Regularity of functions: Genericity and multifractal analysis

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Regularity of functions:

Genericity and multifractal analysis

Dissertation presented by

Céline ESSER

for the degree of Doctor in Sciences

University of Liège – Institute of Mathematics

Liège – October 22, 2014

Advisor: Françoise BASTIN(University of Liège)

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Introduction Weierstraß function. W (x) := +∞ X n=0 ancos(bnπx), a_{∈ (0, 1), ab > 1.} 1 −2 −1 0 1 2 1

Figure:Weierstraß function fora = 0.5andb = 3

Two questions.

• Are theremany such functions? Or is this example atypical?

−→Notions of genericity

• Is it possible tocharacterize the local behaviorof such functions?

−→Hölder exponent and multifractal analysis

Content of the presentation.

1. Notions of genericity

a) Residuality, prevalence and lineability b) Denjoy-Carleman classes

2. Multifractal analysis

a) Hölder regularity and multifractal spectrum b) Multifractal formalism

c) Leaders profile method d) Lν

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Introduction Weierstraß function. W (x) := +∞ X n=0 ancos(bnπx), a_{∈ (0, 1), ab > 1.} Two questions.

spaces

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spaces

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spaces

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Notions of genericity Residuality, prevalence and lineability

Notions of genericity

• Residuality. LetX be a Baire space. A subsetM ofXisresidualinX ifM

contains a countable intersection of dense open sets inX.

• Prevalence (Christensen, 1974 / Hunt, Sauer, Yorke, 1992). LetXbe a

complete metrizable vector space. A Borel subsetM ofXisshyif there exists a

Borel measureµonX with compact support such that

µ(M + x) = 0, x_{∈ X.}

More generally, a subsetV is called shy if it is contained in a shy Borel set. The

complement of a shy set is called aprevalentset.

• Lineability (Aron, Gurariy, Seoane-Sepúlveda, 2005). LetXbe a topological

vector space andµa cardinal number. A subsetM ofXis(dense-)lineableif

M_{∪ {0}}contains an infinite dimensional vector subspace (dense) inX. If the

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Notions of genericity

µ(M + x) = 0, x_{∈ X.}

vector space andµa cardinal number. A subsetM ofXis(dense-)lineableif

dimension of this subspace isµ,M is said to beµ-(dense-)lineable.

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Notions of genericity

µ(M + x) = 0, x_{∈ X.}

vector space andµa cardinal number. A subsetM ofX is(dense-)lineableif

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Notions of genericity Denjoy-Carleman classes

Existence of nowhere analytic functions. An example was given by Cellérier (1890)

by the function f (x) := +∞ X n=1 sin(an_x) n! , x∈ R

whereais a positive integer larger than1.

Results.

• Genericity of the set of nowhere analytic functions inC∞([0, 1]).

• Extension of these results using Gevrey classes.

Question. Similar results in the context of classes of ultradifferentiable functions?

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Results.

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Results.

Question. Similar results in the context of classes of ultradifferentiable functions?

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Denjoy-Carleman classes

An arbitrary sequence of positive real numbersM = (Mk)k∈N0 is called aweight

sequence.

Definition

LetΩbe an open subset of_RandM be a weight sequence. The space_E{M }(Ω)is

defined by

E{M }(Ω) :=f ∈ C∞(Ω) :∀K ⊆ Ωcompact∃h > 0such thatkfkMK,h< +∞ ,

where kfkM K,h:= sup k∈N0 sup x∈K |Dk_{f (x)} | hk_M k .

Iff _{∈ E}{M }(Ω), we say thatf isM-ultradifferentiable of Roumieu typeonΩ.

Particular case. The weight sequences(k!)_k∈N0and((k!)α₎

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Denjoy-Carleman classes

sequence.

Definition

defined by

k∈N0 withα > 1.

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Denjoy-Carleman classes

sequence.

Definition

defined by

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Definition

LetΩbe an open subset of_RandM be a weight sequence. The space_E(M )(Ω)is

defined by

E(M )(Ω) :=f ∈ C∞(Ω) :∀K ⊆ Ωcompact,∀h > 0, kfkMK,h< +∞ .

Iff _{∈ E}(M )(Ω), we say thatf isM-ultradifferentiable of Beurling typeonΩand we

use the representation

E(M )(Ω) = proj ←−−− K⊆Ω proj ←−− h>0 EM,h(K)

to endow_E(M )(Ω)with a structure of Fréchet space.

Questions.

• When do we have_E_{{M }}(Ω)_{⊆ E}(N )(Ω)?

• In that case, “how small” is_E{M }(Ω)inE(N )(Ω)?

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Definition

LetΩbe an open subset of_RandM be a weight sequence. The space_E(M )(Ω)is

defined by

E(M )(Ω) :=f ∈ C∞(Ω) :∀K ⊆ Ωcompact,∀h > 0, kfkMK,h< +∞ .

Iff _{∈ E}(M )(Ω), we say thatf isM-ultradifferentiable of Beurling typeonΩand we

use the representation

E(M )(Ω) = proj ←−−− K⊆Ω proj ←−− h>0 EM,h(K)

to endow_E(M )(Ω)with a structure of Fréchet space.

Questions.

• When do we have_E_{{M }}(Ω)_{⊆ E}(N )(Ω)?

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General assumptions.

• We assume that any weight sequenceM is logarithmically convex, i.e.

M_k2_{≤ M}k−1Mk+1, ∀k ∈ N .

It implies that the space_E{M }(Ω)is an algebra.

• We assume that any weight sequenceM is such thatM0= 1.

• We usually assume that any weight sequenceM is non-quasianalytic, i.e.

+∞

X

k=1

(Mk)−1/k< +∞.

By Denjoy-Carleman theorem, it implies that there exists non-zero functions with

compact support in_E_{{M }}_(R).

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M_k2_{≤ M}k−1Mk+1, ∀k ∈ N .

+∞

X

k=1

(Mk)−1/k< +∞.

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M_k2_{≤ M}k−1Mk+1, ∀k ∈ N .

+∞

X

k=1

(Mk)−1/k< +∞.

compact support in_E_{{M }}_(R).

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Inclusions between Denjoy-Carleman classes

Notation. Given two weight sequencesM andN, we write

M_{N ⇐⇒} lim k→+∞ Mk Nk 1k = 0.

Proposition

LetM, N be two weight sequences and letΩbe an open subset of_R. Then

M_{N ⇐⇒ E}{M }(Ω)⊆ E(N )(Ω)

and in this case, the inclusion is strict.

Keys.

• IfM _N, then there exists a weight sequenceLsuch thatM_{L N}.

• There existsθ_{∈ E}{M }(R)such that|Dkθ(0)| ≥ Mkfor allk∈ N0. In particular,

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Inclusions between Denjoy-Carleman classes

M_{N ⇐⇒} lim k→+∞ Mk Nk 1k = 0.

Proposition

M_{N ⇐⇒ E}{M }(Ω)⊆ E(N )(Ω)

Keys.

• IfM _N, then there exists a weight sequenceLsuch thatM_{L N}.

this function does not belong to_E(M )(R).

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Inclusions between Denjoy-Carleman classes

M_{N ⇐⇒} lim k→+∞ Mk Nk 1k = 0.

Proposition

M_{N ⇐⇒ E}{M }(Ω)⊆ E(N )(Ω)

Keys.

• IfM_N, then there exists a weight sequenceLsuch thatM_{L N}.

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Construction

Definition

We say that a function isnowhere in_E{M }if its restriction to any open and non-empty

subsetΩof_Rnever belongs to_E_{{M }}(Ω).

Proposition

Assume thatM andNare two weight sequences such thatM _N. IfM is

non-quasianalytic, there exists a function of_E(N )(R)which is nowhere inE{M }.

Idea. Construct a sequence(L(p)₎

p∈Nof weight sequences such that

M_L(1)_L(2)_{· · · L}(p)_{· · · N.}

For everyp_{∈ N}, consider a functionfp∈ E{L(p)_}(R)such that|Dkfp(0)| ≥ L

(p) k ,

∀k ∈ N0.If{xp: p∈ N}is a dense subset ofR, consider f (x) =

+∞

X

p=1

fp(x− xp)Φp(x), x∈ R

whereΦpis a compactly supported function well chosen.

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Construction

Definition

Proposition

Assume thatM andN are two weight sequences such thatM _N. IfM is

M_L(1)_L(2)_{· · · L}(p)_{· · · N.}

(p) k ,

+∞

X

p=1

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Construction

Definition

Proposition

M_L(1)_L(2)_{· · · L}(p)_{· · · N.}

(p) k ,

+∞

X

p=1

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Construction

Definition

Proposition

M_L(1)_L(2)_{· · · L}(p)_{· · · N.}

(p) k ,

∀k ∈ N0.

If_{xp: p∈ N}is a dense subset ofR, consider

f (x) =

+∞

X

p=1

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Construction

Definition

Proposition

M_L(1)_L(2)_{· · · L}(p)_{· · · N.}

(p) k ,

+∞

X

p=1

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Generic results

Proposition

Assume thatN andM are two weight sequences such thatM _N. IfM is non

quasianalytic, the set of functions of_E(N )(R)which are nowhere inE{M }is

• prevalent,

• residual,

• c-dense-lineable.

More with countable unions

LetN be a weight sequence and let(M(n)₎

n∈Nbe a sequence of weight sequences

such thatM(n)_N _{for every}_n

∈ N. If there isn0∈ Nsuch that the weight

sequenceM(n0)_{is non quasianalytic, the set of functions of}

E(N )(R)which are

nowhere inS_n∈N_E_{M(n)_}is prevalent, residual andc-dense-lineable inE_{(N )}(R).

Idea. Construct a weight sequenceP such that

[

n∈N

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Generic results

Proposition

• prevalent,

• residual,

Idea. The set of functions of_E(N )(R)which are nowhere inE{M }is the complement of

[ I⊆R [ m∈N [ s∈N f _{∈ E}(N )(R) : sup x∈I|D k_{f (x)} | ≤ smk_M k, ∀k ∈ N0 | {z }

closed set with empty interior

| {z }

proper linear subspace ofE(N )(R)

.

More with countable unions

E(N )(R)which are

nowhere inS

n∈NE{M(n)_}is prevalent, residual andc-dense-lineable inE(N )(R).

[

n∈N

E{M(n)_}(Ω)⊆ E_{{P }}(Ω) (E(N )(Ω).

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Generic results

Proposition

• prevalent,

• residual,

More with countable unions

E(N )(R)which are

nowhere inS_n∈N_E_{M(n)_}is prevalent, residual andc-dense-lineable inE_{(N )}(R).

[

n∈N

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Generic results

Proposition

• prevalent,

• residual,

Idea. Construct for everyt_{∈ (0, 1)}a weight sequenceL(t)_{such that}

M _L(t)_N and L(t)_L(s)ift < s.

Then, we have for everyt_{∈ (0, 1)}

M_L(t2) L(2t3) L(3t4) · · · L(t) N

and we construct as before a function of_E(N )(R)which is nowhere inE{M }.

More with countable unions

LetN be a weight sequence and let(M(n))_n∈Nbe a sequence of weight sequences

such thatM(n)_N for everyn_{∈ N}. If there isn0∈ Nsuch that the weight

sequenceM(n0)_{is non quasianalytic, the set of functions of}E

(N )(R)which are

nowhere inS

n∈NE{M(n)_}is prevalent, residual andc-dense-lineable inE(N )(R).

[

n∈N

E{M(n)_}(Ω)⊆ E{P }(Ω) (E(N )(Ω).

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Generic results

Proposition

• prevalent,

• residual,

More with countable unions

such thatM(n)_N_{for every}_n

E(N )(R)which are

nowhere inS_n∈N_E_{M(n)_}is prevalent, residual andc-dense-lineable inE(N )(R).

[

n∈N

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An important example of ultradifferentiable functions of Roumieu type is given by the

classes ofGevrey differentiable functionsof orderα > 1. They correspond to the

weight sequences

Mk := (k!)α, k∈ N0.

Particular case of Gevrey classes

Letα > 1. The set of functions of_E((k!)α₎(R)which are nowhere inE_{(k!)β_}for every

β_{∈ (1, α)}, is prevalent, residual andc-dense-lineable in_E((k!)α₎(R).

It suffices to take the weight sequencesM(n)₍_n

∈ N) given by

M_k(n):= (k!)βn, k_{∈ N}0,

where(βn)n∈Nis an increasing sequence of(1, α)that converges toα.

Proposition (Schmets, Valdivia, 1991)

β_{∈ (1, α)}is residual in_E((k!)α₎(R).

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weight sequences

Mk := (k!)α, k∈ N0.

Particular case of Gevrey classes

∈ N) given by

M_k(n):= (k!)βn, k_{∈ N}0,

Proposition (Schmets, Valdivia, 1991)

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weight sequences

Mk := (k!)α, k∈ N0.

Particular case of Gevrey classes

∈ N) given by

M_k(n):= (k!)βn, k_{∈ N}0,

Proposition (Schmets, Valdivia, 1991)

β_{∈ (1, α)}is residual in_E((k!)α₎(R).

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Other results.

• Similar results have been obtained with classes of ultradifferentiable functions

defined using weight functions and weight matrices.

Perspectives.

• What about the algebrability?

• Other notions of genericity (such as porosity)?

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Multifractal analysis

c) Leaders profile method d) Lν_spaces

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Multifractal analysis Hölder regularity and multifractal spectrum

Hölder regularity and multifractal spectrum

Recall. Is it possible to characterize the local regularity of an irregular function?

1 -100 -80 -60 -40 -20 0 20 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 1 -25 -20 -15 -10 -5 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 -500 0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Hölder regularity and multifractal spectrum

Definition

Let_{f : R}_{→ R}be a locally bounded function,α_{≥ 0}andx_{∈ R}. The functionf

belongs to theHölder spaceCα_(x)_{if there exist a constant}_{C > 0}_{and a polynomial}

Pof degree strictly smaller thanαsuch that

|f(y) − P (y)| ≤ C|y − x|α

for allyin a neighborhood ofx. Then, theHölder exponenthf(x)offatxis defined

by

hf(x) := sup{α ≥ 0 : f ∈ Cα(x)}.

Weierstraß function.hf(x) =−log a_{log b}, ∀x ∈ R.

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Hölder regularity and multifractal spectrum

Definition

Let_{f : R}_{→ R}be a locally bounded function,α_{≥ 0}andx_{∈ R}. The functionf

belongs to theHölder spaceCα_(x)_{if there exist a constant}_{C > 0}_{and a polynomial}

Pof degree strictly smaller thanαsuch that

|f(y) − P (y)| ≤ C|y − x|α

for allyin a neighborhood ofx. Then, theHölder exponenthf(x)offatxis defined

by

hf(x) := sup{α ≥ 0 : f ∈ Cα(x)}.

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• Sincehf(x)can change widely from a point to another, we will characterize the

size of the sets of points which have the same local regularity. • Theiso-Hölder setsoff areEh:={x ∈ R : hf(x) = h}.

Definition

Themultifractal spectrumdf off is defined by

df(h) := dimHEh, ∀h ∈ [0, +∞],

with the convention thatdimH∅ = −∞.

−→ dfgives a geometrical idea about the distribution of the singularities off

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• Sincehf(x)can change widely from a point to another, we will characterize the

size of the sets of points which have the same local regularity. • Theiso-Hölder setsoff areEh:={x ∈ R : hf(x) = h}.

Definition

Themultifractal spectrumdf offis defined by

df(h) := dimHEh, ∀h ∈ [0, +∞],

with the convention thatdimH∅ = −∞.

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Examples

Riemann function 1 0 1

0− log2(1− p1)− log2(1− p2)1 − log2(p2) 2 − log2(p1)

Sum of two cascades

1 0 1 0 − log2(1− p)1 − log2(p) 2 Cascade 1 0 1

0 − log2(1− p) 1 γ− log2(p) htmax

Threshold of a cascade

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Examples

Riemann function 1 0 1

Sum of two cascades

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Examples

Riemann function 1

0 1

Sum of two cascades

Threshold of a cascade

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Examples

Riemann function 1

0 1

Sum of two cascades

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Multifractal analysis Multifractal formalism

Multifractal formalism

Amultifractal formalismis a method which is expected to give the multifractal spectrum of a function, from “global” quantities which are numerically computable.

Several multifractal formalisms based on a decomposition off _{∈ L}2_{([0, 1])}_{in a}

wavelet basis f = X j∈N0 2j₋₁ X k=0 cj,kψj,k + C

have been proposed to estimatedf, where the mother waveletψbelongs toS(R).

Characterization of the Hölder exponent using wavelet coefficients

Iff is uniformly Hölder, the Hölder exponent off atxis

hf(x) = lim inf

j→+∞k∈{0,...,2infj_−1}

log(_|cj,k|)

log(2−j₊_|k2−j_{− x|)}.

Advantage. Easy to compute and relatively stable from a numerical point of view.

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Multifractal formalism

Characterization of the Hölder exponent using wavelet coefficients

hf(x) = lim inf

j→+∞k∈{0,...,2infj_−1}

log(_|cj,k|)

log(2−j₊_|k2−j_{− x|)}.

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Multifractal formalism

Characterization of the Hölder exponent using wavelet coefficients

hf(x) = lim inf

j→+∞k∈{0,...,2infj_−1}

log(_|cj,k|)

log(2−j₊_|k2−j_{− x|)}.

Advantage. Easy to compute and relatively stable from a numerical point of view.

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• The Frisch-Parisi formalism (1985) and the classical use of Besov spaces lead to

a loss of information (only concave hull and increasing part of spectra can be recovered).

• Wavelet leaders method (S. Jaffard, 2004): Modification of the Frisch-Parisi

formalism using the wavelet leaders of the function instead of wavelet coefficients.

−→Detection of increasing and decreasing parts of concave spectra.

• Introduction of spaces of type_Sν (J.M. Aubry, S. Jaffard, 2005), based on

histograms of wavelet coefficients.

−→Detection of concave and non-concave parts of increasing spectra.

• Combination of the two previous methods to obtain theleaders profile method

and the spaces of type_Lν.

−→Detection of increasing and decreasing parts of concave and non-concave

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spectra.

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formalism using thewavelet leadersof the function instead of wavelet coefficients.

spectra.

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histogramsof wavelet coefficients.

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histogramsof wavelet coefficients.

spectra.

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Wavelet leaders

Standard notation. Forj_{∈ N}0, k∈0, . . . , 2j− 1 ,

λ(j, k) :=_{x ∈ R : 2}jx_{− k ∈ [0, 1[} = k 2j, k + 1 2j ,

and for allj_{∈ N}0,Λjdenotes the set of all dyadic intervals (of[0, 1)) of length2−j.

Ifλ = λ(j, k), we use both notationscj,korcλto denote the wavelet coefficients.

Definition

Thewavelet leadersof a functionf _{∈ L}2_{([0, 1])}_{are defined by}

dλ:= sup λ0_⊆3λ|cλ

0|, λ ∈ Λ_j, j∈ N₀.

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Wavelet leaders

Standard notation. Forj_{∈ N}0, k∈0, . . . , 2j− 1 ,

λ(j, k) :=_{x ∈ R : 2}jx_{− k ∈ [0, 1[} = k 2j, k + 1 2j ,

and for allj_{∈ N}0,Λjdenotes the set of all dyadic intervals (of[0, 1)) of length2−j.

Ifλ = λ(j, k), we use both notationscj,korcλto denote the wavelet coefficients.

Definition

Thewavelet leadersof a functionf _{∈ L}2_{([0, 1])}_{are defined by}

dλ:= sup λ0_⊆3λ|cλ

0|, λ ∈ Λ_j, j∈ N₀.

−→their decay properties are directly related with the Hölder exponent.

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Ifx_{∈ [0, 1)}, letλj(x)denote the dyadic interval of length2−j which containsx.

Hölder regularity and wavelet leaders

Iff is uniformly Hölder, the Hölder exponent off atxis given by

hf(x) = lim inf j→+∞ log dλj(x) log 2−j . Interpretation. dλj(x)∼ 2 −hf(x)j

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Hölder regularity and wavelet leaders

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Hölder regularity and wavelet leaders

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Hölder regularity and wavelet leaders

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Method based on

S

ν

spaces

Thewavelet profileνf of a locally bounded functionfis defined for everyh≥ 0by

νf(h) := lim

ε→0+lim sup_j→+∞

log #_{λ ∈ Λ}j : |cλ| ≥ 2−(h+ε)j

log 2j .

Interpretation.

• There are approximatively2νf(h)j _{coefficients greater in modulus than}₂−hj_.

Properties.

• νf is a right-continuous increasing function.

• νf is independent of the chosen wavelet basis.

• Iff is uniformly Hölder, df(h)≤ dνf(h) := min ( h sup h0_∈(0,h] νf(h0) h0 , 1 ) , _{∀h ≥ 0.}

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Method based on

S

ν

spaces

Thewavelet profileνf of a locally bounded functionfis defined for everyh≥ 0by

νf(h) := lim

log #_{λ ∈ Λ}j : |cλ| ≥ 2−(h+ε)j

log 2j .

Interpretation.

• There are approximatively2νf(h)j _{coefficients greater in modulus than}₂−hj_.

Properties.

• νf is a right-continuous increasing function.

• νf is independent of the chosen wavelet basis.

• Iff is uniformly Hölder, df(h)≤ dνf(h) := min ( h sup h0_∈(0,h] νf(h0) h0 , 1 ) , _{∀h ≥ 0.}

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Definition

Take0_{≤ a < b ≤ +∞}. A functiong : [a, b]_{7→ [0, +∞)}is withincreasing-visibilityifg

is continuous ataandsupy∈(a,x]

g(y)

y ≤

g(x)

x for allx∈ (a, b].

In other words, a functiongis with increasing-visibility if for allx_{∈ (a, b]}, the segment

[(0, 0), (x, g(x))]lies above the graph ofgon(a, x]. 1

0 1

0

Example ofνf (---) anddνf (—)

−→The passage fromνftodνf transforms the functionνfinto a function with

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Definition

g(y)

y ≤

g(x)

0 1

0

increasing-visibility.

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Definition

g(y)

y ≤

g(x)

0 1

0

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Definition

g(y)

y ≤

g(x)

0 1

0

increasing-visibility.

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Definition

g(y)

y ≤

g(x)

0 1

0 hmax

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Particular case

Assumption. Assume that the wavelet coefficients off are given bycλ= µ(λ)where

µis a finite Borel measure on[0, 1].

Notation. Letfβdenote the function with wavelet coefficients given byc β

λ = 2−βjcλ.

In this case, one has

• dfβ(h) = df(h− β)for allh≥ β. • νfβ(h) = νf(h− β)for allh≥ β. Moreover, if inf νf(x)− νf(y) x_{− y} : x, y∈ [hmin, h 0 max], x < y > 0,

wherehmin= inf{α : νf(α)≥ 0},h0max= inf{α : νf(α) = 1}, then there exists

β > 0such that the functionνfβ is with increasing-visibility on[hmin, h0max]. In this

case,dνfβ _{= ν}

fβ approximatesdfβ. Therefore the increasing part ofdfcan be

approximated byνf.

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Particular case

λ = 2−βjcλ.

case,dνfβ _{= ν}

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Particular case

λ = 2−βjcλ.

case,dνfβ _{= ν}

approximated byνf.

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Multifractal analysis Leaders profile method

Wavelet leaders density

Thewavelet leaders densityoffis defined for everyh_{≥ 0}by

e

ρf(h) := lim

log #_{λ ∈ Λ}j: 2−(h+ε)j ≤ dλ< 2−(h−ε)j

log 2j .

Interpretation. There are approximatively2ρef(h)jcoefficients of size2−hj.

Heuristic argument. We consider the pointsxsuch thathf(x) = h.

• dλj(x)∼ 2−hjand there are about2eρf(h)j such dyadic intervals.

• If we cover each singularityxby dyadic intervals of size2−j, from the definition of

the Hausdorff dimension, there are about2df(h)j _{such intervals.}

=_⇒ρ_ef(h) = df(h)

Problems.

• The wavelet leaders density may depend on the chosen wavelet basis.

• The definition of the wavelet leaders density is numerically extremely unstable.

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Wavelet leaders density

e

ρf(h) := lim

log #_{λ ∈ Λ}j: 2−(h+ε)j ≤ dλ< 2−(h−ε)j

log 2j .

=_⇒ρ_ef(h) = df(h)

Problems.

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Wavelet leaders density

e

ρf(h) := lim

log #_{λ ∈ Λ}j: 2−(h+ε)j ≤ dλ< 2−(h−ε)j

log 2j .

=_⇒ρ_ef(h)≥df(h)

Problems.

• The definition of the wavelet leaders density is numerically extremely unstable.

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Wavelet leaders density

e

ρf(h) := lim

log #_{λ ∈ Λ}j: 2−(h+ε)j ≤ dλ< 2−(h−ε)j

log 2j .

=_⇒ρ_ef(h)≥df(h)

Problems.

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Wavelet leaders profile

Lethsbe the smallest positive real number such thatρef(hs) = 1. Thewavelet leaders

profileoff is defined by e νf(h) :=            lim ε→0+lim sup_j→+∞ log #_{λ ∈ Λ}j : dλ≥ 2−(h+ε)j log 2j ifh≤ hs, lim ε→0+lim sup_j→+∞ log #_{λ ∈ Λ}j : dλ≤ 2−(h−ε)j log 2j ifh≥ hs. Properties. • e

νf is independent of the chosen wavelet basis.

•

e

νf takes values in{−∞} ∪ [0, 1], it is increasing and right-continuous on[0, hs],

decreasing and left-continuous on[hs, +∞),_eνf(hs) = 1and the function

h_{∈ [h}s, +∞) 7→ e

νf(h)− 1

h

is decreasing.

• Moreover, any functionνwhich satisfies these properties is the wavelet leaders

profile of a function.

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Wavelet leaders profile

Lethsbe the smallest positive real number such thatρef(hs) = 1. Thewavelet leaders

profileoff is defined by e νf(h) :=            lim ε→0+lim sup_j→+∞ log #_{λ ∈ Λ}j : dλ≥ 2−(h+ε)j log 2j ifh≤ hs, lim ε→0+lim sup_j→+∞ log #_{λ ∈ Λ}j : dλ≤ 2−(h−ε)j log 2j ifh≥ hs. Properties. • e

νf is independent of the chosen wavelet basis.

•

e

νf takes values in{−∞} ∪ [0, 1], it is increasing and right-continuous on[0, hs],

decreasing and left-continuous on[hs, +∞),_eνf(hs) = 1and the function

h_{∈ [h}s, +∞) 7→ e

νf(h)− 1

h

is decreasing.

• Moreover, any functionνwhich satisfies these properties is the wavelet leaders

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Leaders profile method. It is based on the estimation of the multifractal spectrumdf

off by the function_eνf.

Results.

• Our method allows to detect some multifractal spectra thatall other methods

proposed were not able to detect;

• It gives the correct multifractal spectrum for somespecific functions;

• It always gives anupper boundfor the multifractal spectrum;

• From a theoretical point of view, it gives as good results as the wavelet leaders

method in the concave case, andbetter results in the non-concave case;

• From a theoretical point of view, it gives better results than the method based on

the_Sν spaces and in particular, it allows to detect spectra which arenot with

increasing visibility.

• Animplementationof this method has been proposed and tested on several examples.

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Results.

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Results.

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Results.

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Results.

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Results.

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Results.

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Multifractal analysis Lνspaces

L

ν

spaces

Letνbe a function which has the same properties as any wavelet leaders profile.

Definition

Thespace_Lνis the set of functionsf _{∈ L}2_{([0, 1])}_{such that}

e νf ≤ ν.

This space has been endowed with a complete metrizable topology.

Results. If there isαmin> 0such thatν(α) =−∞ifα < αmin, then

• Lνis also separable;

• The set of functionsf such that_eνf = νis residual and dense-lineable inLν.

Perspectives.

• Generic validity of the leaders profile method;

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L

ν

spaces

Definition

e νf ≤ ν.

Perspectives.

• More with oscillating singularities.

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L

ν

spaces

Definition

e νf ≤ ν.

Perspectives.