Services on Demand
Journal
Article
Indicators
 Cited by SciELO
 Access statistics
Related links
 Cited by Google
 Similars in SciELO
 Similars in Google
Share
TEMA (São Carlos)
Online version ISSN 21798451
TEMA (São Carlos) vol.15 no.3 São Carlos Sept./Dec. 2014
https://doi.org/10.1590/S217984512014000300004
Numerical calculations of Hölder exponents for the Weierstrass functions with (min, +)wavelets
M. Gondran^{I}; A. Kenoufi^{II, }^{*}
^{I}University Paris Dauphine, Lamsade, Paris, France. Email: michel.gondran@polytechnique.org
^{II}Scientific COnsulting for Research & Engineering (SCORE), Strasbourg, France. Email: kenoufi@score.fr
ABSTRACT
One reminds for all function f : ^{n }→ the socalled (min, +)wavelets which are lower and upper hulls build from (min, +) analysis [12, 13]. One shows that this analysis can be applied numerically to the Weierstrass and WeierstrassMandelbrot functions, and that one recovers their theoretical Hölder exponents and fractal dimensions.
Keywords: (min, +)wavelets, Hölder exponents,Weierstrass functions.
RESUMO
Lembrando que para todas as funções f : ^{n }→ , as chamadas (min, +)wavelets são construções do fecho inferior e superior, vindos da análise (min, +) [12, 13]. Mostrase que esta análise pode ser aplicada numericamente às funções de Weierstrass e WeierstrassMandelbrot, e que recuperase os seus expoentes de Hölder teóricos e dimensões fractais.
Palavraschave: (min, +)wavelets , expoentes de Hölder, funções de Weierstrass.
1 INTRODUCTION
Genesis of wavelets theory started in 1946 with D. Gabor [9], who introduced the Windowed Fourier Transform (WFT)
for the local spectral analysis of radar signals. The localization is reached due to fast decaying window function g(x) 0. Even if WFT exhibits many powerful and practical features, there are some defects compared to Fourier Transform. The transform (1.1) can not resolve efficiently wavelengths longer than the window g(x) width. Conversely, for signal with high frequencies, short decomposition needs a broad window with a large number of periods. Thus, signal reconstruction in this case adds a large number of terms with comparable amplitudes and hence becomes numerically unstable. Finally, one needs a scheme with a wide window for low frequency signals and a narrow window for high frequency ones. Such a scheme, was independently suggested as a tool for geophysical studies by several authors at the beginning of 1980s [19, 23]. Wavelets Theory (WT) word was introduced in analysis by J. Morlet [17, 11]. It is considered nowadays as a preferable alternative to the Fourier analysis, used where and when the signals are random and comprised of fluctuations of different scales, such as in turbulence phenomena [3]. WT has been immediately followed by several applications in science and engineering, such as signal processing and detection, fractals, selfsimilar objects, selfsimilar random processes, like turbulence and Brownian motion [2]. WT was then mathematically formalized by Grossman & Morlet [11], Goupillaud et al. [10], Daubechies [6] and some other authors. Practically, WT is a separate convolution of the signal in question with a family of functions obtained from some basic one, the basic wavelet called mother wavelet or analysing function, by shifts τ and dilatations a:
An interesting point of view is to consider WT as a realization of a function decomposition with respect to the representation of the affine group [2]: x ax + b.
We refer the reader to useful and interesting articles [1, 8, 20, 4, 7, 5] and both theoretical and practical book [2].
Nevertheless, wavelets decompositions are limited by their linear features. This present article aims to apply for pathological functions such as Weierstrass functions, a nonlinear transform, called (min, +) transform, which has been already defined within (min, +) analysis [12, 13, 18]. This one consists to replace in the scalar product definition of two realvalued functions f and g defined on a domain X, the real number field (, +, ×) with the (min, +) dioid ( ∪{+ ∞}, min, +). The classical scalar product 〈f, g〉 = ∫_{x }_{∈}_{ X} f(x)g(x)dx becomes then the (min, +) scalar product [12]:
The demonstration that it is a scalar product within the (min, +) dioid is straightforward and easy excepted for its linearity.
One has to show that 〈f, g〉_{min +} is distributive according to min, which means
〈f, min{g_{1}, g_{2})〉_{(min,+)} = min{〈f, g_{1}〉_{(min,+)}, 〈f, g_{2}〉_{(min,+)}},
and linear according to the addition of a scalar λ : 〈f (x), λ + g(x)〉_{(min,+)} = λ + 〈f, g〉_{(min,+)}. The linearity is obvious since inf_{x}_{∈}_{X}{f (x) + λ + g(x)} = λ + inf_{x}_{∈}_{X}{f(x) + g(x)}. Distributivity is obtained in two steps. One has first to prove this equality with mean of two inequalities. We start first with the simple relations:
〈f, g_{1}〉_{(min,+)} < f(x) + g_{1}(x), and 〈f, g_{2}〉_{(min,+)} < f(x) + g_{2}(x), ∀x.
This gives min{〈f, g_{1}〉_{(min,+)}, 〈f, g_{2}〉_{(min,+)}} < min{f(x) + g_{1}(x), f(x) + g_{2}(x)}∀x. And since
min{f(x) + g_{1}(x), f(x) + g_{2}(x)} = f (x) + min{g_{1}(x), g_{2}(x)},
one has min{〈f, g_{1}〉_{(min,+)}, 〈f, g_{2}〉_{(min,+)}} < f(x) + min{g_{1}(x), g_{2}(x)}∀x, which yields to the inequality
In a second step, one can write
〈f, min{g_{1}, g_{2}}〉_{(min,+)} < f(x) + min{g_{1}(x), g_{2}(x)} < f(x) + g_{1}(x) ∀x,
which becomes
and in the same manner
〈f, min{g_{1}, g_{2}}〉_{(min,+)} < f(x) + min{g_{1}(x), g_{2}(x)} < f(x) + g_{2}(x) ∀x,
giving now
and then from (1.4) and (1.5)
From relations (1.3) and (1.6), one deduces finally the equality and thus the distributivity.
With this (min, +) scalar product, one obtains a distributionlike theory : the operator is linear and continuous according the dioid structure ( ∪{+ ∞}, min, +), nonlinear and continuous according to the classical structure (, +, ×). The nonlinear distribution δ_{min}(x) defined as
δ_{(min,+)}(x) = {0 if x = 0, +∞ else}
is similar in (min, +) analysis to the classical Dirac distribution. Then, one has
In (min, +) analysis, the LegendreFenchel transform which permits to get Hamiltonian from Lagrangian and which has an important role in physics is similar in (min, +) analysis to the Fourier transform in the classical one [18].
In this article, we explore how (min, +)wavelets decomposition and reconstruction could be an interesting signal processing tool, since (min, +) transforms can be applied to a larger class of functions than the functions treated with classical wavelet transforms, especially to lower semicontinuous functions [12], such as x g(x) · Floor(x) for instance, where g is a continuous function.
In this paper, one focus on Weierstrass and WeierstrassMandelbrot functions which are classical examples of functions continuous everywhere but differentiable nowhere [22].
One introduces in the following Section 2, the (min, +)wavelets decomposition and reconstruction of a signal with mean of (min, +) scalar product. in Section 3, we show how the (min, +)wavelets allow a characterisation of Hölder functions. in Section 4, we apply these results to numerical calculations of Hölder exponents of Weierstrasslike functions and compare them to the theoretical values [21]. This permits to deduce immediately their fractal dimensions.
2 (min, +)WAVELETS
The usual wavelet transform of a function f from ^{n }to is a linear transform defined for all scales a ∈^{+ }and points b ∈^{n}, which can be computed according to the equation (1.2):
where Ψ is a function called mother wavelet or analysing function. It has to be zero average and exhibiting oscillations until a certain order p. This can be written as
In (min, +) analysis, a set of nonlinear transforms has been introduced for lower semicontinuous functions [12, 18], the socalled (min, +)wavelets transforms which are defined for a function f : ^{n }→ and for all a ∈^{+ }and b ∈^{n }such as:
where h is a basis analysing function (upper semicontinuous and infcompact verifying h(0) = 0), like the following functions:
Since T_{f }^{–}(a, x) < f (x) for all a > 0, T_{f }^{–}(a, x) is a lower hull of f(x). For any lower bounded and lower semicontinuous function, one has a reconstruction formula like in the linear wavelets theory [2]:
which can be simplified within the (min, +) theory in
The (min, +)wavelets analysis will be based on simultaneous analysis of lower hulls T_{f }^{–}(a, b), and upper hulls of f represented by T_{f }^{+}(a, b) defined by:
For the upper hulls T_{f }^{+}(a, b), we have a reconstruction formula which is symmetric to lower hulls T_{f }^{–}(a, b) (2.9, 2.8):
which simplifies as well as:
For each analysing function h, one has [13]:
because T_{f} ^{–}(a, x) (respectively T_{f }^{+}(a, x)) are functions decreasing with scales (respectively increasing) and converging to f_{*}(x) (respectively f *(x)), the lower semicontinuous closure of f (respectively upper semicontinuous closure) when the scale tends to 0.
Remark 1. We use the word "wavelet" by analogy with linear wavelets since the decomposition and reconstruction formula are very similar and since one just replaces the usual real number field (, +, ×) with the (min, +) dioid ( ∪{+ ∞}, min, +). Another name can be (min, +) pen or (min, +) hulls.
Remark 2. The shift and scale parameters have the same meaning as in Linear Wavelet Theory: for high frequencies, one needs small scales, and the inverse as well. But the relation between them is not simply proportionally inverse as in linear theory, because it depends on the choice of analyzing function h_{α}, and this introduces nonlinear dependency between scale and frequency. This leads to a relation such as ν = γ(a, α), where ν, and a are respectively the frequency and the scale, and γ a nonlinear function decreasing with a.
Definition 1. (min, +)wavelet is defined as the couple {T_{f }^{–}(a, x), T_{f }^{+}(a, x)}. For all ^{+},the aoscillation of f is defined:
In the case of analysing function h_{∞}, one has
and ΔTf (a, x) = sup_{x–y<a} f (y) – inf_{x–z<a}f (z) corresponds to the aoscillation defined in one dimension by Tricot [21]: osc_{a}f (x) = sup_{y,z}_{∈}_{[x–a,x+a]}[ f (y) – f (z)].
3 CHARACTERISATION OF HÖLDERIAN FUNCTIONS WITH (min, +)WAVELETS ANALYSIS
The calculations of oscillations according to the analysing function and the scale will permit to study the global and local regularity of a function.
First, let's start with the case of global regularity of a Hölderian function for which it exists H (0 < H < 1) and a constant K such as
It is a sufficient but not necessary condition for a function to be continuous. In the case of fractal function, K is related to its fractal dimension.
Theorem 1. The function f is Hölderian with exponent H, 0 < H < 1, if and only if it exists a constant C such as for all a, one of the following condition is verified:
Demonstration:

Demonstration for the case of analysing function h_{∞} is classic [21]: if f verifies (3.17), let's consider some x and y in ^{n}. One can assume that f (x) > f (y). Then, one has
this yields to
f (x) – f (y) < ΔT_{f} (a, x) < Ka^{H }< Kx –y^{H }.
Conversely, let's assume that f (x) – f (y) < K x – y^{H }for all y. Let y_{1} such as f (y_{1}) =sup_{x–z<a}f (z) and y_{2} such as f (y_{2}) =inf_{x–z<a}f (z). One has then
ΔT_{f} (a, x) = f (y_{1}) – f (y_{2}) = f (y_{1}) – f (y) + f (y) – f (y_{2}),
which yields to
ΔT_{f} (a, x) < f (y_{1}) – f (y) + f (y) – f (y_{2}) < 2Ka^{H }.
In the case of analysing functions h_{α}, α > 1, let's suppose first that f verifies (3.18). We consider x and y in ^{n }with f (x) > f (y). Reconstruction equation (2.12) of f (x) can be written as
and the equation of reconstuction for f (y) (2.8)
One deduces
thus
and the optimisation on the scale a implies that it exists K such as f (x) – f (y) < K x –y^{H }.
Conversely, let's assume that f (x) – f (y) < K x –y^{H }for all x and y. Using (2.8) and (2.11), one has
We deduce that
whose optimisation gives (3.18). □
Let's consider now the case of local irregularity at x_{0} where the function is Hölderian: it exists H (0 < H < 1) and a constant K such as
Theorem 2. The function f is Hölderian at point x_{0}, with exponent H, 0 < H < 1, if and only if it exists a constant C such as for all a, one has one of the following conditions:
Demonstration:

In the case of the analysing function h_{∞}, if f verifies (3.20) for all x, let a = x – x_{0}. One has then inequations sup_{x–x0<a}f (z) > f (x) > f (x_{0}) > inf_{x–z < a}f (z) or sup_{x–x0 <a}f (z) > f (x0) > f (x) >inf_{x–z<a}f (z). In both cases one gets
f (x) – f (x_{0}) < ΔT_{f} (a, x) < 2Cx – x_{0}^{H }.
Conversely, let suppose f (x) – f (x_{0}) < K x – x_{0}^{H }for all x, and y_{1} such as f (y_{1}) = sup_{x–z<a}f (z) and y_{2} such as f (y_{2}) =inf_{x–z<a}f (z); One has then
that means
ΔT_{f} (a, x) < K(y_{1} – x^{H }+ x – x_{0}^{H }+ y_{2} – x^{H }+ x – x_{0}^{H }),
this yields to
ΔT_{f} (a, x) < 2K(a^{H }+ x – x_{0}^{H}).

For analysing functions h_{α}, α > 1, we assume first that f verifies (3.21). Let's consider x in ^{n }and the two cases, f (x) > f (x_{0}),and f (x) < f (x_{0}). In the first case, one uses the reconstruction equations
and
For the second case, one uses a symmetric reconstruction method. This yields to
which gives
This implies that it exists a constant K such as
f (x) – f (x_{0}) < Kx – x_{0}^{H }.
Conversely, let suppose that f (x) – f (x_{0}) < Kx – x_{0}^{H }for all x. With mean of (2.8) and (2.11), one has
Since
f (x) – f (y) = f (x) – f (x_{0}) + f (x_{0}) – f (y),
one deduces
which yields to
whose optimisation gives (3.21). □
One gets here a reciprocal relation which is not fully obtained with linear wavelets [15].
4 HÖLDER EXPONENTS CALCULATION FOR WEIERSTRASS FUNCTIONS
We exhibit an application of the (min, +)wavelets analysis to the Weierstrass function in order to compute its Hölder exponent H and its fractal dimension D. This one is a typical example of function continuous everywhere but nowhere differentiable [22]. One consider the general form of Weierstrass functions on [0, 2π]
with ω^{H }> 1and {φ_{m}}_{m>0}, constant or randomly distributed variable.
Those functions are Hölderian (and antiHölderian) with coefficient H and fractal dimension [16, 14]:
One calculates for all scales s = k · scale_{min} with k an integer from 1 to 10 and scale_{min} =10^{–2}, the following function of scale for h_{2} and h_{∞}
For the Weierstrass function, the upper bound of the sum is replaced with a finite constant M =15 which is sufficient for our tests. Thus, the truncated Weierstrass function can be written as
and is represented with its (min, +)wavelets decomposition on Figure 1 for φ_{m} = 0.
We made numerical calculations to determine Hölder exponents. The fractal dimension is then directly given by equation (4.25). Computations were performed for H ∈ , ω =2, for analysing functions h_{∞} and h_{2}, for both cases of zero and random φ_{m} with a uniform probability measure in [0, 2π].
The slope of the linear part of curves for small scales gives the value of Hölder exponent.
Hölder exponents calculations for random phase Weierstrass functions are summarized on Tables 1 and 2. According to equations (3.17, 3.18), the slopes and Hölder exponents are very close to the theoretical value H = for h_{∞} and for h_{2} [21, 14]. The fractal dimension is then given by D = 2 –H = . Same result for H = with a slope of H = for h_{∞} and for h_{2}. They confirm that the Hölder exponents and fractal dimensions of Weierstrass function remain the same in the case of a uniform random phase [16, 14].
The Weierstrass function W(t) = (ω^{–H})^{m }cos(ω^{m}t), verifies
which is not a scaling invariance property [21]. In order to circumvent that, one can build the WeierstrassMandelbrot function
Since
the change of variable m' = m + 1 leads to WM(ωt) = ω^{H}WM(t), which has scaling invariance property. Hölder exponents calculations for a truncated version of this function are exhibited on Figures 2 and 3, confirming thus the validity of (min, +)wavelets decomposition for its Hölder exponents computation.
5 CONCLUSION AND PERSPECTIVES
We have presented in this article a promising tool to determine numerically Hölder exponents of Weierstrasslike functions which are exhibiting fractal properties. It is based on (min, +) analysis and proposes a signal decomposition using the (min, +) scalar product. By analogy with Linear Wavelet Theory, this permits to define (min, +)wavelets , which are lower and upper hulls of a signal at a certain scale.
ACKNOWLEDGEMENTS
We thank Mikhail Altaisky, Thierry Lehner and René Voltz for useful and helpful discussions about turbulence and wavelets.
REFERENCES
[1] A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray & J.F. Muzy. Ondelettes, multifractales et turbulence. Diderot, Paris, (1995). [ Links ]
[2] M.V. Altaisky. Wavelets, Theory, Applications and Implementation. Universities Press, (2005). [ Links ]
[3] M.V. Altaisky. Multiscale theory of turbulence in wavelet representation. Number 410 (3). Doklady Akademii Nauk, (2006). [ Links ]
[4] N.M. Astaf'eva. Wavelet analysis: basic theory and some applications. Uspekhi fizicheskih nauk, 166(11) (1994), 1146–1170, (in Russian). [ Links ]
[5] C.K. Chui. An Introduction to Wavelets. Academic Press Inc., (1992). [ Links ]
[6] I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure. Apl. Math., 41 (1988), 909–996. [ Links ]
[7] I. Daubechies. Ten lectures on wavelets. S.I.A.M., Philadelphie, (1992). [ Links ]
[8] M. Farge. Wavelets and their application to turbulence. Annual review of fluid mechanics, 24 (1992), 395–457. [ Links ]
[9] D. Gabor. Theory of communication. Proc. IEE, 93 (1946), 429–457. [ Links ]
[10] P. Goupillaud, A. Grossmann & J. Morlet. Cycleoctave and related transforms in seismic signal analysis. Geoexploration, 23 (1984/85), 85–102. [ Links ]
[11] A. Grossmann & J. Morlet. Decomposition of Hardy functions into squareintegrable wavelets of constant shape. SIAM J. Math. Anal., 15(4) (1984), 723–736. [ Links ]
[12] M. Gondran. Analyse minplus. C.R. Acad. Sci. Paris, 323 (1996), 371–375. [ Links ]
[13] M. Gondran. Convergences de fonctions à valeurs dans Rk et analyse minplus complexe. C.R. Acad. Sci. Paris, 329 (1999), 783–788. [ Links ]
[14] B.R. Hunt. The hausdorff dimension of graphs of weierstrass functions. Proceedings of the American Mathematical Society, 126 (1998), 791–800. [ Links ]
[15] S. Jaffard. Décompositions en ondelettes, "developments of mathematics 19502000" (JP. Pier ed.). pages 609–634, (2000). [ Links ]
[16] J. MalletParet J.L. Kaplan & J.A. Yorke. The lyapounov dimension of nowhere differentianle attracting torus. Ergod. Th. Dynam. Sys., 4 (1984), 261–281. [ Links ]
[17] J. Morlet, G. Arens & I. Fourgeau. Wave propagation and sampling theory. Geophysics, 47 (1982), 203–226. [ Links ]
[18] M. Minoux & M. Gondran. Graphs, Dioids and Semirings. Springer, (2008). [ Links ]
[19] J. Morlet. Sampling theory and wave propagation. In: Proc. 51st Annu. Meet. Soc. Explor. Geophys., LosAngeles, (1981). [ Links ]
[20] K. Shneider & M. Farge. Wavelet approach for modelling and computing turbulence, volume 199805 of Advances in turbulence modelling. Von Karman Institute for Fluid Dynamics, Bruxelles, (1998). [ Links ]
[21] C. Tricot. Courbes et dimension fractale. SpringerVerlag, (1993). [ Links ]
[22] K. Weierstrass. Über continuirliche funktionen eines reellen arguments, die f¨ur keinen werth des letzteren einen bestimmten differentialquotienten besitzen. Karl Weiertrass Mathematische Werke, Abhandlungen II, Gelesen in der Königl. Akademie der Wissenchaften am 18 Juli 1872, 1967. [ Links ]
[23] V.D. Zimin. Hierarchic turbulence model. Izv. Atmos. Ocean. Phys., 17 (1981), 941–946. [ Links ]
Received on November 9, 2013
Accepted on August 29, 2014
* Corresponding author: Abdel Kenoufi