, . . . , N ; k, l = 1, . . . , M and yˆN − yˆ0 ̸= 1. The SIFS associated with the data {(xi , yˆi ) ∈ R2 : i = 0, . . . N } is now defined as the pool of IFS {{ } } ˆ n,k (x, y)) : n = 1, . . . , N , k = 1, . . . , M R2 ; ω ˆ n,k (x, y) = (Ln (x), G (14) where, the functions ˆ n,k (x, y) = γˆn,k y + qˆn,k (x) G

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(15)

G.P.Kapoor, S.A.Prasad: Super Fractal Interpolation Functions

25

ˆ n,k (x0 , yˆ0 ) = yˆn−1 and G ˆ n,k (xN , yˆN ) = yˆn . These join-up conditions ensure that there satisfy the join-up conditions G 2 ˆ exits a unique attractor Gσ ∈ H(R ) of SIFS (14). The following theorem shows that the integral of SFIF is also an SFIF albeit for interpolation data {(xi , yˆi ) ∈ R2 : i = 0, 1, . . . , N }. Theorem 3 For the interpolation data {(xi , yi ) ∈ R2 : i = 0, . . . N }, let gσ be SFIF corresponding to SIFS (10) for σ ∈ Λ. Then, the integral ∫x gˆσ (x) = yˆ0 +

gσ (t) dt

(16)

x0

is SFIF associated with SIFS (14) for the interpolation data {(xi , yˆi ) : i = 0, . . . , N }. Proof. Using (16) and (9), it is observed that,  gˆσ (Lik ◦ . . . ◦ Li1 (x)) = gˆσ (Lik ◦ . . . ◦ Li1 (x0 )) + 

k ∏



( ) aij γij ,σj  gˆσ (x) − yˆ0

j=1

+

k ∑



k ∏



p=1



Lip−1 ◦...◦Li1 (x)

∫

aij γij ,σj  aip

qip ,σp (t).

j=p+1

Lip−1 ◦...◦Li1 (x0 )

k ( ∏ k ∑

){ i∑ p −1

(17)

Also, by (16), gˆσ (Lik ◦ . . . ◦ Li1 (x0 )) = yˆ0 +

p=1

∫xN +

aij γij ,σj

j=p+1

[ al γl,σp (ˆ yN − yˆ0 )

l=1

] ql,σp (t) dt + aip

x0

Lip−1 ◦...◦Li1 (x0 )

∫

} qip ,σp (t) .

x0

The above identity and (12) give gˆσ (Lik ◦ . . . ◦ Li1 (x0 )) =

(∏ k

) aij γij ,σj yˆ0

j=1

+

k ( ∏ k ∑ p=1

Lip−1 ◦...◦Li1 (x0 )

){

∫

yˆip −1 − aip γip ,σp yˆ0 + aip

aij γij ,σj

j=p+1

} qip ,σp (t) .

(18)

x0

Now, substituting the value of gˆσ (Lik ◦ . . . ◦ Li1 (x0 )) from (18) in (17), it follows that ( ) ( ( ) ) ˆ i ,σ Li ˆ ˆ ˆ gˆσ (Lik ◦ . . . ◦ Li1 (x)) = G ◦ . . . ◦ L (x), G ., . . . G L (x), G (x, g ˆ (x)) . . . . i1 ik−1 ,σk−1 i2 ,σ2 i1 i1 ,σ1 σ k k k−1 Thus, gˆσ is SFIF associated with SIFS (14). Remark 4 Suppose yˆN is given and (12) is defined as γˆn,k

= an γn,k

yˆ0,k

= yˆN −

N ∑

aj

[ x∫N

j=1

= yˆN −

qj,k (t) dt

x0

1−

yˆn,k

N ∑

N ∑

aj γj,k

j=1

x∫N [ ] aj γj,k (ˆ yN − yˆ0,k ) + qj,k (t) dt

j=n+1

qˆn,k (x)

            

]

= yˆn,k − an γn,k yˆN − an

x∫N

x0

qn,k (t) dt.

x

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           

26

International Journal of Nonlinear Science, Vol.19(2015), No.1, pp.20-29

Then the integral of SFIF defined by gˆσ (x) = yˆN −

x∫N

gσ (t) dt is also an SFIF associated with SIFS (14) for the

x

interpolation data {(xi , yˆi ) : i = 0, 1, . . . , N }.

For investigating the smoothness of an SFIF, the following notations and definitions are needed: λ = min{λn,k : n = 1, 2, . . . , N, k = 1, 2, . . . M }, where λn,k are real numbers satisfying 0 < λn,k ≤ 1 C1 = max{

|γn,k | : n = 1, 2, . . . , N, k = 1, 2, . . . , M }, where γn,k are real numbers |In |λ satisfying |γn,k | ≤ 1.

(19)

The modulus of continuity of SFIF gσ (x) is defined as ω(gσ , t) = max max |gσ (x + h) − gσ (x)| |h|≤t

x

and a function f : R → R is said to have Lipschitz exponent δ if |f (x) − f (¯ x)| ≤ K|x − x ¯|δ , where K is any positive constant and 0 < δ ≤ 1. The smoothness of an SFIF in terms of its Lipschitz exponent is given by the following theorem: Theorem 5 Let gσ be a SFIF corresponding to SIFS (10) with qn,k ∈ Lip λn,k , 0 < λn,k ≤ 1. Then, (i) for C1 < 1, gσ ∈ Lip λ (ii) for C1 = 1, ω(gσ , t) = O(|t|λ log |t|) ¯ (iii) for C1 > 1, gσ ∈ Lip λ, ( ) log γn,k ¯ ≤ max where, λ and C1 , λ are given by (19). log an n=1,...,N k=1,...,M

Proof. The method of proof is similar to that in [28], wherein γn is replaced by γn,σn . In general, an SFIF belonging to certain Lipschitz class, need not be differentiable. This, as a natural follow up, leads to identification of conditions for the existence of derivative of a SFIF in the following proposition: Proposition 6 For the interpolation data {(xi , yi ) ∈ R2 : i = 0, 1, . . . N }, let gσ be an SFIF corresponding to SIFS (10) for σ ∈ Λ. Then, gˆσ′ exists and gˆσ′ (x) = gσ (x) if and only if gˆσ is an SFIF associated with SIFS (14) for the interpolation d data {(xi , yˆi ) : i = 0, 1, . . . , N }, where γˆj,k = aj γj,k and dx (ˆ qj,k (x)) = aj qj,k (x) hold. Proof. If gˆσ′ (x) = gσ (x), then gˆσ (x) = yˆ0 +

∫x

gσ (t) dt, so that “ if ” part follows from Theorem 3. Conversely, suppose

x0

gˆσ is an SFIF associated with SIFS (14) for the interpolation data {(xi , yˆi ) : i = 0, . . . , N }. Then,     k k k ∏ ∑ ∏  gˆσ (Lik ◦ . . . ◦ Li1 (x)) =  γˆij ,σj  gˆσ (x) + γˆij ,σj  qˆip ,σp (Lip−1 ◦ . . . ◦ Li1 (x)). p=1

j=1

Since,

d qj,k (x)) dx (ˆ

(20)

j=p+1

= aj qj,k (x) ∫x

∫x qj,k (t)dt = yˆj−1 − aj γj,k yˆ0 + aj

qˆj,k (x) = qˆj,k (x0 ) + aj x0

qj,k (t)dt.

(21)

x0

Substituting (21) and γˆj,k = aj γj,k in (20), gˆσ (Lik ◦ . . . ◦ Li1 (x)) =

k (∏ j=1

k ( ∏ k ) ) ∑ aij γij ,σj gˆσ (x) + aij γij ,σj × p=1

[ × yˆip −1 − aip γip ,σp yˆ0 + aip

j=p+1 Lip−1 ◦...◦Li1 (x)

∫

x0

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] qip ,σp (t)dt .

(22)

G.P.Kapoor, S.A.Prasad: Super Fractal Interpolation Functions

27

For a fixed σ ∈ Λ, it is easily seen that the Read-Bajraktarevi´ c operator Tˆ defined by {( ∏ k k ( ∏ k ) ) ∑ −1 −1 ˆ T (σ, g)(x) = lim aij γij ,σj g(Li1 ◦ . . . ◦ Lik (x)) + aij γij ,σj × k→∞

p=1

j=1

j=p+1

−1 L−1 ip ◦...◦Lik (x)

]} qip ,σp (t)dt

∫

[

× yˆip −1 − aip γip ,σp yˆ0 + aip

(23)

x0

is a contraction map on Gˆ = {f : I → R such that f is continuous, f (x0 ) = yˆ0 and f (xN ) = yˆN }. By (22), the ∫x function gˆσ is a fixed point of Tˆ. Also, by Theorem 3, the function h(x) = yˆ0 + gσ (t) dt is a SFIF associated with x0

SIFS (14) satisfying (22). Consequently, h also is a fixed point of Tˆ. Hence, by uniqueness of fixed point of Read∫x Bajraktarevic operator Tˆ, gˆσ (x) = yˆ0 + gσ (t) dt which implies that gˆσ′ exists and gˆσ′ (x) = gσ (x), since gσ being a x0

SFIF corresponding to SIFS (10), is a continuous function. For the investigation of nth derivative of SFIF, denote Gi,k,j (x, y) = γi,k,j y + qi,k,j (x)

(24)

where, Gi,k,0 (x, y) = Gi,k (x, y), qi,k,0 (x) = qi,k (x), γi,k,0 = γi,k and Gi−1,k,j (xN , yN,k,j ) = Gi,k,j (x0 , y0,k,j ), i = 1, . . . , N, k = 1, . . . , M and j = 0, 1, . . . , n. To determine interpolation data through which derivatives of SFIF passes, let the functions qi,k,j (x) in (24) satisfy: N ∑ p=1

1−

x∫N

ap

x0 N ∑

N ∑

qp,k,j =

ap γp,k,j

p=1

1−

p=1

x∫N

ap

qp,l,j

x0 N ∑

̸= 1,

(25)

ap γp,l,j

p=1

where yˆ0,j , j = 0, 1, . . . , n are arbitrary real numbers. For example, for ai = N1 , γi,k,j = γk,j and qi,k,j = (1−γk )¯ qi,j (x), where q¯i,j (x) are polynomials of degree n − j for i = 1, . . . , N , the condition (25) is satisfied. Then, yˆi,k,j = yˆi,l,j = yˆi,j for i = 1, . . . , N , k, l = 1, . . . , M and j = 0, 1, . . . , n. The SIFS associated with the interpolation data {(xi , yi,j ) : i = 0, 1, . . . , N }, j = 0, 1, . . . , n, is now defined as {{ } } R2 ; ωi,k,j (x, y) = (Li (x), Gi,k,j (x, y)) : i = 1, . . . , N , k = 1, . . . , M . (26) It is observed that SIFS (26) reduces to SIFS (10) if j = 0. The following theorem gives the existence of derivatives of a SFIF. Theorem 7 Let the functions Gi,k,j (x, y) defined in (24) be such that, for some integer n ≥ 0, |γi,k | < ani , qi,k ∈ C n [x0 , xN ], i = 1, 2, . . . , N, k = 1, 2, . . . , M and gσ be an SFIF corresponding to SIFS (26) for j = 0 and σ ∈ Λ. (j) Then, for j = 1, 2, . . . , n, gσ exists and is an SFIF associated with SIFS (26) for the interpolation data {(xi , yi,j ) : i = 0, 1, . . . , N }, where γi,k,j =

γi,k aji

(1)

and qi,k,j (x) =

qi,k,j−1 (x) . ai

Proof. The equation G1,k,j (x0 , y0,j ) = y0,j gives y0,j = y0,j =

(j) q1,k (x0 ) (aj1 −γ1,k )

γ1,k y aj1 0,j

(j)

+

q1,k (x0 ) aj1

which implies

(j) qN,k (xN ) (ajN −γN,k )

. Similarly, GN,k,j (xN , yN,j ) = yN,j gives yN,j = . By Proposition 6, it now follows {{ (j) that, for j = 1, 2, . . . , n, gσ (x) is the SFIF associated with SIFS R2 ; ωi,k,j (x, y) = (Li (x), Gi,k,j (x, y)) : i = } } 1, 2, . . . , N , k = 1, 2, . . . , M .

5

Conclusions

In the present work, the notion of Super Fractal Interpolation Function (SFIF) is introduced for finer simulation of the objects of the nature or outcomes of scientific experiments that reveal one or more structures embedded in to another.

IJNS homepage: http://www.nonlinearscience.org.uk/

28

International Journal of Nonlinear Science, Vol.19(2015), No.1, pp.20-29

Since, in the construction of SFIF, at each level of iteration, an IFSs can be chosen from a pool of several IFS, the desired randomness and variability can be implemented in fractal interpolation of the given data. Thus, SFIF may be used as a tool for better geometrical modeling of objects found in nature and results of certain scientific experiments. Also, a description of investigations on the integral, the smoothness and determination of conditions for existence of derivatives of an SFIF is given in the present work. It is proved that, for an SFIF passing through a given interpolation data, its integral is also an SFIF, albeit for a different interpolation data. The smoothness of an SFIF is given in terms of its Lipschitz exponent. An SFIF gσ , for C1 ̸= 1, belongs to a Lipschitz class and, for C1 = 1, ω(gσ , t) = O(|t|λ log |t|). It is seen that the smoothness of SFIF depend on free variables γn,k as well as on the smoothness of functions qn,k (x) occurring in its definition. Further, sufficient conditions for existence of derivatives of an SFIF are derived in the present paper. Our results on SFIF found here are likely to have wide applications in areas like pattern-forming alloy solidification in chemistry, blood vessel patterns in biology, signal processing, fragmentation of thin plates in engineering, stock markets in finance, wherein significant randomness and variability is observed in simulation of various processes.

Acknowledgement The authors are thankful to the referee for a constructive evaluation of the paper and his/her valuable suggestions. The second author thanks CSIR for Research Grant No: 9/92(417)/2005-EMR-I for the present work.

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