ETNA
Electronic Transactions on Numerical Analysis.
Volume 20, pp. 64-74, 2005.
Copyright  2005, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
FRACTAL TRIGONOMETRIC APPROXIMATION
M. A. NAVASCUES
Abstract. A general procedure to define nonsmooth fractal versions of classical trigonometric approximants
is proposed. The systems of trigonometric polynomials in the space of continuous and periodic functions are extended to bases of fractal analogues. As a consequence of the process, the density of trigonometric fractal
functions in is deduced. We generalize also some classical results (Dini-Lipschitz’s Theorem, for instance)
concerning the convergence of the Fourier series of a function of . Furthermore, a method for real data fitting
is proposed, by means of the construction of a fractal function proceeding from a classical approximant.
Key words. iterated function systems, fractal interpolation functions, trigonometric approximation
AMS subject classifications. 37M10, 58C05
1. Introduction. A classical approach to handle real experimental recordings consists
in their decomposition in signal content and noise. The first component is considered as
deterministic and the noise is studied from a statistical point of view. We give here a global
deterministic method to model both, signal and noise, by means of fractal interpolation. This
method was introduced by M. Barnsley and others ([1], [2], [3], [4], [10]) in the eighties and
provides good techniques for the construction of not necessarily smooth interpolants to real
data. The procedure is based on the theory of Iterated Function Systems and their associated
attractors ([2], [8]).
In former papers, we have proved that Barnsley’s method is a general theory which contains other interpolation techniques as particular cases (see for instance [12], [13]). Another
important fact is that the graph of these interpolants possesses a fractal dimension, and this
number can be used to measure the complexity of a signal, allowing an automatic comparison
of recordings, electroencephalographic for instance ([14]).
A general procedure to define nonsmooth fractal versions of classical trigonometric approximants is proposed. The fractal trigonometric polynomials defined here do not share in
general the properties of differentiability of classical trigonometric functions and they preserve some others like closeness to continuous periodic functions. The systems of trigonometric polynomials, in the space of continuous and periodic functions ,are extended
to bases of fractal analogues. As a consequence of the process, the density of trigonometric
fractal functions in is deduced. This result illustrates the fact that, fractal interpolation functions are everywhere in the space of continuous functions, in a metric sense. In the
reference [14], for instance, we have proved the density of affine fractal functions in .
We generalize also some classical theorems (Dini-Lipschitz’s Theorem, for instance)
concerning the convergence of the Fourier series of a function of . Furthermore, a
method for real data fitting is proposed, by means of the construction of a fractal function
proceeding from a classical approximant.
From an applied point of view, the trigonometric approximants display the spectral content of a signal, providing a representation in the frequency domain which allows its processing and filtering. On the other hand, Besicovitch and Ursell, in the reference [5], proved
that the graph of a smooth function has a fractal dimension of one. As a consequence, the
nonsmoothnes is a required condition in order to obtain an approximation of the geometrical
complexity of arbitrary signals. A conventional interpolant excludes the possibility of using
Received June 15, 2004. Accepted for publication January 24, 2005. Recommended by R. Varga.
Dept. Matemática Aplicada, Centro Politécnico Superior de Ingenieros, Universidad de Zaragoza, C/ Marı́a de
Luna 3, 50018 Zaragoza, Spain (manavas@unizar.es).
64
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FRACTAL TRIGONOMETRIC APPROXIMATION
this parameter for numerical characterizations of experimental signals ([16]).
2. -Fractal Functions. Let !#"%$&!'($*)+),)-$.!#/ be real numbers, and 0213 !#"45!#/6 be
the closed interval that contains them. Let a set of data points 789!:;5<=:?>@0BADCFEHG&1
I
KJLM),)+),NHO be given. Set 0M:P1Q !#:
R-'!#:8 and let ST:.E0VUW0X:YZG*>[74J4M),)+),NDO be
contractive homeomorphisms such that:
ST:Y\!]"^1_!#:`RY'^aST:Y9!#/b1@!#:
(2.1)
c
(2.2)
I jPk
Sd:Y9eK'KgfhSd:Y9eXi^
c`jlkgc
eK'dfDeXi
cnm
eK'eXi@>o0
for some
$pJ .
Let fqJr$lg:?$pJ ; GZ1[JL`K)+),)+N , st1.0uAv ewxL for some fzy{$Pez$Px|$@}qy
continuous mappings, sa:oELs~UC be given satisfying:
and N
sg:Y\!]"85<€"1@<=:
R-'^Zs:Y9!#/‚<=/q1@<=:;vGZ1~JL`M),),)+N
(2.3)
c
c
c
g:
< fh„ †
!‡>o0=†<„2>vC
m
Now define functions, ˆ : \!ƒ5<;T1~S : 9!5ƒs : 9!ƒ5<ƒ GB1[JLM),)+),N .
(2.4)
s:-9!ƒ5<;afhsg:-9!ƒ5„
c
j
T HEOREM 2.1. (Barnsley [1]) The Iterated Function System (IFS) 7‰sŠVˆ‹:ŒE
G&1JL`M),),)+NDO defined above admits a unique attractorI Ž . Ž is the graph of a continuous function ZE80(UC which obeys a9!5:T1_<=: , for GZ1
KJL`M),),)+N .
The previous function is called a Fractal Interpolation Function (FIF) corresponding to
78ST:;\!5Xsg:-\!ƒ<5ƒO /:wg' .
Let ‘ be the set of continuous functions ’Ez !5"85!#/6rUŒ exL such that a9!]"‰21“<=" ;
”a\!#/r61[<=/ . ‘ is a complete metric space respect to the uniform norm. Define a mapping
E8‘•U–‘ by:
m
”
(2.5)
YM\!51.s:YS : R-' \!5X|—‡S : R-' \!5
!‡>H !#:
R-'^!#:8oGZ1[JL`K)+),)+N
”
is a contraction mapping on the metric space ‘ , ˜‡™8˜^š% :
jtc c
”
”‹›
›
(2.6)
˜ …f
˜Kš
šB˜K|f ˜Kš
c c
c c Ÿ
c c
”
where
a unique fixed
šF1†œ|žo7 : Gl1¡JL`K)+),)+NDO . Since ” š–$¢J , possesses
m
point on ‘ , that is to say, there is t>~‘ such that YM\!5|1£a9!5
!2>* !#"45!#/6 . This
function is the FIF corresponding to ˆ‹: and it is the unique ¤>H‘ satisfying the functional
equation ([1]):
(2.7)
R ' \ !5X%—‡S Y
R '
a\!51_sg:-S : Y
: \ !5ƒ¥GB1~J4`K)+),),NZ2!d>v0X:…1[ !#:
R-'^5!#:8
The most widely studied fractal interpolation functions so far, are defined by the IFS
¦
(2.8)
Sd:Y\!5T1@8:
!}§ƒ:
sg:-\!ƒ<T1.g:
<¨}¤©K:Y\!5
: is called a vertical scaling factor of the transformation ˆ : and l1ª ' i K)M)K)X / is
the scale vector of the IFS. Following the equalities (2.1)
(2.9)
:|1
!#:(f•!#:`RY'
! / fh! "
ƒ:%1
!#/z!#:
RY'dfh!]"K!#:
! / f•! "
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M. A. NAVASCUES
Let n>?90
be a continuous function. We consider here ,the case
© : \!5T1.¨—‡S : \!5afD : \!5
(2.10)
«  .
where is continuous, such that w\! " T1_< " , 9! / T1_< / and ‚1.
This case is proposed by Barnsley, in the reference ([1]), as generalization of any continuous function. It is easy to check that the condition (2.3) is fulfilled. By this method one can
define fractal analogues of any continuous function (see Fig. 2.1).
4
2
0
-2
-4
0 0.2 0.4 0.6 0.8 1
4
2
0
-2
0 0.2 0.4 0.6 0.8 1
F IG . 2.1. The left figure represents the graph of the function ¬­®]¯r°†±K²³`´¶µ·¸‰¹^­¶ºX±ƒ®#¯ . The right
graph represents the corresponding » -fractal, with ¼ª½;¾o¿[ºXÀMÁo¿p‰ÀKÁo¿[ÃÃÃg¿[º , Ä a line in the
interval Å ¾LƃºÇ and »YÈr°¤¾wà ÂÊÉ`˨°lº‰ÆƒÃÃÃÆ5Á .
D EFINITION 2.2. Let YÌ be the continuous function defined by the IFS (2.8), (2.9) and
(2.10). YÌ is the -fractal function associated to  with respect to and the partition Í .
Following (2.7) and (2.10),  Ì verifies the fixed point equation:
m
R ' 9 !5
 Ì \!5d1@a\!5}¤ :  Ì fHX-—‡S : Y
(2.11)
!d>v0 :
YÌ interpolates to  at ! : as, using (2.1), (2.11) and Barnsley’s Theorem:
m
I
(2.12)
 Ì 9!#:€1&a\!#:}V:Y Ì fHX-—b\!#/rT1@a9!#:
Gv1 KJLM)K)M)XN
From (2.11) it is easy to deduce that:
˜M Ì fH˜Mš
jtc c
j~c c
šB˜K Ì fDw˜Kš
and
˜M Ì fH˜Mš
(2.13)
j
I
If D1 , then (2.11) Y̅1. .
Let Ï Ì be the operator of ‘ ; Ï Ì…1
JÎf
šZ˜K Ì fD˜Kšp}&˜M…fDw˜Kš|
c c
š
c c
˜M…fDL˜Mš
š
ÐÌ Ñ Ò :
Ï ‡
Ï Ð‡
Ì Ñ Ò EF‘

U
ÓU
‘
 Ì
Y
Ð Ì Ñ Ò depends on and Í but sometimes we will omit the subindices in order to simplify the
Ï ‡
notation.
P ROPOSITION 2.3. For fixed Í
(2.14)
and , Ï Ì satisfies the Lipschitz condition:
›
˜ Ï Ì Ygf Ï Ì K ˜ š
j
JÎf
J
c c
š
˜M?f
›
˜š
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FRACTAL TRIGONOMETRIC APPROXIMATION
m
! >Z0 :
Ì . By the equation (2.11), T
R-'
 Ì \!5T1.a\!5}Vg:Y Ì fHX-—‡S : 9!5
›
›
›
Ì \!5T1 9!5}¤g:Y Ì fHX-—‡S : R-' 9!5
›
›
Proof. Let Ï Ì-Y1@YÌ , Ï ÌY 1
and
›
 Ì 9!5af
then
ÔMÕ8ր׶ØLÙ¶Ú
c
›
 Ì \!5f
and
˜M Ì f
›
Ì \!5T1.a\!5af
›
Ì 9!5
j
Ì ˜ š
c`j
›
˜M…f
›
˜M|f
›
9!5}V:Y Ì f
c c
˜ š }
c c
˜ š }
R '
Ì —‡S : Y
\ !5
š ˜M Ì f
š ˜K Ì f
›
›
Ì ˜ š
Ì ˜ š
from which the result is deduced.
T HEOREM 2.4. Ï Ì…1 Ï Ð‡
Ì Ñ Ò is a continuous operator of ‘ .
Proof. It is an inmediate consequence of the former
c proposition.
c
P ROPOSITION 2.5. Let gÛÜ>nC / be such that gÛ šÝ$~J and gÛ[U
infinity. Then ÏÎÌ4ß YUQ uniformly as Þ tends to infinity.
Proof. By the inequality (2.13):
˜ Ï Ì ß YfD˜ š
I
as Þ
tends to
c
c
gÛ š
c
c ˜M…fDw˜ š
J f Û š
Î
j
from which the result is deduced.
To construct non-smooth interpolating functions one can proceed in the following way.
Let  be a classical (smooth) interpolant of the data. Choose a m nowhere differentiable function
(for instance, a Weierstrass function ([9])) and g: non-null G . As  is smooth, -Ì can not
be differentiable in every point because if it were, for any !d>v0 , SŠ:Y\!5‡>Z0X: and the equation
(2.11) can be written as
J
w\!51& Ì \!5}
:
?fD Ì —‡ST:;9!5
As a consequence, would be differentiable at ! (see Fig. 2.2).
×
3. Fractal Linear Operator. If we choose ‹1t|—‡e where e is continuous, increasing
I
and such that ew9!]"T1@!]" and ew9!#/rT1_!#/ (for instance, eL\!5T1’9àwá fl‰J â9àáqflJ‰ for ãBä
I
in the interval KJX ), then the operator of 08 which assigns Ì ( -fractal of  respect to
¨—‡e and Í ) to the function 
å
å
Ì 1
m
is linear as, by (2.11) !T>v0X: :
R ' \ !5
 Ì \!51&a\!5}V :  Ì fH¨—‡eM-—‡S : Y
Ї
Ì Ñæ
›
›
Ì \!5T1
›
›
R ' 9 !5
9!5-}¤ : Ì f
—‡eM—‡S : Y
Multiplying the first equation by ã and the second by ç , the uniqueness of the solution of the
fixed point equation defining the FIF gives:
ã|}Hç
›
Ì 1&ã= Ì }Hç
›
Ì
m
ã-çn>ZC
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M. A. NAVASCUES
4
2
0
-2
-4
0 0.2 0.4 0.6 0.8 1
4
2
0
-2
-4
0 0.2 0.4 0.6 0.8 1
F IG . 2.2. The left figure represents the graph of the function ¬­®]¯r°†±K² ³`´¶µ ·¸‰¹^­¶ºX±ƒ®#¯ . The right
graph represents the corresponding » -fractal, with ¼’½
¾u¿_ºXÀMÁ¨¿§Â‰ÀKÁ‚¿_ÃÃÀ¿_º , Ä the Weierstrass
é
í
function ĉ­®#¯-°Hè8é€êvè
´=ë&ìíî é ´]ï ¹ð\Ë-­\ñ ®]¯ (with suitable è8é , è ´ in order to verify the hypotheses) and
» È °H¾wà º{É`˨°PºKƃÃÃÃÆ]Á .
Besides, applying the equation (2.13) for Š1.¨—‡e , one has
å
from which it is clear
that,
å
˜
Ì YK˜ š
j
c c
š
c c ˜M˜Kš
J f
Î
š
c c
š
c c ˜M˜ š }&˜M˜ š
J f
Î
š
and as a consequence,
(3.2)
j
Ì YafH˜Mš
˜
(3.1)
å
˜
å
Ì ˜
j
c c
Jd}
š
c c ˜M˜ š
J f
Î
š
c c
Jd}
š
c c
J f
Î
š
j
and so Ì is a linear and bounded operator. From here on we consider this particular case
( ‡1&¨—‡e ).
4. Fractal Trigonometric Polynomials. We consider here the space of w -periodic
Õ ÕYÔ
continuous functions
w1t7ZE€+fÎ5;;UQC
Ô
Ô
Ô
Ÿ
;eXò^GY!]ó¶G
Ô
ò
a5fÎg1&a\ƒO
Ô
Ô
Let ôMÛ be the set of trigonometric polynomials of degree (or order) at most Þ , linearly
spanned by the set 78JLÔ ó¶G9<ƒeXÔ ò \<;ƒ Ô ó¶Gw<XeXÔ ò w
< XM™M™K™MÔ ó¶G9Þ2<;ƒeMÔ ò 9Þ2<;O .
(4.1)
ôMÛ&1($&74J4 ó¶G\<XeXò 9<ƒ óG<;ƒeMò <;ƒM™K™M™K ó¶G\Þo<ƒeXò \Þo<ƒOuä
This family constitutes a basis for ô Û . This system is orthogonal with respect to the inner
i
product (5.1). In fact it is a complete
system in õ ([7]).
å
Let Í EfÎZ1_!]"u$§!'b$.™K™M™$l!#/&1P be a partition of the interval +fÎ5; .
D EFINITION 4.1. ô€Û Ì 1
-fractal
trigonometric
polynomials
of
Ì\ôMÛq is the set of
Ô
Ô
Ô
Ô
degree atÔ most Þ . å
Ô
Ô
å Ô
P ROPOSITION 4.2. ô=Û Ì is spanned by 74JLeXò ÌY\!5å X ó¶GÌ9!5ƒM™K™M™K ó¶GÌY\Þo!5ƒeXò Ì9Þ2!5ƒO
where eXò Ì+ö4!5T1
ÌeMò ,ö4!5 and ó¶GÌ-,ö4!51
Ì ó¶G+ö4!5 .
Proof.
The
constant
functions
are
fixed
points
of Ì ,as the equation (2.11), for a\!5Š1
m
÷
!d>o0
 Ì \!51
÷
R ' \ !5
}Vg:- Ì fH¨—‡eK—‡S : -
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FRACTAL TRIGONOMETRIC APPROXIMATION
m å
÷
å
69
å
is satisfied by YÌ\!51 Ô
Ô !T>v0 and by Ôthe uniqueness
Ô of the solution Ì-Y1. .
Ì\! Û where ! Û >hô Û . By the linearity of Ì , !5ÛÌ is a linear
If !5ÛÌ >•ô€Û Ì , then !5ÛÌ 1
combination of 78JLeMò ÌY9!5ƒ ó¶GÌ\!5ƒK™M™K™M ó¶GÌ-9Þ2!5XeXò ́\Þo!5O .
C ONSEQUENCE 4.3. x4óÞD\ô=Û Ì Š$_}qy .
This fact allows the existence of a finite uniform distance from n>2 to ô Û Ì :
x Û Ì 1.x5ô Û Ì T1_ó¶G7
˜M|f•! ÛÌ ˜Mš
Ô
Ô
(4.2)
Ÿ
! ÛÌ >vô Û Ì O
I
In ø 6, we approach the problem of finding a basis for ô;Û Ì . The case Ü1
gives the
classical case of smooth óG and eXò functions.
The Theorem of Uniform Approximation (Weierstrass) for -periodic functions asserts
that, any _>¤w can be uniformly approximated by trigonometric polynomials (see for
instance [11]).
I
T HEOREM 4.4. Let H>• be given. For Ô all ùuä
, any partition Í of the interval
0(1’ fÎ; with Nh}%J points (Nä&J ) and any function e verifying Ithe conditions prescribed,
/ such that
Ô
there exists an -fractal trigonometric polynomial
in C
Ì\!5 with P«1
c
ÔI
Proof. For any ùVä
approximation of , ú
c
Ì \!5 l
$ ù
a9!5af
I
, let us consider ùƒâLpä
Ô that,
\!5Î>2ô Û ,such
. Applying the theorem of uniform
c
c
a\!5gf 9!5 l
$ ùXâw
!‡>v0
I
/
For a partition Í we choose V>oC
small enough to verify
Ô
Ô
Ô , _«1
c c
c
cj
š
c c ˜ ˜Mš¢$§ùXâL
(4.4)
9!5af Ì \!5
JÎf
š
(4.3)
Then, by (4.3) and (4.4) we obtain the result.
T HEOREM 4.5. For fixed Í and e , the set of fractal trigonometric polynomials
û
Ÿ
7K! ÛÌ
Ÿ
!#Û[>2ôMÛ
/
H>H9C
Ÿ^c c
Ÿ
š¢$&J £
Þ >oNHO
is dense in .
Proof. It is an inmediate consequence of the former theorem.
I
We exclude *1
because
in this case !ÛÌ 1£!#Û and the fact is known. This result
Ô
I
confirms that it is possible to choose F«1
and Þü>*N such that there exists a fractal
close
to
trigonometric polynomial ÛÔÌ arbitrarily
Ô
Ô any n>2Ô . That is to say, the set
û Ø4ý
78JLeXò Ì \<;ƒ ó¶G Ì \<XM™K™M™‰ ó¶G Ì \Þo<XeXò Ì \Þo<;™K™M™+O
Ì
/ Ÿ‰c c šª$pJ Ÿ _«1 I O , is fundamental respect to the uniform norm. ([6]).
where þn1p7‰¤>vC
5. Fractal Fourier Series. We consider here with the inner product
The system
(5.2)
›
$l
(5.1)
7
J
J
Ô
ó¶G\!5X
J
R
Ô
›
a\!5 \ !55xL!
äz1.ÿ
eXò \!5X
J
Ô
ó¶G!5X
J
Ô
eXò w!5ƒM™K™M™+O
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M. A. NAVASCUES
is orthonormal and complete ([7], [15]). The Fourier series of  is
4"
}
a\!5
(5.3)
where
š
1
Ô
÷
5! }§
÷
ó¶G 5! 5
Ô
J
1
eXò
g' Ô
ÿ
÷
a\!55eXò 5! 5xL!
R
Ô
R
J
÷
a\!5 ó¶G 5! #x4!
ÿ
In general, the Fourier series of an element is merely the sum of its projections on a system
i
of orthonormal elements. Fourier expansions converge in the mean of order 2 (õ -norm) to
the elements that give rise to them. That is to say, if
(5.4)
8"
}
Û \!5d1
then,
˜M|f
Û
' 9
i
Û ˜i 1 ÿ
eXò
Ô
÷
5! -}V
…f
R
Ô
÷
óG 5! 5
I
i
Û Lx !U
as Þ¡U–y .
Pointwise and uniform convergence of the Fourier series of  is not verified in general. A
collection of theorems concerning this topic can be consulted in ([7], [6], [15]). For instance,
we remark the following result.
k ›
I
I
T HEOREM 5.1. (Dini-Lipschitz) If a\!5‡>2 , and if 6 w ò wU
as qU
, then
the Fourier series of  converges uniformly to  . ( is the modulus of continuity of  ).
Let us consider the operator gÛ~EwUFôMÛ ,such that Û¨Y is defined by
where
Û%YX9!5T1 m
Û(9!5
!T>D fÎ;
Û is the Fourier sum of order Þ of  (5.4). In the reference ([6]), it is proved that Û
is a bounded operator and the following inequality holds:
k ›
j
ò \ÞZ‡$t˜ Û(˜
i
Û is a projection on ô Û as Û — Û 1 Û .
(5.5)
}
k ›
ò 9Þv
The error committed by the finite Fourier sum can be bounded in several ways. For
Ô
Ô
instance, if
x Û Y1.x=5ôMÛrT1Pó¶G7`˜M%f
it can be proved that (see [17])
(5.6)
˜ Ûb?fD˜Kš
j
}
˜Mš2
>vôMÛuO
k ›
ò \ÞZ55x Û Y
The theorems of Jackson give upper bounds for the quantity x Û Y . For instance:
T HEOREM 5.2. (Jackson’s Theorem [6]) For all Z>o
x Û Y
j
6 ÞÜ }_J ETNA
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FRACTAL TRIGONOMETRIC APPROXIMATION
where is the modulus of continuity of  . The coefficient 1 of 6 Û
one independent of  and Þ .
As a consequence, if B>2
å
j
˜ Ûr…fH˜Mš
(5.7)
k ›
ò 9Þv 6
}
' is the best possible
ކ}_J
It can be observed that, from this inequality, the Dini-Lipschitz theorem is deduced.
Ì(— Û be the operator such that å
Let ‡Û Ì 1
Û Ì Y1
Ì Û%Y5
( -fractal Fourier finite sum of  ). ŠÛ Ì is a linear and bounded operator and by (3.2) and (5.5)
j
˜ ÛÌ ˜
˜ Û Ì ?fD˜Kš
c c
k ›
Jd}
š
c c } ò \ÞZ5
J f
Î
š
j
c c
Jd}
š
c c ˜ Ûr…fH˜Mšt}
J f
Î
š
j
and applying (5.7)
(5.8) ˜
Û Ì ?fD˜Kš
j
JÎf
J
c c
š
j
˜ Û Ì …fH Ì ˜Kšp}&˜M Ì fH˜Mš
55J‡}
c c
š…M }
c c
š
c c ˜K˜Mš
J f
Î
š
k ›
ò \ÞZ5 6
c c
}§
šZ˜K˜Mš%
ކ}@J
In this way, the theorem of Dini-Lipschitz can bek generalized to the fractal series.
›
I
I
T HEOREM 5.3. Let @>Vw such that 6 w ò wuU
as oU
and let Û be a
I
sequence of scale vectors, such that Û U
as ÞÝU y ,then the Û -fractal Fourier series
of  converges uniformly to  as ÞªU{y .
I
I
Proof. As  is continuous on fÎ; , 6 bU
as HU
([6]). The hypotheses and
(5.8) give the result.
6. Trigonometric Fractal Interpolation. We approach here the problem of trigonometric interpolation by means of fractal techniques. Let Í EYfÎD1t!"%$@!'‚$’™K™M™Y$@!#iۆ$
iÛ '
!#iÛ '|13 be given correponding to the data 789! 5< O Y" where <="o1¥<=iÛ ' . Let us
consider the fundamental functions of interpolation
Ô ([17], [7])
(6.1)
For öL
÷
1
I
KJLK™M™M™MÞh
iÛ
9!5T1 i Û Y" Ñ  Ô óG ' i' 9!g fh! 5
Ñ Y"  ó¶G i \! fh! 5
Ô \! 1!
.Ô Each function
" ,isÔ a linear combination of
Ô
JLeXò \!5XM™M™K™XeMò 9Þ2!5X ó¶G\!5XM™M™K™M ó¶G9Þ2!5
and hence is an element of ô‰Û ([7]).
The function,
(6.2)
\!5T1
iÛ < \!5
Y"
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72
M. A. NAVASCUES
,is also an element of ô Û and is the unique solution in this space for the interpolation problem
\! T1P< ÷
”
I
1
MJ4M™M™K™Xwޕ)
If Û represents the trigonometric interpolation operator, which assigns to a function  its
iÛ
trigonometric interpolant with respect to 789! a\! O Y" , then by (6.2)
j
”
˜ Û ˜ š
(6.3)
k
˜M˜ š ˜ Û ˜ š
where
k
iÛ c 9!5 c
Y"
Û \!51
We define the -fractal trigonometric interpolant as
Ì
å
iÛ < Ì
Ì 9!5T1
Ì X\!5d1
-"
9!5
as Ì \! r1 9! r1#
is the -fractal function of
with respect to Í .
The function Ì passes through the points 9! <
(2.12)). Besides
å
The functions 7
O i-Û "
Ì 1
Ì —
”
Û Y
are orthogonal with respect to the form:
›
 1
iÛ
Y"
›
a\! 9 ! and hence a basis of ô‰Û . This property is inherited by 7
nodes and so
where
Ì
O as Ì interpolates to
Ì Ì T1 iÛ 9! 9! 1 iÛ YÔ "
Y"
Ô
Ì 1
iÛ ã Ì
Y"
at the
å
is the delta of Kronecker. If Ìo>oô€Û Ì , by the linearity of the operator
(see ø 2
Ì ,
the orthogonality of Ì implies the linear independence and hence 7 Ì O iÛ Y" constitutes a
I
basis of ô Û Ì of -fractal trigonometric polynomials with respect to the partition Í . For •1
,
Ö
we retrieve the standard basis .
I
7. Fitting Method by Fractal Approximants. Let 78 ! < Xö(1
KJLK)M)M)M O be a collection of data. Let us define a function of approximation to the data (for instance, a minimax
approximation or a least squares fitting curve) a9!5 . Consider an interval 0 containing the
m abscissas and let Í be a partition of 0 , Í EL¨1@!#"q$l!'z$p)+),)$l!#/&1@ such that ! 1 « !
óö ,
!]"u$ !]" and !5/[ä ! . We look for an -fractal function -Ì\!5 . To choose g: we consider all
%$
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'&
the abscissas ! K™M™K™X !
(fixed point) (2.11):
73
(
FRACTAL TRIGONOMETRIC APPROXIMATION
j
in 0 : : ! :
R-'
)
<
Approximating YÌ by 
<
!
)
j
+*
! : for ó‹13JLM™K™M™X . Then, by the equation
)
)
)
)
R ' ! 1.a ! -}V :  Ì fH¨—‡eK—‡S : -
),
R ' ! a ! }V:Y2fD%—‡eM-—‡S : -
We choose : by a least squares procedure:
. a ! ) gf < )
'
-
Þ2ó¶G … : 1
)
R ' ! 5 i
}¤ : ?fH¨—‡eM—TS : Y
Differentiating the former expression we obtain:
g:|1~f ë
By the Schwartz’s inequality
. ' a ! ) af < ) X?fH%—‡eM—dS :R-' ! ) R-' ) i
ë . ' 5…f¤%—‡eM-—‡S : ! 5
Õ
c
Õ
where
1’a !
c
j
˜ X˜ i
˜ ;˜Mi
0/
'& f < '& K™M™K™Ma ! ( gf < ( /%1[2fD%—‡eM-—‡S
)
:
:
RY'
!
'& ƒM™K™M™KK?fH¨—‡eM—‡S
:
RY'
!
( )
We must choose the order of the approximant
c c  in such a way that the differences between
a ! and < are small enough to obtain : $.J .
REFERENCES
[1] M.F. B ARNSLEY , Fractal functions and interpolation, Constr. Approx., 2 (1986), No. 4, pp. 303-329.
[2] M.F. B ARNSLEY , Fractals Everywhere, Academic Press, Inc., 1988.
[3] M. F. B ARNSLEY AND A. N. H ARRINGTON , The Calculus of Fractal Interpolation Functions, J. Approx.
Theory, 57 (1989), pp. 14-34.
[4] M.F. B ARNSLEY, D. S AUPE AND E.R. V RSCAY, EDS ., Fractals in Multimedia, I.M.A., 132, Springer, 2002.
[5] A. S. B ESICOVITCH AND H. D. U RSELL , On dimensional numbers of some continuous curves, in Classics
on Fractals, Edgar, G. A., eds., Addison-Wesley, 1993, pp. 171-179.
[6] E.W. C HENEY , Introduction to Approximation Theory, McGraw Hill Book Co.,New york-toronto,London.,
1966.
[7] P. J. D AVIS , Interpolation and Approximation, Dover Publ., 1975.
[8] K.J. FALCONER , Fractal Geometry, Mathematical Foundations and Applications, J. Wiley
Sons,Ltd.,
Chichester, 1990.
[9] G.H. H ARDY , Weierstrass non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), No. 3, pp. 301325.
[10] J.E. H UTCHINSON , Fractals and Self Similarity, Indiana Univer. Math. J., 30 (1981), No. 5, pp. 713-747.
[11] E. I SAACSON AND H.B. K ELLER , Analysis of Numerical Methods, J. Wiley Sons, 1966.
[12] M.A. N AVASCU ÉS AND M.V. S EBASTI ÁN , Some results of convergence of cubic spline fractal interpolation
functions, Fractals, 11 (2003), No. 1, pp. 1-7.
[13] M.A. N AVASCU ÉS AND M.V. S EBASTI ÁN , Generalization of Hermite functions by fractal interpolation, J.
of Approx. Theory, 131 (2004), No. 1, pp. 19-29.
1
1
ETNA
Kent State University
etna@mcs.kent.edu
74
M. A. NAVASCUES
[14] M.A. N AVASCU ÉS AND M.V. S EBASTI ÁN , Fitting curves by fractal interpolation: an application to the
quantification of cognitive brain processes, in Thinking in Patterns: Fractals and Related Phenomena in
Nature. Novak, M.M., eds., World Scientific Publishing Co., Inc., Teaneck, NJ, 2004, pp. 143-154.
[15] G. S ANSON , Orthogonal Functions, Robert E. Krieger Publishing Co., Huntington, N.Y., 1977.
[16] M.V. S EBASTI ÁN AND M.A. N AVASCU ÉS ET AL , An alternative to correlation dimension for the quantification of bioelectric signal complexity, WSEAS Trans. on Biol. & Biomed., 1 (2004), No. 3, pp. 357-362.
[17] J. S ZABADOS AND P. V ERTESI , Interpolation of Functions, World Scientific Publishing Co., Inc., Teaneck,
NJ, 1990.