On the averaging procedure over the Cantor set
The procedure of averaging a smooth function over the normalized density of the Cantor set (A. Le Mehaute, R.R. Nigmatullin, L. Nivanen. Fleches du temps et geometric fractale. Paris: “Hermes”, 1998, Chapter 5) has been shown not to reduce exactly the convolution to the classical fractional integral...
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irk-123456789-798982015-04-07T03:01:52Z On the averaging procedure over the Cantor set Stanislavsky, A.A. Weron, K. Anomalous diffusion, fractals, and chaos The procedure of averaging a smooth function over the normalized density of the Cantor set (A. Le Mehaute, R.R. Nigmatullin, L. Nivanen. Fleches du temps et geometric fractale. Paris: “Hermes”, 1998, Chapter 5) has been shown not to reduce exactly the convolution to the classical fractional integral of Riemann-Liouville type. Although the asymptotic behavior of the self-similar convolution kernel is very close to the product of a power and a log-periodic function, this is not obviously enough to claim the direct relationship between the fractals and the fractional calculus. 2001 Article On the averaging procedure over the Cantor set / A.A. Stanislavsky, K. Weron // Вопросы атомной науки и техники. — 2001. — № 6. — С. 245-246. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 05.40.-a, 05.45.-a, 05.46.-k. http://dspace.nbuv.gov.ua/handle/123456789/79898 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Anomalous diffusion, fractals, and chaos Anomalous diffusion, fractals, and chaos |
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Anomalous diffusion, fractals, and chaos Anomalous diffusion, fractals, and chaos Stanislavsky, A.A. Weron, K. On the averaging procedure over the Cantor set Вопросы атомной науки и техники |
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The procedure of averaging a smooth function over the normalized density of the Cantor set (A. Le Mehaute, R.R. Nigmatullin, L. Nivanen. Fleches du temps et geometric fractale. Paris: “Hermes”, 1998, Chapter 5) has been shown not to reduce exactly the convolution to the classical fractional integral of Riemann-Liouville type. Although the asymptotic behavior of the self-similar convolution kernel is very close to the product of a power and a log-periodic function, this is not obviously enough to claim the direct relationship between the fractals and the fractional calculus. |
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Article |
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Stanislavsky, A.A. Weron, K. |
| author_facet |
Stanislavsky, A.A. Weron, K. |
| author_sort |
Stanislavsky, A.A. |
| title |
On the averaging procedure over the Cantor set |
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On the averaging procedure over the Cantor set |
| title_full |
On the averaging procedure over the Cantor set |
| title_fullStr |
On the averaging procedure over the Cantor set |
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On the averaging procedure over the Cantor set |
| title_sort |
on the averaging procedure over the cantor set |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
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Anomalous diffusion, fractals, and chaos |
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http://dspace.nbuv.gov.ua/handle/123456789/79898 |
| citation_txt |
On the averaging procedure over the Cantor set / A.A. Stanislavsky, K. Weron // Вопросы атомной науки и техники. — 2001. — № 6. — С. 245-246. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT stanislavskyaa ontheaveragingprocedureoverthecantorset AT weronk ontheaveragingprocedureoverthecantorset |
| first_indexed |
2025-07-06T03:50:38Z |
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2025-07-06T03:50:38Z |
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1836868005159501824 |
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ON THE AVERAGING PROCEDURE OVER THE CANTOR SET
A.A. Stanislavsky 1 , K. Weron 2
1 Institute of Radio Astronomy, Kharkov, Ukraine
e-mail: alexstan@ira.kharkov.ua
2 Institute of Physics, Wroclaw University of Technology, Wroclaw 50-370, Poland
e-mail: karina@rainbow.if.pwr.wroc.pl
The procedure of averaging a smooth function over the normalized density of the Cantor set
(A. Le Mehaute, R.R. Nigmatullin, L. Nivanen. Fleches du temps et geometric fractale. Paris: “Hermes”, 1998,
Chapter 5) has been shown not to reduce exactly the convolution to the classical fractional integral of Riemann-
Liouville type. Although the asymptotic behavior of the self-similar convolution kernel is very close to the product
of a power and a log-periodic function, this is not obviously enough to claim the direct relationship between the
fractals and the fractional calculus.
PACS: 05.40.-a, 05.45.-a, 05.46.-k.
1. INTRODUCTION
The result [1,2] of convolution of a smooth function
with the normalized density of the Cantor set in the limit
N → Ґ has caused a great interest [3,4] (and references
therein). The point is that a clear relationship between
fractal geometry and fractional calculus has been long
sought by the scientific community. In the paper [1]
such a relation has been asserted to be found. The
criticism expressed in [3] has stated the approach of [1]
under serious doubts and has required the
reconsideration of the previous result in [2].
Nevertheless, the detailed analysis of the modification
has shown [5] that the procedure of averaging a smooth
function on fractal sets does not allow one to obtain the
kernel corresponding to the fractional integral.
The goal of this paper is to present the supplementary
results verifying the main conclusion of [5].
2. THE CONVOLUTION
OVERTHE CANTOR SET
Let a value ( )J t be related with a function ( )f t by
the convolution operation
0
( ) ( )* ( ) ( ) ( ) ,
t
J t K t f t K t f dτ τ τ= = −т (1)
where ( )K t is the memory function determined on the
segment [ ]0,T so that
0
( ) 1
T
K t dt =т . The binary Cantor
sets with fractal dimension ln 2 / ln(1/ )ν ξ= is built
iteratively on the interval [ ]0,T by deleting, at the first
step, the middle part of length 2(1 )Tξ− , where 2 νξ −=
, 0 1/ 2ξ< Ј . Each following step repeats the previous
one on the all remaining intervals. The height of each
Cantor bar on any stage of the construction provides the
conservation of normalization. The result ( )K t of the
procedure on the N -th stage has been given by the
recurrence relation [2] with the Laplace image of
( )
2 ( )NK t taking the following form
( ) ( )
2 2
0
( ) exp( ) ( )
N NK p pt K t dt
Ґ
= − =т
( )
2
1 exp( ) [ (1 )]
N
N
N
pT Q pT
pT
ξ ξ
ξ
− −= − ,
where
1
( )
2
0
1 exp( )( )
2
nN
N
n
zQ z ξ−
=
+ −= Х (2)
and (1 )z pT ξ= − . The product (2) converges due to the
numerator tending exponentially to 2 for 1n ? . In the
limit N → Ґ the Laplace image of
( ) ( )
2 2( ) ( )* ( )N NJ t K t f t= is of the form
2 2( ) [ (1 )] ( )J p Q pT f pξ= − ,
where 2 ( )Q z is the limit of the product (2). It satisfies
the functional equation 2 2( ) ( ){1 exp( )}/ 2Q z Q z zξ= + −
that reduces to the scaling relation 2 2( ) ( ) / 2Q z Q zξ≈
for 1z ? .
In the Cantor sets, having M bars ( 1Mξ < ) in each
stage of its construction, the corresponding recurrence
relation [2] leads to
1( )
0
1 exp[ /( 1)]( ) ,
1 exp[ /( 1)]
nNN N
M n
n
z M MK p M
z M
ξ
ξ
−
−
=
− − −=
− − −Х
(1 )z pT ξ= − .
In the framework proposed in [2] the Cantor sets with
an exponential damping or random localization of
random bars were also considered. From the particular
cases it follows that the memory function is written as
µ $( )
0
( ) lim ( ) ( ) ( ),
N n
N
n
K p G z G z g zξ
ҐΣ
→ Ґ
=
= = = Х (3)
(1 )z pT ξ= − ,
where $( )g x is an entire function (without zero), and
$
1
ln ( ) / !k
k
k
g x c x k
Ґ
=
= е , where kc are some constants.
Assume that the product (3) converges. The expansion
of $ln ( )g x can be obtained by means of analytic
computer calculations. So, in the case of binary Cantor
set (2) we have
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 245-246. 245
-x
2 4 6
8 10
12 14
16
1+eln = -1/2 x + 1/8 x - 1/192 x +1/2880 x -
2
-17/645120 x +31/14515200 x -
- 691/3832012800 x 5461/348713164800 x -
-929569/669529276416000 x + .
Ч Ч Ч Ч
Ч Ч
+Ч Ч
Ч L
The product (3) will be also an entire function, which
has no zero in the whole of finite complex plane, and
can be represented in the form of ( ) exp[ ( )]G z A z= ,
where ( )A z is an entire function [6]. The function
( )G z has the essential singular point at infinity and
becomes vanishing small for Re z → Ґ . This indicates
its non-analytic asymptotic behavior.
3. EXPONENTIAL FORM OF THE
MEMORY FUNCTION
By taking logarithm, the product ( )K p
Σ can be
represented in the form of the convergent series
$
0 0
( , ) ln ( ) ( )n
n n
A z g z h nξ ξ
Ґ Ґ
= =
= =е е . Here z plays the role
of a parameter. By means of the Poisson summation
formula
1
2
0 10 0
( ) (0) ( ) 2 ( )cos(2 ) ,
n m
h n h h x dx h x mx dxπ
Ґ ҐҐ Ґ
= =
= + +е ет т
where $( ) [exp( ln ) ]h x g x zξ= , and the change of
variables exp( ln )y x ξ= , we have
$ $1 2 / ln1
2
1
1( , ) ( ) ln ( ) .
ln
im
m
A z g z y g yz dyπ ξξ
ξ
ҐҐ
− +
= − Ґ
= + е т (4)
Since 1
2 exp( ) ( )
m
ims sπ δ
Ґ
= − Ґ
=е is the Dirac δ -function,
and $ln (0) 0g = , the expression
$1 2 / ln
0
1 ln ( )
ln
im
m
y g yz dyπ ξ
ξ
ҐҐ
− +
= − Ґ
е т
also equals 0, too. Then the remaining terms of (4)
become
$1
2
1
1( , ) ( )
ln ![ 2 / ln ]
k
k
m k
c zA z g z
k k im
ξ
ξ π ξ
Ґ Ґ
= − =Ґ
= −
+е е . (5)
Using the identity
1 1
2 2
1 ln coth[ ln ]
[ 2 / ln ]m
k
k im
ξ ξ
π ξ
Ґ
= − Ґ
=
+е ,
and interchanging the summations in the double series
by virtue of its convergence, the expression (5) reduces
to the following form
1
( , )
!(1 )
k
k
k
k
c zA z
k
ξ
ξ
Ґ
=
=
−е . (6)
Thus, we obtain ( ) exp[ ( , )]K p A z ξ
Σ
= . By the direct
substitution of the solution into the functional equation
$( ) ( ) / ( )G z G z g zξ = it is easy to check the correctness
of the result. It should be observed that the expansion of
$ln ( )g x defines completely, in some sense, the
exponential form of ( )K p
Σ . If $ln ( )g z is an entire
function of degree lim n
n kcα → Ґ= , ( , )A z ξ will be
also an entire function with the same degree.
Since $lim ( ) ( ) 1x g x g x const→ Ґ = = < (in particular,
for binary Cantor set lim [1 exp( )] / 2 1/ 2x x→ Ґ + − = , and
for Cantor set with M bars the limit is 1/ M ), the
functional equation $( ) ( ) / ( )G z G z g zξ = reduces to the
scaling relation ( ) ( ) /G z G z gξ ≈ which has the unique
non-trivial solution ( )Bz L zµ , where B is a constant,
ln(1/ ) / ln(1/ )gµ ξ= , ( ) ( )L z L zξ = is a log-periodic
function. A more precise asymptotic representation of
( )K p
Σ can be reached with the help of Euler-Maclaurin
formula [2]. Although in some cases the asymptotic
behaviour (after averaging of the log-periodic term) can
be well approximated by the power function with non-
integer exponent for 1z > , the analytic background of
( )K p
Σ cannot be fully ignored.
4. CONCLUDING REMARKS
We have shown that the exponential form of ( )K p
Σ
is a direct consequence of properties of the memory
function. In the Laplace-image space the convolution
kernel for the averaging procedure over the Cantor set is
an analytic (entire transcendental) function, and the
kernel of fractional calculus is non-analytic. None of
non-analytic functions is impossible to represent as a
product of analytic functions on the whole of complex
plane. Therefore, the approach of [2] does not give the
correct procedure for passing from fractal geometry
(Cantor set) to the fractional integral of Riemann-
Liouville type. Thus, the treatment (both mathematical
and physical) of [2] to the notion of fractional integral in
terms of fractals requires the revision.
REFERENCES
1. R.R. Nigmatullin. Fractional integral and its physical
interpretation // TMF. 1992, v. 90, №3, p. 354-368.
2. A. Le Mehaute, R.R. Nigmatullin, L. Nivanen.
Fleches du temps et geometric fractale. Chapter 5.
Paris: “Hermes”, 1998, 352 p.
3. R.S. Rutman. On the paper by R.R. Nigmatullin
“Fractional integral and its physical interpretation” //
TMF. 1994, v. 100, №3, p. 476-478; R.S. Rutman.
On physical interpretations of fractional integration
and differentiation // TMF. 1995, v. 105, №3, p.
393-404.
4. A. I.Olemskoi, A.Ya. Flat. Aplication of fractals in
condensed-matter physics // Physics Uspekhi 1993,
v. 163, № 12, p. 1-50; F.Y. Ren, J.R. Liang,
X.T. Wang. The determination of the diffusion
kernel on fractals and fractional diffusion equation
for transport phenomena in random media // Phys.
Lett. 1999, v. A252, p. 141-150; Z.G. Yu. Flux and
memory measure on net fractals // Phys. Lett. 1999,
v. A257, p. 221-225; A.A. Stanislavsky. Memory
effects and macroscopic manifestation of
randomness // Phys. Rev. 2000, v. E61, № 5, p.
246
4752-4759; A.I. Olemskoi, V.F. Klepikov. The
theory of spatiotemporal pattern in nonequilibrium
systems // Phys. Rep. 2000, v. 338, p. 571-677.
5. A.A. Stanislavsky, K. Weron. Exact solution for the
averaging procedure over the Cantor set // Physica.
2002, v. A303, № 1-2, p. 57-66.
6. M.A. Lavrent'ev, B.V. Shabat. Methods of the theory of
Functions of complex variable. M.: “Nauka”, 1987, 688
p.
247
ON THE AVERAGING PROCEDURE OVER THE CANTOR SET
A.A. Stanislavsky, K. Weron
Institute of Radio Astronomy, Kharkov, Ukraine
e-mail: alexstan@ira.kharkov.ua
1. INTRODUCTION
2. THE CONVOLUTION
OVERTHE CANTOR SET
3. EXPONENTIAL FORM OF THE
MEMORY FUNCTION
4. CONCLUDING REMARKS
REFERENCES
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