M. A. NAVASCU ´ES AND M. V. SEBASTI ´AN

*Received 12 December 2005; Revised 5 May 2006; Accepted 14 June 2006*

Fractal methodology provides a general frame for the understanding of real-world phe-
nomena. In particular, the classical methods of real-data interpolation can be generalized
by means of fractal techniques. In this paper, we describe a procedure for the construc-
tion of smooth fractal functions, with the help of Hermite osculatory polynomials. As a
consequence of the process, we generalize any smooth interpolant by means of a fam-
ily of fractal functions. In particular, the elements of the class can be defined so that the
smoothness of the original is preserved. Under some hypotheses, bounds of the interpo-
lation error for function and derivatives are obtained. A set of interpolating mappings
associated to a cubic spline is defined and the density of fractal cubic splines inᏴ^{2}[a,*b]*

is proven.

Copyright © 2006 M. A. Navascu´es and M. V. Sebasti´an. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited.

**1. Introduction**

Fractal interpolation techniques provide good deterministic representations of complex phenomena. Barnsley [2,3] and Hutchinson [8] were pioneers in the use of fractal func- tions to interpolate sets of data. Fractal interpolants can be defined for any continuous function defined on a real compact interval. This method constitutes an advance in the techniques of approximation, since all the classical methods of real-data interpolation can be generalized by means of fractal techniques (see, e.g., [5,10,12]).

Fractal interpolation functions are defined as fixed points of maps between spaces of functions using iterated function systems. The theorem of Barnsley and Harrington (see [4]) proves the existence of diﬀerentiable fractal interpolation functions. However, in some cases, it is diﬃcult to find an iterated funcion system satisfying the hypothe- ses of the theorem, mainly whenever some specific boundary conditions are required (see [4]). In this paper, we describe a very general way of constructing smooth fractal

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 78734, Pages1–20 DOI 10.1155/JIA/2006/78734

functions with the help of Hermite osculatory polynomials. The proposed method solves the problem with the help of a classical interpolant. The fractal solution is unique and the constructed interpolant preserves the prefixed boundary conditions. The procedure has a computational cost similar to that of the classical method.

As a consequence of the process, we generalize any smooth interpolant by means of a family of fractal functions. Each element of the class preserves the smoothness and the boundary conditions of the original. Under some hypotheses, bounds of the interpolation error for function and derivatives are obtained. Assuming some additional conditions on the scaling factors, the convergence is also preserved.

In the last section, a set of interpolating mappings associated to a cubic spline is de- fined, in the general frame of functions whose second derivative has an integrable square.

In particular, the density of fractal cubic splines inᏴ^{2}[a,b] is proven.

**2. Construction of smooth fractal interpolants**

**2.1. Fractal interpolation functions. Let***t*0*< t*1*<**···**< t**N* be real numbers, and *I**=*
[t0,t* _{N}*]

*⊂*Rthe closed interval that contains them. Let a set of data points

*{*(t

*,*

_{i}*x*

*)*

_{i}*∈*

*I*

*×*R:

*i*

*=*0, 1, 2,

*. . .,N*

*}*be given. Set

*I*

_{n}*=*[t

_{n}*1,t*

_{−}*] and let*

_{n}*L*

*:*

_{n}*I*

*→*

*I*

*,*

_{n}*n*

*∈ {*1, 2,. . .,N

*}*, be contractive homeomorphisms such that

*L*_{n}^{}*t*0

*=**t*_{n}* _{−}*1,

*L*

_{n}^{}

*t*

_{N}^{}

*=*

*t*

*, (2.1)*

_{n}*L*

*n*

*c*1

*−**L**n*

*c*2*≤**l*^{}*c*1*−**c*2 *∀**c*1,c2*∈**I* (2.2)
for some 0*≤**l <*1.

Let*−*1*< s**n**<*1,*n**=*1, 2,. . .,N, and*F**=**I**×*R, let*N*be continuous mappings, let*F**n*:
*F**→*Rbe given satisfying

*F*_{n}^{}*t*0,x0

*=**x*_{n}* _{−}*1,

*F*

_{n}^{}

*t*

*,x*

_{N}

_{N}^{}

*=*

*x*

*,*

_{n}*n*

*=*1, 2,. . .,N, (2.3)

*F*

*n*(t,x)

*−*

*F*

*n*(t,

*y)*

^{}

*≤*

*s*

*n*

*|*

*x*

*−*

*y*

*|*,

*t*

*∈*

*I,x,y*

*∈*R

*.*(2.4) Now define functions

*w** _{n}*(t,x)

_{=}^{}

*L*

*(t),*

_{n}*F*

*(t,x)*

_{n}^{}

_{∀}*n*

*1, 2,. . .,*

_{=}*N,*(2.5) and consider the following theorem.

Theorem 2.1 [2,3]. The iterated function system (IFS)*{**F,w**n*:*n**=*1, 2,. . .,N*}**defined*
*above admits a unique attractorG.Gis the graph of a continuous function* *f* :*I**→*R*which*
*obeys* *f*(t* _{i}*)

*=*

*x*

_{i}*fori*

*=*0, 1, 2,

*. . .*,N.

The previous function *f* is called a fractal interpolation function (FIF) corresponding
to*{*(L*n*(t),*F**n*(t,x))*}*^{N}_{n}* _{=}*1.

*f*:

*I*

*→*Ris the unique function satisfying the functional equa- tion

*f*^{}*L** _{n}*(t)

^{}

*=*

*F*

_{n}^{}

*t,f*(t)

^{},

*n*

*=*1, 2,. . .,N,

*t*

*∈*

*I,*(2.6)

or

*f*(t)*=**F**n*

*L*^{−}_{n}^{1}(t),*f* *◦**L*^{−}_{n}^{1}(t)^{}, *n**=*1, 2,. . .,N,*t**∈**I**n**=*
*t**n**−*1,t*n*

*.* (2.7)

LetᏲbe the set of continuous functions *f* : [t0,t*N*]*→*Rsuch that *f*(t0)*=**x*0,*f*(t*N*)*=*
*x**N*. Define a metric onᏲby

*d(f*,*g)**= **f* *−**g**∞**=*max^{}*f*(t)*−**g*(t)^{};*t**∈*
*t*0,t*N*

*∀**f*,g*∈*Ᏺ. (2.8)
Then (Ᏺ,*d) is a complete metric space.*

Define a mapping*T*:Ᏺ*→*Ᏺby

(T f)(t)*=**F*_{n}^{}*L*^{−}_{n}^{1}(t),*f* *◦**L*^{−}_{n}^{1}(t)^{} *∀**t**∈*

*t*_{n}* _{−}*1,

*t*

_{n}^{},

*n*

*=*1, 2,. . .,N. (2.9) Using (2.1)–(2.4), it can be proved that (T f)(t) is continuous on the interval [t

*n*

*−*1,t

*n*] for

*n*

*=*1, 2,. . .,N and at each of the points

*t*1,t2,

*. . .,t*

_{N}*1.*

_{−}*T*is a contractive mapping on the metric space (Ᏺ,d),

*T f* *−**Tg**∞**≤ |**s**|**∞**f* *−**g**∞*, (2.10)
where*|**s**|**∞**=*max*{|**s**n**|*; *n**=*1, 2,*. . .,N**}*. Since*|**s**|**∞**<*1,*T*possesses a unique fixed point
onᏲ, that is to say, there is *f* * _{∈}*Ᏺsuch that (T f)(t)

_{=}*f*(t) for all

*t*

*[t0,t*

_{∈}*]. This func- tion is the FIF corresponding to*

_{N}*w*

*.*

_{n}The most widely studied fractal interpolation functions so far are defined by the fol- lowing IFS:

*L**n*(t)*=**a**n**t*+*b**n*,

*F**n*(t,x)*=**s**n**x*+*q**n*(t) (2.11)

with

*a*_{n}*=*

*t*_{n}*−**t*_{n}* _{−}*1

*t*_{N}*−**t*0

, *b*_{n}*=*

*t*_{N}*t*_{n}* _{−}*1

*−*

*t*0

*t*

_{n}^{}

*t*

_{N}*−*

*t*0

*.* (2.12)

*s**n* is called the vertical scaling factor of the iterated function system and *s**=*(s1,
*s*2,. . .,s* _{N}*) is the scale vector of the transformation. If

*q*

*(t) are linear for*

_{n}*t*

*∈*[t0,t

*] then the FIF is called aﬃne (AFIF) (see [2,11]). The cubic FIF (see [10,13]) is constructed using*

_{N}*q*

*n*(t) as a cubic polynomial.

In many cases, the data are evenly sampled, then
*h**=**t**n**−**t**n**−*1,

*t**N**−**t*0*=**Nh.* (2.13)

In the particular case,*s**n**=*0 for all*n**=*1, 2,. . .,*N, then*

*F**n*(t,x)*=**q**n*(t) (2.14)

and *f*(t)*=**q**n**◦**L*^{−}_{n}^{1}(t) for all*t**∈**I**n*.

**2.2. Diﬀerentiable fractal interpolation functions. In this section, we study the con-**
struction of smooth fractal interpolation functions. The theorem of Barnsley and Har-
rington [4] proves the existence of diﬀerentiable FIFs and gives the conditions for their
existence. We look for IFS satisfying the hypotheses of this theorem.

Theorem 2.2 (Barnsley and Harrington [4]). Let*t*0*< t*1*< t*2*<**···**< t**N* *andL**n*(t),*n**=*
1, 2,. . .,N, the aﬃne function*L**n*(t)*=**a**n**t*+*b**n**satisfying the expressions (2.1)-(2.2). Leta**n**=*
*L*^{}* _{n}*(t)

*=*(t

_{n}*−*

*t*

_{n}*1)/(t*

_{−}

_{N}*−*

*t*0

*) andF*

*(t,x)*

_{n}*=*

*s*

_{n}*x*+

*q*

*(t),*

_{n}*n*

*=*1, 2,

*. . .,N, verifying (2.3)-(2.4).*

*Suppose for some integerp**≥**0,**|**s**n**|**< a*^{p}*n**, andq**n**∈*Ꮿ* ^{p}*[t0,t

*N*

*];n*

*=*1, 2,. . .,

*N. Let*

*F*

*nk*(t,x)

*=*

*s*

*n*

*x*+

*q*

*n*

^{(k)}(t)

*a*^{k}* _{n}* ,

*k*

*=*1, 2,. . .,

*p,*(2.15)

*x*0,k

*=*

*q*

^{(k)}

_{1}

^{}

*t*0

*a** ^{k}*1

*−*

*s*1

, *x**N,k**=**q*^{(k)}_{N}^{}*t*_{N}^{}
*a*^{k}_{N}*−**s**N*

, *k**=*1, 2,. . .,*p.* (2.16)
*IfF*_{n}_{−}_{1,k}(t* _{N}*,x

*)*

_{N,k}*=*

*F*

*(t0,x*

_{nk}_{0,k}

*) withn*

*=*2, 3,. . .,N

*andk*

*=*1, 2,. . .,

*p, then*

*L**n*(t),*F**n*(t,x)^{}^{N}_{n}_{=}_{1} (2.17)

*determines a FIF* *f* *∈*Ꮿ* ^{p}*[t0,

*t*

*N*

*] and*

*f*

^{(k)}

*is the FIF determined by*

*L** _{n}*(t),

*F*

*(t,x)*

_{nk}^{}

^{N}

_{n}

_{=}_{1}(2.18)

*fork*

*=*1, 2,. . .,

*p.*

From here on, we consider a uniform partition in order to simplify the calculus. In this case,

*a**n**=* 1

*N.* (2.19)

If we consider a generic polynomial*q** _{n}*, for instance, the equality proposed in the the-
orem implies the resolution of systems of equations. Sometimes the system has no solu-
tion, mainly whenever some boundary conditions are imposed on the function (see [4]).

We will proceed in a diﬀerent way. In order to define an IFS satisfyingTheorem 2.2, we consider the following mappings:

*L**n*(t)*=**a**n**t*+*b**n*,

*F**n*(t,x)*=**s**n**x*+*q**n*(t), (2.20)

where

*q**n*(t)*=**g**◦**L**n*(t)*−**s**n**b(t),* (2.21)
*g*is a continuous function satisfying

*g*^{}*t*_{i}^{}*=**x** _{i}*,

*i*

*=*0, 1,. . .,N, (2.22)

and*b(t) is a real continuous function,b**=**g*, such that
*b*^{}*t*0

*=**x*0, *b*^{}*t**N*

*=**x**N**.* (2.23)

The IFS satisfies the hypotheses (2.1)–(2.5) of Barnsley’s theorem (see [11]). In [11], we proved some properties of this fractal function.

*Definition 2.3. Considerg**∈*Ꮿ(I) and a partition of the closed interval*I**=*[t0,t*N*],Δ:
*t*0*< t*1*<**···**< t**N*. Let*b*be defined as before and let*s**=*(s1,. . . s*N*) be the scaling vector of
the IFS defined by (2.11) and (2.21).

The corresponding FIF*g*_{Δb}* ^{s}* ,

*g*

_{b}*,*

^{s}*g*

_{Δ}

*or simply*

^{s}*g*

*is called*

^{s}*s-fractal function ofg*with respect to the partitionΔand the function

*b.*

Theorem 2.4 (see [11]). The*s-fractal functiong*_{b}^{s}*ofgwith respect to*Δ*andbsatisfies the*
*inequality*

*g*_{b}^{s}*−**g*^{ }_{∞}*≤* *|**s**|**∞*

1*− |**s**|**∞**g**−**b**∞*, (2.24)

*where**|**s**|**∞**=*max1_{≤}*n**≤**N**{|**s**n**|}**. Besides,g*_{b}^{s}*interpolates tog, that is to say,*

*g*_{b}^{s}^{}*t*_{n}^{}*=**g*^{}*t*_{n}^{} *∀**n**=*0, 1,. . .,*N.* (2.25)
*Consequence 2.5. Ifs**=*0, then*g*_{b}^{s}*=**g*.

*Remark 2.6. By (2.7), for allt**∈**I**n*,*n**=*1, 2,. . .,N,
*g*_{b}* ^{s}*(t)

*=*

*g(t) +s*

*n*

*g*_{b}^{s}*−**b*^{}*◦**L*^{−}_{n}^{1}(t). (2.26)
The first step is to check which conditions should satisfy*b(t) in order to fulfill the*
hypotheses of the theorem of Barnsley and Harrington.

Let us consider*p**≥*0,*|**s**n**|**<*1/N* ^{p}*, and

*q*

*n*(t)

*∈*Ꮿ

*[t0,t*

^{p}*N*],

*n*

*=*1, 2,. . .,N.

The prescribed conditions are
*F**n**−*1,k

*t**N*,*x**N,k*

*=**F**nk*
*t*0,x0,k

, (2.27)

where*n**=*2, 3,. . .,N,*k**=*1, 2,*. . .,p.*

We have from the assumptions (2.15) of the theorem,
*F**nk*(t,x)*=**s**n**x*+*q**n*^{(k)}(t)

*a*^{k}_{n}*.* (2.28)

In this particular case,

*q** _{n}*(t)

*=*

*g*

*◦*

*L*

*(t)*

_{n}*−*

*s*

_{n}*b(t)*(2.29) as

*L*

*n*(t)

*=*(1/N)t+

*b*

*n*and

*L*

^{}*(t)*

_{n}*=*1/N

*=*

*a*

*n*, we have for all

*k*

*=*0, 1,. . .,

*p,*

*q*_{n}^{(k)}(t)*=* 1

*N*^{k}*g*^{(k)}^{}*L**n*(t)^{}*−**s**n**b*^{(k)}(t) (2.30)

so that (2.27) becomes
*N*^{k}*s**n**−*1

*g*^{(k)}^{}*t*_{N}^{}*−**N*^{k}*s*_{N}*b*^{(k)}^{}*t*_{N}^{}

1*−**N*^{k}*s*_{N}^{−}*s**n**−*1*N*^{k}*b*^{(k)}^{}*t**N*

*=**N*^{k}*s*_{n}*g*^{(k)}^{}*t*0

*−**N*^{k}*s*1*b*^{(k)}^{}*t*0

1*−**N*^{k}*s*1 *−**s*_{n}*N*^{k}*b*^{(k)}^{}*t*0

*.*

(2.31)

If we consider constant scale factors*s**n**=**s*1for all*n**=*1,. . .,N,
*g*^{(k)}^{}*t*_{N}^{}*−**b*^{(k)}^{}*t*_{N}^{}*=**g*^{(k)}^{}*t*0

*−**b*^{(k)}^{}*t*0

*.* (2.32)

A suﬃcient condition in order to satisfy this equality is
*b*^{(k)}^{}*t*0

*=**g*^{(k)}^{}*t*0

,

*b*^{(k)}^{}*t*_{N}^{}*=**g*^{(k)}^{}*t*_{N}^{} (2.33)

for*k**=*0, 1, 2,. . .,*p. In this case, we look for a functionb*which agrees with*g* at the ex-
tremes of the interval until the*pth derivative.*

The conditions (2.33) will be satisfied if a Hermite interpolating polynomial*b*is con-
sidered, with nodes*t*0,*t**N*and*p*derivatives at the extremes. In this case, (see [14]),

*b(t)**=**H**g*(t)*=*
*p*
*k**=*0

*g*^{(k)}^{}*t*0

ᏸ0k(t) +
*p*
*k**=*0

*g*^{(k)}^{}*t**N*

ᏸ*Nk*(t). (2.34)

The functionsᏸ*ik*are defined by means of intermediate*l** _{ik}*, for

*i*

*=*0,

*N*and 0

*≤*

*k*

*≤*

*p,*

*l*0k(t)

*=*

*t**−**t*0

*k*

*k!*

*t**−**t**N*

*t*0*−**t*_{N}*p+1*

, *l**Nk*(t)*=*
*t**−**t**N*

*k*

*k!*

*t**−**t*0

*t*_{N}*−**t*0

*p+1*

(2.35) so that

ᏸ0p(t)*=**l*0p(t),

ᏸ*N p*(t)*=**l** _{N p}*(t), (2.36)

and for*k**=**p**−*1,*p**−*2,*. . ., 0,*

ᏸ0k(t)*=**l*0k(t)*−*
*p*
*ν**=**k+1*

*l*^{(ν)}_{0k}^{}*t*0
ᏸ0*ν*(t),

ᏸ*Nk*(t)*=**l**Nk*(t)*−*
*p*
*ν**=**k+1*

*l*^{(}_{Nk}^{ν}^{)}^{}*t**N*

ᏸ*Nν*(t).

(2.37)

The mappingsᏸ*ik*satisfy

ᏸ^{(σ)}_{ik}^{}*t*_{j}^{}*=*

⎧⎨

⎩1 if*i**=**j,k**=**σ*,

0 otherwise. (2.38)

The degree of*H**g*(t) is 2p+ 1. The function*g* is an interpolant of the data such that
*g**∈*Ꮿ* ^{p}*.

According to the theorem of Barnsley and Harrington, the IFS associated with the*kth*
derivative of a FIF is expressed by

*L**n*(t)*=* 1
*Nt*+*b**n*,

*F**nk*(t,x)*=**N*^{k}*s*1*x*+*N*^{k}*q*^{(k)}* _{n}* (t),

*k*

*=*0, 1, 2,

*. . .,p.*

(2.39) In our case,

*q**n*(t)*=**g**◦**L**n*(t)*−**s*1*b(t),* (2.40)
where*b(t) is a Hermite interpolating polynomial of degree 2p*+ 1 of*g*. The derivatives of
*q** _{n}*(t) become

*q*^{(k)}* _{n}* (t)

*=*1

*N*^{k}*g*^{(k)}^{}*L**n*(t)^{}*−**s*1*b*^{(k)}(t), *k**=*0, 1, 2,*. . .,p,* (2.41)
so that the IFS defining the*kth derivative ofg*_{b}* ^{s}*is (2.15),

*L**n*(t)*=* 1
*Nt*+*b**n*,

*F** _{nk}*(t,x)

*=*

*N*

^{k}*s*1

*x*+

*g*

^{(k)}

*◦*

*L*

*(t)*

_{n}*−*

*N*

^{k}*s*1

*b*

^{(k)}(t),

*k*

*=*0, 1, 2,. . .,

*p,*

(2.42)
that is to say, the map*q** _{nk}*corresponding to

*F*

*is*

_{nk}*q** _{nk}*(t)

*=*

*g*

^{(k)}

*◦*

*L*

*(t)*

_{n}*−*

*N*

^{k}*s*1

*b*

^{(k)}(t),

*k*

*=*0, 1, 2,. . .,

*p,*(2.43) so that the

*kth derivative of thes-fractal function ofg*with respect to

*s*and

*b,g*

_{b}*, agrees with the fractal function of*

^{s}*g*

^{(k)}with respect to the scaling vector

*N*

^{k}*s*and

*b*

^{(k)}(Definition 2.3):

*g*_{b}^{s}^{}^{(k)}*=*

*g*^{(k)}^{}^{N}* _{b}*(k)

^{k}*,*

^{s}*k*

*=*0, 1, 2,. . .,

*p.*(2.44) Proposition 2.7. (g

_{b}*)*

^{s}^{(k)}

*interpolates tog*

^{(k)}

*at the nodes of*Δ

*, for 0*

*≤*

*k*

*≤*

*p.*

*Proof. The ordinates of (g*_{b}* ^{s}*)

^{(k)}at the extremes of the interval are given in the theorem of Barnsley and Harrington. Applying (2.16), (2.33), and (2.41),

*g*_{b}^{s}^{}^{(k)}^{}*t*0

*=**x*0,k*=**q*^{(k)}1

*t*0

*a*^{k}_{1}*−**s*1

*=* 1
*a*^{k}_{1}*−**s*1

1

*N*^{k}*g*^{(k)}^{}*L*1
*t*0

*−**s*1*b*^{(k)}^{}*t*0

*=* 1
1*−**s*1*N*^{k}

*g*^{(k)}^{}*t*0

*−**s*1*N*^{k}*b*^{(k)}^{}*t*0

*=**g*^{(k)}^{}*t*0

*.*

(2.45)

In the same way,

*g*_{b}^{s}^{}^{(k)}^{}*t*_{N}^{}*=**g*^{(k)}^{}*t*_{N}^{}*.* (2.46)

Now, applying the fixed point equation (2.26) corresponding to*kth IFS att**n*,
*g*_{b}^{s}^{}^{(k)}^{}*t**n*

*=**F**nk*

*L*^{−}_{n}^{1}^{}*t**n*

,^{}*g*_{b}^{s}^{}^{(k)}*◦**L*^{−}_{n}^{1}^{}*t**n*

*=**N*^{k}*s*1

*g*_{b}^{s}^{}^{(k)}*◦**L*^{−}_{n}^{1}^{}*t*_{n}^{}+*g*^{(k)}^{}*t*_{n}^{}*−**N*^{k}*s*1*b*^{(k)}*◦**L*^{−}_{n}^{1}^{}*t*_{n}^{}*=**g*^{(k)}^{}*t*_{n}^{}

(2.47) since

*L*^{−}_{n}^{1}^{}*t**n*

*=**t**N*,
*g*_{b}^{s}^{}^{(k)}^{}*t**N*

*=**g*^{(k)}^{}*t**N*

*=**b*^{(k)}^{}*t**N*

*.* (2.48)

The properties of*g*_{b}* ^{s}*are as the following.

(i) (g_{b}* ^{s}*)

^{(k)}interpolates to

*g*

^{(k)}at the nodes of the partitionΔ, for 0

*≤*

*k*

*≤*

*p.*

(ii)*g*_{b}* ^{s}*may be close to

*g*(choosing suitably the scale vector according to (2.24)).

(iii)*g*_{b}* ^{s}*preserves the

*p-smoothness ofg*. (iv)

*g*

_{b}*preserves the boundary conditions of*

^{s}*g*.

(v) If*s**=*0,*g*_{b}^{s}*=**g, that is to say,g*is a particular case of*g*_{b}* ^{s}*.

*Note. Despite the similarity betweeng*_{b}* ^{s}* and

*g*, in general, they do not agree. In fact, if

*s*

*=*0 and

*b*

*=*

*g*, then

*g*

_{b}

^{s}*=*

*g*.

Let us assume that*g*_{b}^{s}*=**g. Ifs*1*=*0, applying (2.26) for*L**n*(t)*∈**I**n*,
*g**◦**L** _{n}*(t)

*=*

*g*

*◦*

*L*

*(t) +*

_{n}*s*1(g

*−*

*b)(t),*

*g*(t)*=**b(t)* (2.49)

for all*t**∈**I.*

**2.3. Uniform bounds. In order to bound the distance between***g* and*g*_{b}* ^{s}*, we consider a
theorem of Ciarlet et al. concerning Hermite interpolation.

Given a partitionΔ:*t*0*< t*1*<**···**< t**N*of an interval [t0,t*N*],*I**n**=*[t*n**−*1,t*n*] for 1*≤**n**≤*
*N, the Hermite function space (see [14])H*_{Δ}* ^{p+1}*(p

*∈*

*N*) is defined by

*H*_{Δ}^{p+1}*=*

*ϕ*:^{}*t*0,t*N*

*−→*R;*ϕ**∈*Ꮿ^{p}^{}*t*0,*t**N*

,*ϕ**|**I**n**∈*ᏼ2p+1

, (2.50)

whereᏼ2p+1is the space consisting of all polynomials of degree at most 2p+ 1.

Theorem 2.8 (Ciarlet et al. [6]). Let*g**∈*Ꮿ* ^{r}*[t0,t

_{N}*] withr*

*≥*2

*p+ 2, let*Δ

*be any partition*

*of [t*0,t

_{N}*], let*Δ:

*t*0

*< t*1

*<*

*···*

*< t*

_{N}*, and letϕ(t) be the unique interpolation ofg(t) inH*

_{Δ}

^{p+1}*,*

*that is,g*

^{(l)}(t

*n*)

*=*

*ϕ*

^{(l)}(t

*n*

*), for all 0*

*≤*

*n*

*≤*

*N, 0*

*≤*

*l*

*≤*

*p. Then*

*g*^{(k)}*−**ϕ*^{(k)}^{ }_{∞}*≤* Δ^{2p+2}^{−}^{k}

2^{2p+2}^{−}^{2k}*k!(2p*+ 2*−*2k)!^{ }*g*^{(2p+2)}^{ }* _{∞}* (2.51)

*for allk*

*=*0, 1,. . .,

*p+ 1.*

In the case in study, we consider a single subinterval of length*T**=**b**−**a. To bound*
the diﬀerence between the *kth derivative ofg* and the*kth derivative ofg*_{b}* ^{s}*, we can use

Theorem 2.4,

*g*_{b}^{s}^{}^{(k)}*−**g*^{(k)}^{ }_{∞}*=* *g*^{(k)}^{}^{N}* _{b}*(k)

^{k}

^{s}*−*

*g*

^{(k)}

^{ }

_{∞}*≤*

*N*

^{k}^{}

*s*1

1*−**N*^{k}^{}*s*1 *g*^{(k)}*−**b*^{(k)}^{ }_{∞}*.* (2.52)
Considering that*b(t)**=**ϕ(t) is the Hermite interpolating polynomial of degree 2p*+
1 of*g*, theorem of Ciarlet et al. can be used in order to bound*g*^{(k)}*−**b*^{(k)}*∞*, so that
applying (2.51), (2.52) and considering*g**∈*Ꮿ^{(2p+2)},

*g*_{b}^{s}^{}^{(k)}*−**g*^{(k)}^{ }_{∞}*≤* *N*^{k}^{}*s*1

1*−**N*^{k}^{}*s*1 *g*^{(k)}*−**b*^{(k)}^{ }_{∞}

*≤* *N*^{k}^{}*s*1

1*−**N*^{k}^{}*s*1 *T*^{2p+2}^{−}^{k}

2^{2p+2}^{−}^{2k}*k!(2p*+ 2*−*2k)!^{ }*g*^{(2p+2)}^{ }* _{∞}*,

*k*

*=*0, 1,. . .,

*p.*

(2.53)
**2.4. An operator of**Ꮿ* ^{p}*(I). From here on, we denote by

*g*

*the*

^{s}*s-fractal function ofg*

*∈*Ꮿ

*(I) with respect to a fixed partitionΔof the interval, a scaling vector*

^{p}*s*with constant coordinates

*s*

*n*

*=*

*s*1for all

*n*

*=*1, 2,. . . Nand

*b(t)*

*=*

*H*

*g*(t) defined in the preceding sections.

For fixedΔ, let us consider the operator ofᏯ* ^{p}*(I) which assigns

*g*

*to the function*

^{s}*g,*

Ᏸ^{s}*p*(g)*=**g*^{s}*.* (2.54)

Theorem 2.9. Ᏸ^{s}_{p}*is a linear, injective, and bounded operator of*Ꮿ* ^{p}*(I).

*Proof. The operator is linear as by (2.26) for allt**∈**I**n*,
*f** ^{s}*(t)

*=*

*f*(t) +

*s*

*n*

*f*^{s}*−**H**f*

*◦**L*^{−}_{n}^{1}(t),

*g** ^{s}*(t)

*=*

*g*(t) +

*s*

_{n}^{}

*g*

^{s}*−*

*H*

_{g}^{}

*◦*

*L*

^{−}

_{n}^{1}(t). (2.55) Multiplying the first equation by

*λ*and the second by

*μ*and considering that

*λH** _{f}*+

*μH*

_{g}*=*

*H*

_{λ f}_{+μg}, (2.56)

the function

*λ f** ^{s}*+

*μg*

*(2.57)*

^{s}satisfies the equation corresponding to

(λ f+*μg)*^{s}*.* (2.58)

By the uniqueness of the solution, the linearity is proved.

To prove the injectivity, let us consider that*g*^{s}*=*0. In this case, for all*t**∈**I**n*by (2.26),
0*=**g*(t)*−**s*1*H**g**◦**L*^{−}_{n}^{1}(t) (2.59)
but this equation is satisfied by*g*(t)*=*0 and due to the uniqueness of the solution*g**=*0.

We considerᏯ* ^{p}*(I) endowed with the norm

*f*Ꮿ

*(I)*

^{p}*=*

*p*

*k**=*0*f*^{(k)}*∞*. Using the defi-
nition of*H**g*(t) (2.34),

*H**g*^{(}* ^{j)}*(t)

^{ }

_{∞}*=*sup

^{}

*p*

*k*

*=*0

*g*^{(k)}^{}*t*0

ᏸ^{(j)}_{0k}(t) +*g*^{(k)}^{}*t**N*

ᏸ^{(}_{Nk}* ^{j)}*(t)

^{}

_{},

*H*

*g*

^{(j)}(t)

^{ }

_{∞}*≤*sup

*t**∈**I**g*Ꮿ* ^{p}*(I)

*p*
*k**=*0

ᏸ^{(j)}_{0k}(t)^{}+^{}ᏸ^{(j)}* _{Nk}*(t)

^{}

*.*

(2.60)

Let us consider

*λ*_{p}*=* sup

*t**∈**I*
0*≤**j**≤**p*

*p*
*k**=*0

ᏸ^{(j)}_{0k}(t)^{}+^{}ᏸ^{(}_{Nk}* ^{j)}*(t)

^{}

(2.61)

then

*H*_{g}^{ }_{Ꮿ}*p*(I)*≤ **g*Ꮿ* ^{p}*(I)

*λ*

*,*

_{p}*g*

*−*

*H*

*g*

Ꮿ* ^{p}*(I)

*≤*

*λ**p*+ 1^{}*g*Ꮿ* ^{p}*(I)

*.*(2.62) On the other hand, usingTheorem 2.4, (2.52), and

*N*^{j}^{}*s*1

1*−**N*^{j}^{}*s*1*≤* *N*^{p}^{}*s*1

1*−**N*^{p}^{}*s*1 (2.63)

for 0*≤**j**≤**p, one has*

*g*^{s}*−**g*^{ }_{Ꮿ}*p*(I)*≤* *N*^{p}^{}*s*1

1*−**N*^{p}^{}*s*1 *g**−**H**g*

Ꮿ* ^{p}*(I), (2.64)

by (2.64) and (2.62),

*g*^{s}^{ }_{Ꮿ}*p*(I)*− **g*Ꮿ* ^{p}*(I)

*≤*

*N*

^{p}^{}

*s*1 1

*−*

*N*

^{p}^{}

*s*1

*λ**p*+ 1^{}*g*Ꮿ* ^{p}*(I),

*g*

^{s}^{ }

_{Ꮿ}

*p*(I)

*≤*1 +

*λ*

*p*

*N*

^{p}^{}

*s*1

1*−**N*^{p}^{}*s*1 *g*Ꮿ* ^{p}*(I)

*.*

(2.65)

As a consequence,Ᏸ^{s}* _{p}*is bounded and

Ᏸ^{s}*p* *≤*1 +*λ**p**N*^{p}^{}*s*1

1*−**N*^{p}^{}*s*1 *.* (2.66)

**2.5. Convergence in**Ꮿ* ^{p}*(I). Let

*x*

*∈*Ꮿ

*(I) be an original function providing the data and let*

^{p}*g*

_{Δ}

_{N}*∈*Ꮿ

*(I) be an interpolant of*

^{p}*x*on the partitionΔ

*N*. We consider the fractal function

*g*

_{Δ}

^{s}

^{N}*of*

_{N}*g*

_{Δ}

*with respect to the partitionΔ*

_{N}*N*, the scale vector

*s*

*with constant coordinates, and the function*

^{N}*b*defined by the equality (2.34).

*Theorem 2.10. Ifx**∈*Ꮿ* ^{p}*(I) is an original function and

*g*

_{Δ}

_{N}*is ap-smooth interpolant ofx*

*with respect to the partition*Δ

*N*

*, consider a scaling vectors*

^{N}*with constant coordinates such*

*that*

*|*

*s*

^{N}_{1}

*|*

*<*1/N

^{p}*, then*

*x**−**g*_{Δ}^{s}^{N}_{N}^{ }_{Ꮿ}*p*(I)*≤* *x**−**g*_{Δ}_{N}^{ }_{Ꮿ}*p*(I)+ *N*^{p}^{}*s*^{N}_{1}^{}
1*−**N*^{p}^{}*s** ^{N}*1

*λ** _{p}*+ 1

^{ }

*g*

_{Δ}

_{N}^{ }

_{Ꮿ}

*p*(I)

*.*(2.67)

*Proof. One has*

*x**−**g*_{Δ}^{s}^{N}_{N}^{ }_{Ꮿ}*p*(I)*≤* *x**−**g*_{Δ}_{N}^{ }_{Ꮿ}*p*(I)+^{ }*g*_{Δ}_{N}*−**g*_{Δ}^{s}^{N}_{N}^{ }_{Ꮿ}*p*(I), (2.68)
by (2.64),

*x**−**g*_{Δ}^{s}^{N}_{N}^{ }_{Ꮿ}*p*(I)*≤* *x**−**g*_{Δ}_{N}^{ }_{Ꮿ}*p*(I)+ *N*^{p}^{}*s** ^{N}*1

1*−**N*^{p}^{}*s*^{N}_{1}^{}^{ }*g*_{Δ}_{N}*−**H**g*_{ΔN}

Ꮿ* ^{p}*(I)

*.*(2.69)

Using (2.62), the result is obtained.

*Consequence 2.11. If one considers a scaling vector such that* *|**s** ^{N}*1

*|*

*<*1/N

*, for*

^{p+r}*r >*0 fixed, the fractal interpolant

*g*

_{Δ}

^{s}

^{N}*converges inᏯ*

_{N}*(I) towards the original*

^{p}*x*if

*g*

_{Δ}

*does (as*

_{N}*N*tends to

*∞*).

*Note. The constantλ**p*does not depend onΔ*N*but only on the extremes of the interval.

**3. Fractal cubic splines**

In this section, we study the particular case*p**=*2, considering the following IFS:

*L**n*(t)*=**a**n**t*+*b**n*,

*F** _{n}*(t,x)

*=*

*s*

_{n}*x*+

*q*

*(t), (3.1)*

_{n}where*a*_{n}*=*1/N,*s*_{n}*=**s*1for all*n**=*1, 2,. . .,*N,*

*q** _{n}*(t)

*=*

*g*

*◦*

*L*

*(t)*

_{n}*−*

*s*1

*b(t),*(3.2)

where*g*is a cubic spline with respect to a uniform partitionΔ*N*(g*=**σ*_{Δ}* _{N}*) and

*b*

*=*

*H*

*is a Hermite interpolating polynomial satisfying the described conditions (2.33) with*

_{g}*p*

*=*2 (b(t) is a polynomial of degree 5).

We use a classical result of splines in order to findᏯ^{2}(I) bounds of the interpolation
error.

Theorem 3.1 (Hall and Meyer [7]). Let *f* *∈*Ꮿ^{4}[a,b] and*|**f*^{(4)}(t)*| ≤**Lfor allt**∈*[a,b].

*Let**f**∞**=*sup*|**f*(t)*|**whent**∈*[a,b]. LetΔ*N**= {**a**=**t*0*< t*1*<**···**< t**N**=**b**}**be a partition*
*of the interval [a,b], with constant distances between nodes* *h**=**t**n**−**t**n**−*1*. Letσ*_{Δ}_{N}*be the*
*spline function that interpolates the values of the function* *f* *at the pointst*0,*t*1,. . .,t_{N}*∈*Δ*N**,*
*beingσ*_{Δ}_{N}*type I or II. Then*

*f*^{(r)}*−**σ*_{Δ}^{(r)}_{N}^{ }_{∞}*≤**C**r**Lh*^{4}^{−}* ^{r}* (r

*=*0, 1, 2) (3.3)

*withC*0

*=*5/384,

*C*1

*=*1/24,

*C*2

*=*3/8. The constants

*C*0

*andC*1

*are optimum.*

*Remark 3.2. A spline is type I if its first derivatives ata*and*b*are known. A spline is type
II if it can be explicitly represented by its second derivatives at*a*and*b.*

*Theorem 3.3. Let* *x(t) be a function verifyingx(t)**∈*Ꮿ^{4}[t0,*t**N**] and**|**x*^{(4)}(t)*| ≤**Lfor all*
*t**∈*[t0,t_{N}*]. Letσ*_{Δ}* _{N}*(t) be a cubic spline (with respect to the uniform partitionΔ

*N*

*) and let*

*b(t) be a Hermite interpolating polynomial of degree 5 ofσ*

_{Δ}

_{N}*at the extremes of the interval.*

*Choose a scaling vector with constant coordinates such that**|**s** ^{N}*1

*|*

*<*1/N

^{2}

*, then*

*x*

*−*

*σ*

_{Δ}

^{s}

^{N}

_{N}^{ }

_{Ꮿ}2(I)

*≤*

*M*

*h*+

*N*

^{2}

^{}

*s*

*1*

^{N}1*−**N*^{2}^{}*s*^{N}_{1}^{}

*λ*2+ 1^{}*M**h*+*x*Ꮿ^{2}(I)

, (3.4)

*whereM**h**=**C*0*Lh*^{4}+*C*1*Lh*^{3}+*C*2*Lh*^{2}*,C*0*,C*1*,C*2*are the constants of Hall and Meyer theo-*
*rem,T**=**t**N**−**t*0*=**Nh, andλ*2*is defined in (2.61).*

*Proof. Applying*Theorem 2.10,

*x**−**σ*_{Δ}^{s}^{N}_{N}^{ }_{Ꮿ}2(I)*≤**M**h*+ *N*^{2}^{}*s** ^{N}*1
1

*−*

*N*

^{2}

^{}

*s*

^{N}_{1}

^{}

*λ*2+ 1^{ }*σ*_{Δ}_{N}^{ }_{Ꮿ}2(I)*.* (3.5)

Using now theorem of Hall and Meyer,

*σ*_{Δ}_{N}^{ }_{Ꮿ}2(I)*≤**M** _{h}*+

*x*Ꮿ

^{2}(I) (3.6)

and the result is obtained.

**3.1. Convergence in**Ᏼ^{2}[a,b]. In this subsection, we weaken the hypotheses about the
original function*x. The set*

Ᏼ^{2}[a,b]*=*

*x*: [a,*b]**−→*R;*x** ^{}*absolutely continuous;

*x*

^{}*∈*

*L*

^{2}(a,

*b)*

^{}(3.7) is a Hilbert space with respect to the inner product

*x,y** =*
2
*j**=*0

*x*^{(j)},*y*^{(j)}^{}* _{L}*2(a,b), (3.8)

where

*f*,g*L*^{2}(a,b)*=*
_{b}

*a* *f*(t)g(t)dt
1/2

(3.9) and the norm

*x*Ᏼ^{2}[a,b]*=*
_{2}

*j**=*0

*x*^{2}* _{j}*
1/2

, (3.10)

where

*x**j**=*
*b*

*a*

*x*^{(j)}(t)^{}^{2}*dt*
1/2

*=* *x*^{(}^{j)}^{ }* _{L}*2(a,b) for

*j*

*=*0, 1, 2. (3.11)