Convergence of Cubic Spline Super Fractal Interpolation Functions

0.15 MB·English

by  G. P. Kapoor

Convergence of Cubic Spline Super Fractal
Interpolation Functions
2
1
0
2
G.P. Kapoor1 and Srijanani Anurag Prasad2

n
a
J Department of Mathematics and Statistics

9
1 Indian Institute of Technology Kanpur

] Kanpur 208016 India
S
D
[email protected] [email protected]
.
h
t
a Abstract
m
[
Inthepresentwork,thenotionofCubicSplineSuperFractalInterpolationFunction

1 (SFIF)isintroducedtosimulateanobjectthatdepictsonestructureembeddedintoan-
v
otheranditsapproximationpropertiesareinvestigated. Itisshownthat,foranequidis-
7
9 tant partition points of [x0,xN], the interpolating Cubic Spline SFIF gσ(x) ≡ gσ(0)(x)
9 (j)
and their derivatives g (x) converge respectively to the data generating function
3 σ
. y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2−j+ǫ(0 < ǫ < 1),j = 0,1,2,
1
as the norm h of the partition of [x ,x ] approaches zero. The convergence results
0 0 N
2 for Cubic Spline SFIF found here show that any desired accuracy can be achieved in
1
the approximation of a regular data generating function and its derivatives by a Cubic
:
v Spline SFIF and its corresponding derivatives.
i
X
r
a Key Words: Fractal Interpolation Function, Spline, Super Fractals, Convergence
Mathematics Subject Classification: 28A80,41A05
1
1 Introduction
Barnsley [1] introduced Fractal Interpolation Function (FIF) using the theory of Iterated
Function System (IFS). Later, Barnsley [2, 3, 4] introduced the class of super fractal sets
constructed byusing multipleIFSstosimulatesuch objects. Massopust [5]constructed super
fractal functions and V-variable fractal functions by joining pieces of fractal functions which
are attractor of finite family of IFss.
FIF, constructed as attractor of a single Iterated Function System (IFS) by virtue of self-
similarity 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
modelsuchobjects. Asolutionoffractalinterpolationproblembasedonseveral IFStomodel
such objects is introduced in [6] by introducing the notion of Super Fractal Interpolation
Function (SFIF). The construction of SFIF use more than one IFS wherein, at each level
of iteration, an IFS is chosen from a pool of several IFS. This approach ensured desired
randomness and variability needed to facilitate better geometrical modeling of objects found
in nature and results of certain scientific experiments.
Spline functions, introduced by Schoenberg [7], find vast applications in areas like data
fitting [8], computer aided geometric design [9, 10], numerical solutions of differential equa-
tions [11], etc. A piecewise polynomial function ϑ on an interval [x ,x ], which is composed
0 N
of subintervals [x ,x ], i = 1,2,...,N, is called a Spline of order n if (i) ϑ(x) is a continu-
i−1 i
ous polynomial of degree atmost n−1 in each subintervals [x ,x ], i = 1,2,...,N, and (ii)
i−1 i
the derivatives ϑ(m), 0 ≤ m ≤ n−2, are continuous on [x ,x ]. A Cubic Spline is a Spline of
0 N
degree 3. For a data set {x } of n+1 points, a Cubic Spline is constructed with n piecewise
i
cubic polynomials between the data points. If ϑ represents a Cubic Spline approximating
the function y ∈ C4[x ,x ], then ϑ is twice continuously differentiable and ϑ(x ) = y(x ).
0 N i i
2
Navascues and Sebastian [12] considered a Cubic Spline FIF as a generalization of classical
Spline and obtained estimates on error in approximation of the data generating function by
a Cubic Spline FIF. However, their Cubic Spline FIF was constructed using a single IFS and
so it is not equipped enough to simulate an object that depicts one structure embedded into
another. To approximate an object by a spline-like FIF, the concept of Cubic Spline SFIF
is introduced in the present work and its approximation properties are investigated. The
convergence results for Cubic Spline SFIF found here show that any desired accuracy can
be achieved in the approximation of a regular data generating function and its derivatives
by a Cubic Spline SFIF and its corresponding derivatives.
The organization of the present chapter is as follows: In Section 2, a brief review on the
construction of Super Fractal Interpolation Function for a given finite set of data is given.
The notion of Cubic Spline SFIF is introduced in Section 3. It is proved in this section
that, for an equidistant partition points of [x ,x ], the interpolating Cubic Spline SFIF
0 N
g (x) ≡ g(0)(x) and their derivatives g(j)(x) converge respectively to the data generating
σ σ σ
function y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2−j+ǫ(0 < ǫ < 1),j = 0,1,2,
as the norm h of the partition of [x ,x ] approaches zero.
0 N
2 Construction of SFIF
In this section, a brief introduction on the construction of Super Fractal Interpolation Func-
tion (SFIF) is given.
Let S = {(x ,y ) ∈ R2 : i = 0,1,...N} be the set of given interpolation data. The
0 i i
contractive homeomorphisms L : I → I for n = 1,...,N, are defined by
n n
L (x) = a x+b (2.1)
n n n
where, a = xn−xn−1 andb = xNxn−1−x0xn. Fork = 1,2,...,M, M > 1andn = 1,2,...,N,
n xN−x0 n xN−x0
3
let the functions G : I ×R → R defined by
n,k
G (x,y) = γ y +q (x) (2.2)
n,k n,k n,k
satisfy the join-up conditions
G (x ,y ) = y and G (x ,y ) = y . (2.3)
n,k 0 0 n−1 n,k N N n
Here, γ are free parameters chosen such that |γ | < 1 and γ 6= γ for k 6= l.
n,k n,k n,k n,l
The Super Iterated Function System (SIFS) that is needed to construct SFIF corresponding
to the set of given interpolation data S is defined as the pool of IFS
0
R2;ω : n = 1,2,...,N , k = 1,2,...,M (2.4)
n,k
n o
(cid:8) (cid:9)
where, the functions ω : I ×R → I ×R are given by
n,k
ω (x,y) = (L (x),G (x,y)) for all (x,y) ∈ R2 (2.5)
n,k n n,k
By (2.3), it is observed that ω are continuous functions.
n,k
TointroduceaSFIFassociatedwithSIFS(2.4),let{W : H(R2) → H(R2), k = 1,2,...,M},
k
be a collection of continuous functions defined by
N
W (G) = ω (G), where ω (G) = {ω (x,y) for all (x,y) ∈ G}. (2.6)
k n,k n,k n,k
n=1
[
Then, {H(R2); W ,...,W }isahyperbolicIFS,sinceh(W (A),W (B)) ≤ max γ h(A,B),
1 M k k n,k
1≤n≤N
where h is Hausdorff metric on H(R2). Hence, by Banach fixed point theorem, there exists
an attractor A ∈ H(H(R2)).
4
Let Λ be the code space on M natural numbers 1,2,...,M . In the construction of SFIF,
for a σ = σ σ ... ∈ Λ, let the action of SIFS (2.4) at the iteration level j be defined by
1 2
S = W (S ), where S is the set of given interpolation data. For a fixed σ ∈ Λ, define
j σj j−1 0
G ≡ lim W ◦...◦W (S ) = lim S . (2.7)
σ k→∞ σk σ1 0 k→∞ k
The following proposition is instrumental for precise definition of a SFIF:
Proposition 2.1 Let G be defined by (2.7). Then, G is the attractor of SIFS (2.4) for
σ σ
σ = σ σ ...σ ... ∈ Λ and is graph of a continuous function g : I → R such that g (x ) =
1 2 k σ σ n
y for all n = 0,1,...,N.
n
Super Fractal Interpolation Function (SFIF) is defined using Proposition 2.1 as follows:
Definition 2.1 The Super Fractal Interpolation Function (SFIF) for the given
interpolation data {(x ,y ) : i = 0,1,...,N} is defined as the function g whose graph G is
i i σ σ
the attractor of SIFS (2.4).
3 Cubic Spline SFIF
The convergence of Cubic Spline SFIF is investigated here using the conditions of differ-
entiability found in [6]. Throughout in this section, the interpolation data {(x ,y ) : n =
n n
0,1,...,N} is assumed to be such that x − x = h, for n = 1,2,...,N and x = 0.
n n−1 0
Also, throughout in the sequel, the SIFS (2.4) is chosen such that γ = γ , n = 1,2,...,N,
n,k k
for some γ , 0 < γ < 1, k ∈ {1,2,...,M}.
k k
Definition 3.1 A SFIF g , associated with SIFS (2.4), is called Cubic Spline SFIF if
σ
q , given in (2.2), are cubic polynomials for n = 1,2,...,N and k = 1,2,...,M.
n,k
5
It is observed that, if q (x) = q x3 +q x2 +q x+q , the coefficients q , i =
n,k n,k,3 n,k,2 n,k,1 n,k,0 n,k,i
0,1,2,3,dependuponγ dueto(2.3), necessitating inthesequel, theuseofnotationq (γ ,x)
k n k
in place of q (x). Throughout in this section, it is assumed that, for some A ≥ 0, the
n,k 0
polynomials q satisfy
n
|q (γ ,x)−q (γ ,x)|
n k n l
≤ A , (3.1)
|γ −γ | 0
k l
for n = 1,2,...,N, k,l = 1,2,...,M and x ∈ [x ,x ].
0 N
Let γ be such that |γ | < 1 , β = max |γ −γ | < 1 and ς = ς ς ...ς ... ∈ Λ be such
k0 k0 k0 l k0 1 2 j
1≤l≤M
that ς = k for all j ∈ N. Consider the family of continuous functions
j 0
G = {f : I → R such that f is continuous,f(x ) = y and f(x ) = y } (3.2)
0 0 N N
with metric d (f,g) = max|f(x)−g(x)|. For Read-Bajraktarevic operator T : Λ×G → G
G
x∈I
defined by
T(σ,g)(x) = lim G L−1(x),G L−1 ◦L−1(x),G .,...
k→∞ ik,σk ik ik−1,σk−1 ik−1 ik ik−2,σk−2
(cid:26) (cid:18)
(cid:16)
(cid:0)
G (L−1 ◦...◦L−1(x),g(L−1 ◦...◦L−1(x)))... (3.3)
i1,σ1 i1 ik i1 ik
(cid:19)(cid:27)
(cid:17)
(cid:1)
where, Λ is the code space on M natural numbers 1,2,...,M and G is given by (3.2), the
following proposition gives a bound on kT(σ,g)(x)−T(ς,g)(x)k for σ,ς ∈ Λ:
Proposition 3.1 Let g ∈ G and inequality (3.1) be satisfied. Then,
A B
kT(σ,g)−T(ς,g)k ≤ β kgk + 0 + 0 (3.4)
∞ k0 ∞ 1−γ 1−β
(cid:26) ∗ k0(cid:27)
whereσ,ς ∈ Λ, max |q (γ ,x)| = B , γ = max |γ | < 1 andβ = max |γ −γ | < 1.
n k0 0 ∗ l k0 l k0
x∈[x0,xN] 1≤l≤M 1≤l≤M
n=1,2,...,N
6
Proof By the definition of T(σ,g)(x) (c.f. (3.3)),
|T(σ,g)(x)−T(ς,g)(x)|
k
≤ lim |γ −γ | |g L−1 ◦...◦L−1(x) |
k→∞( σj k0 ! i1 ik
j=1
Y (cid:0) (cid:1)
k k
+ |γ | q γ ,L−1 ◦...◦L−1(x) −q γ ,L−1 ◦...◦L−1(x)
σj ip σp ip ik ip k0 ip ik
!
Xp=1 j=Yp+1 (cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
k−1 k (cid:12) (cid:12)
+ |γ −γ | |q γ ,L−1 ◦...◦L−1(x) | . (3.5)
σj k0 ip k0 ip ik
! )
p=1 j=p+1
X Y (cid:0) (cid:1)
Since q (γ ,x) are cubic polynomials defined on compact set [x ,x ], there exists a B > 0
n k0 0 N 0
such that max |q (γ ,x)| = B . Therefore, by (3.1) and (3.5), it follows that
n k0 0
x∈[x0,xN]
n=1,2,...,N
k
|T(σ,g)(x)−T(ς,g)(x)| ≤ lim |γ −γ | kgk
k→∞( σj k0 ! ∞
j=1
Y
k k k−1 k−1
+ |γ | |γ −γ | A + |γ −γ | B .
σj σp k0 0 σj k0 0
! ! )
p=1 j=p+1 p=1 j=p+1
X Y X Y
Since β = max |γ −γ | < 1 and γ = max |γ | < 1, (3.4) follows from the above
k0 l k0 ∗ l
1≤l≤M 1≤l≤M
inequality.
The following proposition gives a bound on kg −g k, σ,ς ∈ Λ:
σ ς
Proposition 3.2 Let g ,g ∈ G and (3.1) be satisfied. Then,
σ ς
β A B
kg −g k ≤ k0 kg k + 0 + 0 (3.6)
σ ς ∞ (1−γ ) ς ∞ 1−γ 1−β
∗ (cid:18) ∗ k0(cid:19)
where γ ,β and B are as in Proposition 3.1.
∗ k0 0
7
Proof Since (3.1) is satisfied, g = T(σ,g ) and g = T(ς,g ) for Read-Bajraktarevic oper-
σ σ ς ς
ator T, defined by (3.3),
A B
kg −g k ≤ γ kg −g k +β kg k + 0 + 0 .
σ ς ∞ ∗ σ ς ∞ k0 ς ∞ 1−γ 1−β
(cid:18) ∗ k0(cid:19)
The inequality (3.6) now follows from the above inequality.
Remark 3.1 For γ = 0, γ = β . Thus, inequality (3.6) for γ = 0 implies
k0 ∗ k0 k0
γ A + B
kg −g k ≤ ∗ kg k + 0 0 .
σ ς ∞ 1−γ ς ∞ 1−γ
∗ (cid:18) ∗ (cid:19)
By Hall and Meyer’s theorem [13], kg k ≤ K h4 +J . Consequently, Proposition 3.2, for
ς ∞ 0 0
γ = 0 gives,
k0
γ A +B
kg −g k ≤ ∗ K h4 +J + 0 0 . (3.7)
σ ς ∞ 1−γ 0 0 1−γ
∗(cid:18) ∗ (cid:19)
Using inequality (3.7), the order of approximation of data generating function y(x) by SFIF
g is given by the following theorem:
σ
Theorem 3.1 Let y(x) ∈ C4[x ,x ] be a data generating function and g ∈ G be a SFIF
0 N σ
associated with SIFS (2.4) such that γ (h) = max |γ | ≤ h2+s , for some s, 0 < s < 1,
∗ i=1,2,...,N i |I|2+s
where h = |x −x |, i = 1,2,...,N and |I| = |x −x |. Then, for 0 < ǫ < s,
i i−1 N 0
ky −g k = o(h2+ǫ). (3.8)
σ ∞
provided (3.1) holds.
8
Proof Since (3.1) holds, an application of inequality (3.7) gives,
ky −g k ≤ ky −g k+kg −g k
σ ∞ ς ς σ ∞
γ (h) A +B
≤ K h4 + ∗ K h4 +J + 0 0
0 0 0
1−γ (h) 1−γ (h)
∗ (cid:18) ∗ (cid:19)
1 γ (h)(A + B )
≤ K h4 +γ (h)J + ∗ 0 0 . (3.9)
1−γ (h) 0 ∗ 0 1−γ (h)
∗ (cid:18) ∗ (cid:19)
Using |γ (h)| ≤ h2+s, the inequality (3.9) implies
∗ T2+s
|I|(2+s) J h(2+s) (A +B ) h(2+s)
ky −g k ≤ K h4 + 0 + 0 0 . (3.10)
σ ∞ |I|(2+s) −h(2+s) 0 |I|(2+s) |I|(2+s) −h(2+s)
( )
(cid:18) (cid:19) (cid:18) (cid:19)
The order of approximation error given by (3.8) follows from the above inequality.
Remark 3.2 It follows from inequality (3.10) that, in fact, ky −g k = O(h2+s).
σ ∞
Remark 3.3 If M = 1 in SIFS (2.4), then g reduces to a FIF. The convergence result for
σ
a Cubic Spline FIF [12] follows as a particular case of Theorem 3.1.
The order in approximation of derivatives of data generating function by corresponding
derivatives of SFIF is now investigated. It is known [6] that g(1)(x) and g(2)(x) are SFIFs
σ σ
associated with SIFSs R2; ω (x,y) = (L (x),G (x,y)) : i = 1,2,...,N ,k =
i,k,j i i,k,j
n
1,2,...,M for j = 1,2(cid:8)respectively. Here, the functions G (x,y) and G (x,(cid:9)y) are
i,k,1 i,k,2
o
given by
G (x,y) = Nγ y +Nq(1)(γ ,x) and G (x,y) = N2γ y +N2q(2)(γ ,x).
i,k,1 k i k i,k,2 k i k
Let, for some A ≥ 0, the polynomials q satisfy
j n
|q(j)(γ ,x)−q(j)(γ ,x)|
n k n l
≤ A , j = 0,1,2, (3.11)
j
|γ −γ |
k l
for all n = 1,2,...,N, k,l = 1,2,...,M and x ∈ [x ,x ].
0 N
9
For j = 1,2, define the Read-Bajraktarevic operator T : Λ×G → G by
j
T (σ,g)(x) = lim G L−1(x),G L−1 ◦L−1(x),G .,...
j k→∞ ik,σk,j ik ik−1,σk−1,j ik−1 ik ik−2,σk−2,j
(cid:26) (cid:18)
(cid:16) (cid:16)
G (L−1 ◦...◦...L−1(x),g(L−1 ◦...◦...L−1(x))) (3.12)
i1,σ1,j i1 ik i1 ik
(cid:19)(cid:27)
(cid:17)(cid:17)
where, Λ is the code space on M natural numbers 1,2,...,M and G is given by (3.2). To find
the order of approximation of derivatives of data generating function y(x) by corresponding
derivatives of SFIF g , the bounds on kT (σ,g)(x) − T (ς,g)(x)k similar to (3.4) and the
σ j j
bounds on kg(j)−g(j)k,j = 1,2, similar to (3.6) are needed. Such a bound on kT (σ,g)(x)−
σ ς j
T (ς,g)(x)k for σ,ς ∈ Λ, is given by the following proposition:
j
Proposition 3.3 Let g ∈ G and inequality (3.11) be satisfied. Then, for j = 1,2,
A NjB
kT (σ,g)−T (ς,g)k ≤ Njβ kgk + j + j (3.13)
j j k0 ∞ 1−Njγ 1−Njβ
( ∗ k0)
where, σ,ς ∈ Λ, max |q(j)(γ ,x)| ≤ B , γ = max |γ | < 1
n k0 j ∗ l N2
x∈[x0,xN] 1≤l≤M
n=1,2,...,N
and β = max |γ −γ | < 1 .
k0 l k0 N2
1≤l≤M
Proof By (3.12),
|T (σ,g)(x)−T (ς,g)(x)|
j j
k
≤ lim Njk |γ −γ | |g L−1 ◦...◦L−1(x) |
k→∞ σn k0 ! i1 ik
(cid:26) n=1
Y (cid:0) (cid:1)
k k
+ Nj(k−m+1) |γ | ×
σn
!
m=1 n=m+1
X Y
×|q(j) γ ,L−1 ◦...◦L−1(x) −q(j) γ ,L−1 ◦...◦L−1(x) |
im σm im ik im k0 im ik
k−1 (cid:0) k (cid:1) (cid:0) (cid:1)
+ Nj(k−m+1) |γ −γ | |q(j) γ ,L−1 ◦...◦L−1(x) | . (3.14)
σn k0 im k0 im ik
!
m=1 n=m+1 (cid:27)
X Y (cid:0) (cid:1)
10