Journal of Basic & Applied Sciences, 2014, 10, 173-176 173
ISSN: 1814-8085 / E-ISSN: 1927-5129/14 © 2014 Lifescience Global
Fractal Functions of Discontinuous Approximation
M.A. Navascués
*
Dpto. de Matemática Aplicada, Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza, C/ María de
Luna, 3. 50018 Zaragoza, Spain
Abstract: A procedure for the definition of discontinuous real functions is developed, based on a fractal methodology.
For this purpose, a binary operation in the space of bounded functions on an interval is established. Two functions give
rise to a new one, called in the paper fractal convolution of the originals, whose graph is discontinuous and has a fractal
structure in general. The new function approximates one of the chosen pair and, under certain conditions, is continuous.
The convolution is used for the definition of discontinuous bases of the space of square integrable functions, whose
elements are as close to a classical orthonormal system as desired.
Keywords: Discontinuous Functions, Interpolation, Approximation, Functional Spaces, Fractals.
INTRODUCTION
Since the advent of the fractal theory (and its
predecessors) the emphasis has been on continuous
functions (from Weierstrass map to wavelets), being
the discontinuous case minimal. In fact there are only a
few examples of mathematical models with
discontinuities (Haar basis [1], for instance), deserving
a deeper and additional study. In this paper we develop
a procedure for the definition of discontinuous real
mappings, based on a fractal methodology.
We set a binary operation in the space of bounded
functions on a compact interval B ( I ) (although the
construction can be done in the space of measurable
essentially bounded mappings L
( I ) as well). Two
functions ( f and b ) give rise to a new one f b
(called in this paper fractal convolution of the originals),
whose graph has a fractal structure in general.
The convolution is defined as fixed point of a
nonlinear contractive operator T of B ( I ), where the
images Tg are piecewisely defined by means of a
partition of the interval and a scale function. The
convolution may interpolate one of the initial pair, if
additional conditions on the nodes of the partition are
imposed. With more stringent requirements, the result
f b is continuous.
In the last part of the text, the convolution is used
for the definition of discontinuous bases of the space of
square integrable functions L
2
( I ) , close to a classical
orthonormal system if desired.
*Address correspondence to this author at the Dpto. de Matemática Aplicada,
Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza, C/ María de
Luna, 3. 50018 Zaragoza, Spain; Tel: 34976761980;
Fax: 34976761886; E-mail: manavas@unizar.es
MSC: 58C05, 26A15, 26A18, 65D05, 28A80.
2. A FRACTAL CONVOLUTION
Let us consider the space B ( I ) of bounded
functions on a compact interval I = [ c, d ] and functions
f , b, B ( I ) such that
< 1, where
= sup{| (t ) |: t I }.
The mapping will be called scale function. Let
: a = t
0
< t
1
< ... < t
N
= b be a partition of the interval
I , and L
n
= a
n
t + b
n
contractive affinities such that
L
n
(t
0
) = t
n1
, L
n
(t
N
) = t
n
.
Let us consider fixed f , b, and with the
conditions prescribed and define the operator
T = T
f , b, ,
: B ( I ) B ( I ), defined as
Tg(t ) = f (t ) + (t )( g b)
°
L
n
1
(t ),
(1)
for t I
n
, n = 1,2,..., N . The intervals I
n
are defined as
I
n
= (t
n1
, t
n
] for n = 2, …, N , and I
1
= [t
0
, t
1
] . Let us see
that T is a contraction defined on the Banach space
( B ( I ),
) . This is proved considering that, if
g, g B ( I ) , and t I
n
,
| Tg(t ) T g (t )|=| (t ) || ( g g )
°
L
n
1
(t ) |,
and thus
Tg T g
g g
.
Consequently, T admits a unique and contracting
fixed point
f
.
Definition 2.1. The map
f = f b is the fractal
convolution of f and b with respect to the partition
and the scale function .