#  Slow, ordinary and rapid points for Gaussian Wavelets Series and
application to Fractional Brownian Motions

C. Esser111 Université de Liège, Département de mathématique – zone Polytech
1, 12 allée de la Découverte, Bât. B37, B-4000 Liège<EMAIL_ADDRESS>and L. Loosveldt222Corresponding author at: Université du Luxembourg, UR en
Mathématiques, Maison du nombre, 6 avenue de la Fonte, L-4364, Esch-sur-
Alzette, Luxembourg<EMAIL_ADDRESS>
###### Abstract

We study the Hölderian regularity of Gaussian wavelets series and show that
they display, almost surely, three types of points: slow, ordinary and rapid.
In particular, this fact holds for the Fractional Brownian Motion. We also
show that this property is satisfied for a multifractal extension of Gaussian
wavelet series. Finally, we remark that the existence of slow points is
specific to these functions.

Keywords: Random Wavelets Series, Fractional Brownian motion, modulus of
continuity, slow/ordinary/rapid points

2020 MSC: 42C40, 26A16, 60G15, 60G22, 60G17

## Introduction

Let $B$ denote the standard Brownian motion on $\mathbb{R}$. The Khinchin law
of the iterated logarithm [23] allows to control the behavior of $B$ at a
given point, in the sense that for every $t\in\mathbb{R}$, it holds

$\limsup_{r\to 0}\frac{|B(t+r)-B(t)|}{\sqrt{|r|\log\log|r|^{-1}}}=\sqrt{2}$
(1)

on an event of probability one. As a direct application of Fubini’s theorem,
one obtains that almost surely, the set of points $t\in\mathbb{R}$ such that
(1) holds, called ordinary points, has full Lebesgue measure. This contrasts
with the uniform Hölder condition obtained by Paul Lévy in 1937 which states
that the uniform modulus of continuity of $B$ is of larger order: almost
surely, one has

$\limsup_{r\to
0}\sup_{t\in[0,1]}\frac{|B(t+r)-B(t)|}{\sqrt{|r|\log|r|^{-1}}}=\sqrt{2}.$

In particular, there exist exceptional points, called fast points, where the
law of the iterated logarithm fails. In 1974, Oray and Taylor studied how
often this exceptional behavior holds and proved especially that the Hausdorff
dimension of the set

$\left\\{t\in[0,1]:\limsup_{r\to
0}\frac{|B(t+r)-B(t)|}{\sqrt{|r|\log|r|^{-1}}}\geq\lambda\sqrt{2}\right\\}$

is given almost surely by $1-\lambda^{2}$ for every $\lambda\in[0,1]$, see
[35].

In the meanwhile, Kahane proposed in [22] an easy way to study the regularity
and irregularity properties of the Brownian motion. Its method relies on the
expansion of $B$ on $[0,1]$ in the so-called Faber-Schauder system. If
$\Lambda$ is the triangular function

$\Lambda\,:\,x\mapsto\left\\{\begin{array}[]{cl}x&\text{ if $\frac{1}{2}\leq
x<1$}\\\\[8.61108pt] \ 1-x&\text{ if $0\leq x<\frac{1}{2}$}\\\\[8.61108pt] \
0&\text{ otherwise,}\end{array}\right.$

and $\xi$, $\xi_{j,k}$ ($j\in\mathbb{N}_{0}$ and $k\in\\{0,\ldots,2^{j}-1\\}$)
are independent $\mathcal{N}(0,1)$ random variables then, we have

$\displaystyle
B(t)=\sum_{j=0}^{+\infty}\sum_{k=0}^{2^{j}-1}\xi_{j,k}2^{-j/2}\Lambda(2^{j}t-k)+\xi
t$ (2)

where the convergence is almost surely uniform for $t\in[0,1]$. Working with
this expression, Kahane recovered the law of the iterated logarithm and the
estimation of the modulus of continuity of the Brownian motion. Furthermore,
Kahane obtained the existence of a third category of points, presenting a
slower oscillation. These points, called slow points, satisfy the condition

$\limsup_{r\to 0}\frac{|B(t+r)-B(t)|}{\sqrt{|r|}}<+\infty.$

The law of iterated logarithm and the study of the set of fast points has
naturally been studied and extended since then for more general classes of
gaussian processus such as gaussian processes with stationary increments, see
e.g. [29, 34, 9, 30, 33, 24]. In particular, given a fractional Brownian
motion $B_{h}$ of index $h\in(0,1)$, one has almost surely

$\limsup_{r\to
0}\frac{|B_{h}(t+r)-B_{h}(t)|}{r^{h}\sqrt{\log\log|r|^{-1}}}=\sqrt{2}$

and

$\limsup_{r\to
0}\sup_{t\in[0,1]}\frac{|B_{h}(t+r)-B_{h}(t)|}{r^{h}\sqrt{\log|r|^{-1}}}=\sqrt{2}.$

In 1999, Meyer, Sellan and Taqqu introduced a famous decomposition of the
fractional Brownian motion using a Lemarié-Meyer or sufficiently smooth
Daubechies wavelet $\psi$, which decorrelates the high frequencies [32]. More
precisely, any fractional Brownian motion $B_{h}$ of Hurst index $h\in(0,1)$
can be written as

$B_{h}(t)=\sum_{j\in\mathbb{N}}\sum_{k\in\mathbb{Z}}2^{-hj}\xi_{j,k}\psi_{h+1/2}(2^{j}t-k)+R(t)$
(3)

where $R$ is a smooth process,
$(\xi_{j,k})_{(j,k)\in\mathbb{N}\times\mathbb{Z}}$ is a sequence of
independent $\mathcal{N}(0,1)$ random variables, and $\psi_{\alpha}$ is
defined by

$\hat{\psi}_{\alpha}(\xi)=\frac{1}{|\xi|^{\alpha}}\hat{\psi}(\xi).$

Note that such a function leads to a biorthogonal wavelet basis, see Section
1.

Motivated by the study of fractional Brownian motions using this particular
decomposition, we develop in this paper a systematic study of the different
pointwise behaviors of random wavelets series of the form

$f_{h}=\sum_{j\in\mathbb{N}}\sum_{k\in\mathbb{Z}}\xi_{j,k}2^{-hj}\psi(2^{j}\cdot-k)$
(4)

where $(\xi_{j,k})_{(j,k)\in\mathbb{N}\times\mathbb{Z}}$ denote a sequence of
independent $\mathcal{N}(0,1)$ random variables, $h\in(0,1)$ is fixed and
$\psi$ is any compactly supported or smooth wavelet, see Section 1 for a
precise definition. Note that, even if the expression (4) is very similar to
(2), dealing with it requires much more technical arguments. Indeed, due to
the symmetry of the function $\Lambda$, most of the terms vanish in the
expansion of the increments $B(t)-B(s)$ if $s$ and $t$ are closed enough. This
fact can not be used anymore while working with an arbitrary wavelet and
compensations of different terms may occurs.

Concerning the regularity of the function $f_{h}$, one can show, see
Proposition 1.3 below, that, for all $s,t\in\mathbb{R}$,

$\mathbb{E}[(f_{h}(t)-f_{h}(s))^{2}]\leq C|s-t|^{2h},$

for some deterministic constant $C>0$. Applying Kolmogorov continuity Theorem
for gaussian processes, one can deduce that almost surely, for every
$t\in\mathbb{R}$, one has

$\limsup_{r\to 0}\frac{|f_{h}(t+r)-f_{h}(t)|}{r^{h-\varepsilon}}<+\infty$

for every $\varepsilon>0$. The aim of this paper is to characterize more
precisely the pointwise behavior of such a wavelet series, in the spirit of
what is known for the Brownian motion. Recently, generalized Hölder spaces has
been introduced to address this kind of questions [25, 27] as well as the
regularity of solutions of partial differential equations [26]. This article
is the continuation of the work done in [2, 4, 22] for the Brownian motion.

As a consequence of our results, we get that almost surely, the Hölder
exponent of the random wavelet series $f_{h}$ is $h$ while it does not belong
to the uniform Hölder space of order $h$. Nevertheless, if $t$ is a slow
point, $f_{h}$ belongs to the pointwise Hölder space of order $h$ at $t$. One
can therefore wonder if this feature is common or if it is specific to the
functions under study in this paper. To address this question, in Section 6 we
recall two commonly used notions of genericity: the prevalence and the Baire
category point of view. We obtain that, in both sense, the existence of slow
points is a specific property of gaussian random wavelets series.

The paper is structured as follows: in Section 1, before stating our main
result, we recall the most important tools used in the paper: discrete wavelet
transform and modulus of continuity as well as some fundamental inequalities.
Section 2 is devoted to exploring the regularity of the gaussian random
wavelet serie (4) by identifying three precise pointwise estimates. In Section
3, we focus on the irregularity that we deduced from the asymptotic behavior
of the wavelet coefficients. Then, using the obtain results of regularity and
irregularity, we prove our main result concerning the existence of slow,
ordinary and rapid points in Section 4. This fact is extended to multifractal
gaussian random series in Section 5. Finally, Section 6, focuses on the
results of generecity for slow points. In this paper, $C$ stands for a
deterministic constant not necessary the same in different lines.

##  1 Preliminaries and statement of the main result

In this section, we present the notions needed for the statement of the main
theorem and the first result concerning the convergence of the wavelets series
defined in (4).

Let us first briefly recall some definitions and notations about wavelets and
biortogonal wavelets (for more precisions, see e.g. [15, 31, 28, 13]). Under
some general assumptions, there exist two functions $\phi$ and $\psi$, called
wavelets, which generate two orthonormal bases of $L^{2}(\mathbb{R})$, namely

$\\{\phi(\cdot-k)\\}_{k\in\mathbb{Z}}\cup\\{\psi(2^{j}\cdot-k):j\in\mathbb{N},k\in\mathbb{Z}\\}$

and

$\\{\psi(2^{j}\cdot-k):j\in\mathbb{Z},k\in\mathbb{Z}\\}.$

Any function $f\in L^{2}(\mathbb{R})$ can be decomposed as follows,

$f=\sum_{k\in\mathbb{Z}}C_{k}\phi(\cdot-k)+\sum_{j\in\mathbb{N}}\sum_{k\in\mathbb{Z}}c_{j,k}\psi(2^{j}\cdot-k)=\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}c_{j,k}\psi(2^{j}\cdot-k)$

where

$c_{j,k}=2^{j}\int_{\mathbb{R}}f(x)\psi(2^{j}x-k)\,dx$

and

$C_{k}=\int_{\mathbb{R}^{n}}f(x)\phi(x-k)\,dx.$

Let us remark that we do not choose the $L^{2}(\mathbb{R})$ normalization for
the wavelets, but rather an $L^{\infty}(\mathbb{R})$ normalization, which is
better fitted to the study of the Hölderian regularity. Amoung the families of
wavelet basis that exist, we are mostly interested in two classes: The
Lemarié-Meyer wavlets for which $\phi$ and $\psi$ belong to the Schwartz class
$\mathcal{S}(\mathbb{R})$, or Daubechies wavelets for which $\phi$ and $\psi$
are compactly supported functions. In both cases, the first moment of the
wavelet $\psi$ vanishes.

The setting in which we work is more general that just orthogonal wavelet
basis, so that it allows to cover the important example supply by the
fractional Brownian motion. Biorthogonal wavelet bases are a couple of two
Riesz wavelet bases of $L^{2}(\mathbb{R})$ generated respectively by $\psi$
and $\widetilde{\psi}$ and such that

$2^{j/2}2^{j^{\prime}/2}\int_{\mathbb{R}}\psi(2^{j}x-k)\widetilde{\psi}(2^{j^{\prime}}x-k^{\prime})dx=\delta_{j,j^{\prime}}\delta_{k,k^{\prime}}.$

In that case, any function $f\in L^{2}(\mathbb{R})$ can be decomposed as

$f=\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}c_{j,k}\psi(2^{j}\cdot-k)$

where

$c_{j,k}=2^{j}\int_{\mathbb{R}}f(x)\widetilde{\psi}(2^{j}x-k)dx.$

Wavelet basis and biorthogonal wavelet basis give a powerful tool to study the
regularity and irregularity of functions or signals belonging to numerous
functional spaces, see e.g. [21, 31, 6, 19, 1, 20, 12, 17, 25, 27]. In this
paper, they will be used to get irregularity properties in Section 3.

Let us now present two lemmata that allow to prove the uniform convergence on
any compact set of the series defined in (4), where $\psi$ comes from any
wavelet basis or biorthogonal wavelet basis. The first one is very classical
and the second one gives informations about the asymptotic behavior of a
sequence of i.i.d gaussian random variables.

######  Lemma 1.1.

There exists a constant $C_{1}>0$ such that, for all $x\in\mathbb{R}^{d}$

$\sum_{k\in\mathbb{Z}}\frac{1}{(1+|x-k|)^{4}}\leq C_{1}.$

######  Lemma 1.2.

[3, 5] Let $(\xi_{j,k})_{(j,k)\in\mathbb{N}\times\mathbb{Z}}$ be a sequence of
independent $\mathcal{N}(0,1)$ random variables. There are an event
$\Omega^{*}$ of probability $1$ and a positive random variable $C_{2}$ of
finite moment of every order such that, for all $\omega\in\Omega^{*}$ and
$(j,k)\in\mathbb{Z}^{2}$, the inequality

$\displaystyle|\xi_{j,k}(\omega)|\leq C_{2}(\omega)\sqrt{\log(3+j+|k|)}$ (5)

holds.

Let $f_{h}$ denote the process defined in (4). For all $j\in\mathbb{N}$, we
set

$f_{h,j}=\sum_{k\in\mathbb{Z}}\xi_{j,k}2^{-hj}\psi(2^{j}\cdot-k).$

and remark that, if inequality (5) holds then, thanks to Lemma 1.1 and the
fast decay of the wavelet and its derivative (or using the compactness of the
support of $\psi$), the sum in the right-hand side converges uniformly on any
compact set, as well as the sum

$\displaystyle\sum_{k\in\mathbb{Z}}\xi_{j,k}2^{(1-h)j}D\psi(2^{j}\cdot-k).$
(6)

Therefore, for all $j$, $f_{h,j}$ is continuously differentiable with
derivative $Df_{h,j}$ given by (6). In particular, the process $f_{h}$ is well
defined and bounded on the event $\Omega^{*}$ of probability $1$. In the
following, we will hence work on this event without mentioning it explicitly.

Using similar arguments, the following Proposition gives us a first
information concerning the regularity of the function $f_{h}$.

######  Proposition 1.3.

Let $f_{h}$ be the random wavelet series defined in (4). If $\psi$ is
continously differentiable, there exists a constant $C>0$ such that, for all
$s,t\in\mathbb{R}$

$\mathbb{E}[(f_{h}(t)-f_{h}(s))^{2}]\leq C|s-t|^{2h}.$

###### Proof.

From the independence of the centered random variables in
$(\xi_{j,k})_{(j,k)\in\mathbb{N}\times\mathbb{Z}}$, we have

$\mathbb{E}[(f_{h}(t)-f_{h}(s))^{2}]=\sum_{j\in\mathbb{N}}\sum_{k\in\mathbb{Z}}2^{-2hj}(\psi(2^{j}t-k)-\psi(2^{j}s-k))^{2}.$

Let us fix $t$ and assume that $s$ and $\nu\in\mathbb{Z}$ are such that

$2^{-\nu}<|t-s|\leq 2^{-\nu+1}.$

For all $j\leq\nu$, we set

$F_{j,t}\,:\,x\mapsto\sum_{k\in\mathbb{Z}}2^{-2hj}(\psi(2^{j}t-k)-\psi(2^{j}x-k))^{2}.$

Let us then remark that $F_{j,t}$ is continuously differentiable on any
compact set. If $\psi$ is compactly supported, this is obvious. Otherwise, it
comes from the fast decay of $\psi$ and Lemma 1.1. In any case, by the mean
value theorem, there exist $x_{1}$ between $s$ and $t$ such that

$\displaystyle|F_{j,t}(s)-F_{j,t}(t)|$
$\displaystyle=\left|(s-t)\sum_{k\in\mathbb{Z}}2^{(1-2h)j}2(\psi(2^{j}t-k)-\psi(2^{j}x_{1}-k))D\psi(2^{j}x_{1}-k)\right|$
$\displaystyle\leq
C\left|(s-t)\sum_{k\in\mathbb{Z}}2^{(1-2h)j}(\psi(2^{j}t-k)-\psi(2^{j}x_{1}-k))\right|.$

Applying again the mean-value theorem to the function

$g_{j}\,:\,x\mapsto\sum_{k\in\mathbb{Z}}2^{(1-2h)j}\psi(2^{j}x-k),$

we get

$|F_{j,t}(s)-F_{j,t}(t)|\leq C|s-t|^{2}2^{(2-2h)j}$

because

$\sum_{k\in\mathbb{Z}}D\psi(2^{j}t-k)$

can be uniformly bounded, by the fast decay and Lemma 1.1 or using the compact
support of $\psi$. With the same argument, we also get, for all $j>\nu$,

$\displaystyle\left|\sum_{k\in\mathbb{Z}}2^{-2hj}(\psi(2^{j}t-k)-\psi(2^{j}s-k))^{2}\right|$
$\displaystyle\leq
2^{-2hj}\sum_{k\in\mathbb{Z}}4((\psi(2^{j}t-k))^{2}+(\psi(2^{j}s-k))^{2})$
$\displaystyle\leq C2^{-2hj}.$

Putting all of these together, we have

$\displaystyle\mathbb{E}[(f_{h}(t)-f_{h}(s))^{2}]$
$\displaystyle=\sum_{j\leq\nu}\sum_{k\in\mathbb{Z}}2^{-2hj}(\psi(2^{j}t-k)-\psi(2^{j}s-k))^{2}$
$\displaystyle+\sum_{j>\nu}\sum_{k\in\mathbb{Z}}2^{-2hj}(\psi(2^{j}t-k)-\psi(2^{j}s-k))^{2}$
$\displaystyle\leq
C(|s-t|^{2}\sum_{j\leq\nu}2^{(2-2h)j}+\sum_{j>\nu}2^{-2hj})$
$\displaystyle\leq C(|s-t|^{2}2^{(2-2h)\nu}+2^{-2h\nu})$ $\displaystyle\leq
C|s-t|^{2h}.$

∎

From the last proposition and Kolmogorov continuity Theorem for gaussian
processes, we know that the sample path of $f_{h}$ are almost surely locally
Hölder-continuous of order $h-\varepsilon$ for every $\varepsilon>0$. Our aim
in this paper is to give more precise information concerning the regularity of
$f_{h}$. In order to state it, we recall finally that a modulus of continuity
is an increasing function $\omega:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying
$\omega(0)=0$ and for which there is $C>0$ such that $\omega(2x)\leq
C\omega(x)$ for all $x\in\mathbb{R}^{+}$. Wavelet characterizations of
regularity require the following additional regularity property for moduli of
continuity, see [21]: A modulus of continuity $\omega$ is regular if there is
$N\geq 0$ such that

$\begin{cases}&\displaystyle\sum_{j=J}^{\infty}2^{Nj}\omega(2^{-j})\leq
C2^{NJ}\omega(2^{-J})\\\
&\\!\\!\displaystyle\sum_{j=-\infty}^{J}2^{(N+1)j}\omega(2^{-j})\leq
C2^{(N+1)J}\omega(2^{-J})\end{cases}$ (7)

for all $J\geq 0$. Our main result will use three different regular moduli of
continuity:

  * •

the modulus of continuity $\omega_{r}$ of the rapid points is defined by

$\omega_{r}^{(h)}(x)=|x|^{h}\sqrt{\log|x|^{-1}}$

  * •

the modulus of continuity $\omega_{o}$ of the ordinary points is defined by

$\omega_{o}^{(h)}(x)=|x|^{h}\sqrt{\log\log|x|^{-1}}$

  * •

the modulus of continuity $\omega_{s}$ of the slow points is defined by

$\omega_{s}^{(h)}(x)=|x|^{h}.$

######  Theorem 1.4.

Almost surely, the random wavelets series defined in (4) satisfies the
following property for every non-empty interval $I$ of $\mathbb{R}$:

  * •

For almost every $t\in I$,

$\limsup_{s\to t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega_{o}^{(h)}(|s-t|)}<+\infty$
(8)

and if $\omega$ is a modulus of continuity such that
$\omega=o(\omega_{o}^{(h)})$, then

$\limsup_{s\to t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega(|s-t|)}=+\infty,$ (9)

Such points are called _ordinary points_.

  * •

There exists $t\in I$ such that

$\limsup_{s\to t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega_{r}^{(h)}(|s-t|)}<+\infty$
(10)

and if $\omega$ is a modulus of continuity such that
$\omega=o(\omega_{r}^{(h)})$, then

$\limsup_{s\to t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega(|s-t|)}=+\infty,$ (11)

Such points are called _rapid points_.

  * •

There exists $t\in I$ such that

$\limsup_{s\to
t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega_{s}^{(h)}(|s-t|)}<+\infty.$ (12)

and if $\omega$ is a modulus of continuity such that
$\omega=o(\omega_{s}^{(h)})$, then

$\limsup_{s\to t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega(|s-t|)}=+\infty.$ (13)

Such points are called _slow points_.

######  Remark 1.5.

Theorem 1.4 is stated in full generality but let us already emphasize that it
can be improved while considering compactly supported wavelets. Indeed, in
this case, one can show the strict positiveness of the limits in (8), (10) and
(12), see Remark 4.2 below.

Theorem 1.4 generalises the famous result of Kahane about the different
pointwise behaviors of the Brownian motion, see [22, Theorem 3 in Chapter 16].
Moreover, it can be applied to the random wavelet series in (3), hence it
proves the existence of the three types of points for the fractional Brownian
motion.

######  Remark 1.6.

A good tool to determine the regularity of a locally bounded function $f$ at a
point $t$ is to compute its Hölder exponent $h_{f}(t)$. If $\alpha>0$, we say
that $f$ belongs to the pointwise Hölder space $C^{\alpha}(t)$ if there exists
a polynomial $P_{t}$ of degree strictly less than $\alpha$ and a constant
$C>0$ such that, for all $j\in\mathbb{N}$,

$\sup_{x\in B(x_{0},2^{-j})}|f(x)-P_{t}(x)|\leq C2^{-\alpha j}.$

It is straightforward to show that, as soon as $\alpha<\beta$,
$C^{\beta}(t)\subseteq C^{\alpha}(t)$. Therefore, we define the pointwise
Hölder exponent of $f$ at point $t$ by

$h_{f}(t)=\sup\\{\alpha>0\,:\,f\in C^{\alpha}(t)\\}.$

Theorem 1.4 tells us in particular that, if $t$ is an ordinary or rapid point,
then, for all $\varepsilon>0$, $f_{h}\in C^{h-\varepsilon}(t)$ and
$f_{h}\notin C^{h}(t)$ which gives $h_{f_{h}}(t)=h$. On the contrary, if $t$
is a slow point, $f_{h}\in C^{h}(t)$ while, for all $\varepsilon>0$,
$f_{h}\notin C^{h+\varepsilon}(t)$ and so, again, $h_{f_{h}}(t)=h$. In fact,
the finiteness of the limits (8), (10) and (12) means that the function $f$
belongs to a generalized pointwise Hölder space [25, 27] associated to the
corresponding modulus of continuity.

The next sections are dedicated to the proof of this result. Any open interval
in $\mathbb{R}$ can be written as a countable union of dyadic intervals. Then,
to prove Theorem 1.4, it is sufficient to show that, for all dyadic interval
of the form $\lambda_{j,k}=[k2^{-j},(k+1)2^{-j}[$ with
$j\in\mathbb{N},k\in\mathbb{Z}$, there exist an event $\Omega_{j,k}$ of
probability $1$ such that, for all $\omega\in\Omega_{j,k}$, almost every
$t\in\lambda_{j,k}$ is ordinary and there exist $t_{r}\in\lambda_{j,k}$ which
is rapid and $t_{s}\in\lambda_{j,k}$ which is slow. For the sake of simpleness
in notation, we will only do the proofs in full details for
$\lambda_{0,0}=[0,1)$. In fact, after dilatations and translations, our proofs
hold true for any arbitrary dyadic interval. Note also that the proofs will be
done in the case where the wavelet $\psi$ is in the Schwartz class; it can
easily be adapted and simplified if $\psi$ is compactly supported.

##  2 Regularity properties

In this section, we establish inequalities (8), (10) and (12). Concerning
rapid and ordinary points, the conduct of the proof is similar. First we use
Lemma 1.2 to bound the coefficients in (4). Then we use the fast decay of the
wavelet to measure the contribution of the coefficients associated to dyadic
intervals that are far away from the point of interest $t$ in the difference
$|f_{h}(s)-f_{h}(t)|$. Concerning the slow points, we take advantage of a
procedure initiated by Kahane in [22] to identify points for which we can
obtain more precise information concerning the coefficients of the “closest”
intervals while we still use Lemma 1.2 and the fast decay for the “furthest”
one.

###  2.1 Rapid points

To prove the existence of rapid points, we apply Lemma 1.2 and get an uniform
modulus of continuity for the function $f_{h}$. We deal with the coefficients
associated to furthest intervals thanks to the following lemma.

######  Lemma 2.1.

There exists a constant $C_{1}>0$ such that, for all $j\in\mathbb{N}_{0}$ and
$x\in(0,1)$,

$\sum_{|k|>2^{j+1}}\frac{\sqrt{\log(3+j+|k|)}}{(1+|2^{j}x-k|)^{5}}\leq C_{1}$

###### Proof.

Let us fix $j\in\mathbb{N}_{0}$ and $x\in(0,1)$. As $|k|>2^{j+1}$, obviously,
$\frac{k}{2^{j}}\notin(-2,2)$ so let $n\notin\\{0,\pm 1,-2\\}$ be such that
$n2^{j}\leq k<(n+1)2^{j}$. Now, as $x\in(0,1)$, we have

$|2^{j}x-k|\geq 2^{j}|x-n|-1\geq 2^{j}(|n|-1)-1\geq\frac{|k|}{2}-1.$

Thus, for all such $j,k$ and $x$, we have

$\frac{\sqrt{\log(3+j+|k|)}}{(1+|2^{j}x-k|)}\leq
2\frac{\sqrt{\log(3+2|k|)}}{|k|}$

and we conclude using the boundedness of the function
$x\mapsto\frac{\sqrt{\log(3+2x)}}{x}$ on $[1,+\infty[$ and Lemma 1.1. ∎

######  Proposition 2.2.

Almost surely, there exists a constant $C_{1}>0$ such that, for all
$t,s\in(0,1)$ we have

$|f_{h}(s)-f_{h}(t)|\leq C_{1}|t-s|^{h}\sqrt{\log|t-s|^{-1}}.$

In particular, one has almost surely

$\limsup_{s\to t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega_{r}^{(h)}(|s-t|)}<+\infty$

for every $t\in(0,1)$.

###### Proof.

Let us assume that $t,s$ and $\nu\in\mathbb{N}$ are such that

$2^{-\nu}<|t-s|\leq 2^{-\nu+1}.$

For all $j\leq\nu$, by the mean value theorem, there exists $x$ between $s$
and $t$ such that

$\displaystyle|f_{h,j}(t)-f_{h,j}(s)|\leq|t-s||Df_{h,j}(x)|$ (14)

and, using the fast decay of $D\psi$ and (5), it follows that

$\displaystyle|Df_{h,j}(x)|$ $\displaystyle\leq
CC_{2}2^{(1-h)j}\left(\sum_{|k|\leq
2^{j+1}}\frac{\sqrt{\log(3+j+|k|)}}{(1+|2^{j}x-k|)^{4}}\right.$
$\displaystyle\left.\quad\quad+\sum_{|k|>2^{j+1}}\frac{\sqrt{\log(3+j+|k|)}}{(1+|2^{j}x-k|)^{5}}\right).$

We bound the second sum by Lemma 2.1 while, for the first sum, we have

$\sum_{|k|\leq 2^{j+1}}\frac{\sqrt{\log(3+j+|k|)}}{(1+|2^{j}x-k|)^{4}}\leq
C\sum_{|k|\leq 2^{j+1}}\frac{\sqrt{j}}{(1+|2^{j}x-k|)^{4}}\leq C\sqrt{j}$

by Lemma 1.1. Thus, we obtain

$\displaystyle|\sum_{j\leq\nu}(f_{h,j}(t)-f_{h,j}(s))|$ $\displaystyle\leq
CC_{2}|t-s|2^{(1-h)\nu}\sqrt{\nu}$ $\displaystyle\leq
CC_{2}(\omega)|t-s|^{h}\sqrt{\log|t-s|^{-1}}.$

Now, if $j>\nu$, by splitting the sums in the same way, we also have

$|f_{h,j}(t)|\leq CC_{2}2^{-hj}\sqrt{j}\text{ and }|f_{h,j}(s)|\leq
CC_{2}2^{-hj}\sqrt{j}$

which obviously leads to

$|\sum_{j>\nu}f_{h,j}(t)|\leq CC_{2}|t-s|^{h}\sqrt{\log|t-s|^{-1}}$

and

$|\sum_{j>\nu}f_{h,j}(s)|\leq CC_{2}|t-s|^{h}\sqrt{\log|t-s|^{-1}}.$

The conclusion follows immediately. ∎

###  2.2 Ordinary points

To establish the existence of ordinary points, we need to introduce some
notations. If $j\in\mathbb{N}_{0}$, and $t\in(0,1)$, we denote by $k_{j}(t)$
the unique positive integer in $\\{0,\ldots,2^{j}-1\\}$ such that
$t\in[k_{j}(t)2^{-j},(k_{j}(t)+1)2^{-j})$. We also define, for all
$n\in\mathbb{N}$ the set

$\kappa_{j}^{t}(n)=\\{k\in\mathbb{Z}\,:\,|k-k_{j}(t)|\leq n\\}$

The main proof of this section consists in applying Lemma 1.2 to a sequence of
independent $\mathcal{N}(0,1)$ random variables indexed from $k_{j}(t)$. Then,
Lemmata 2.3 and 2.4 help us to deal with the coefficients associated with the
furthest intervals.

######  Lemma 2.3.

There exists a constant $C_{1}>0$ such that, for all $j\in\mathbb{N}_{0}$,
$t\in(0,1)$ and $x\in(0,1)$ such that $|t-x|\leq 2^{-j+1}$ we have

$\sum_{k\notin\kappa_{j}^{t}(j)}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(3+|2^{j}x-k|)(1+|2^{j}x-k|)^{4}}\leq
C_{1}$

###### Proof.

Let us fix $j\in\mathbb{N}_{0}$, $t\in(0,1)$ and $x\in(0,1)$ such that
$|t-x|\leq 2^{-j+1}$. If $k\notin\kappa_{j}^{t}(j)$, we immediately have
$|2^{j}x-k_{j}(t)|\leq 3$ and thus $|2^{j}x-k|\geq|k_{j}(t)-k|-3$. Therefore,

$\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(3+|2^{j}x-k|)}\leq\frac{\sqrt{\log(3+2|k-k_{j}(t)|)}}{|k_{j}(t)-k|}$

and the conclusion follows just like in Lemma 2.1. ∎

######  Lemma 2.4.

There exists a constant $C_{1}>0$ such that, for all $j\in\mathbb{N}_{0}$,
$t\in(0,1)$ and $s\in(0,1)$ such that $|2^{j}s-k_{j}(t)|\leq j$ we have

$\sum_{k\notin\kappa_{j}^{t}(2j)}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(1+|2^{j}s-k|)^{5}}\leq
C_{1}$

###### Proof.

Let us fix $j\in\mathbb{N}_{0}$ $t\in(0,1)$ and $s\in(0,1)$ such that
$|2^{j}s-k_{j}(t)|\leq j$. If $k\notin\kappa_{j}^{t}(2j)$, of course,
$|2^{j}s-k_{j}(t)|<\frac{|k-k_{j}(t)|}{2}$ and thus
$|2^{j}s-k|\geq\frac{|k-k_{j}(t)|}{2}$. It follows that

$\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{1+|2^{j}s-k|}\leq
2\frac{\sqrt{\log(3+2|k-k_{j}(t)|)}}{|k-k_{j}(t)|}$

and, again, we conclude just like in Lemma 2.1. ∎

######  Proposition 2.5.

Almost surely, for almost every $t\in(0,1)$,

$\limsup_{s\to
t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega_{o}^{(h)}(|t-s|)}<+\infty.$

###### Proof.

Let us fix $t\in(0,1)$, and $j\in\mathbb{N}_{0}$ and let $0\leq
k_{j}(t)<2^{j}$ be such that $t\in[k_{j}(t)2^{-j},(k_{j}(t)+1)2^{-j}[$. A
simple modification of Lemma 1.2 insure us the existence of a positive random
variable $C_{t}$ of finite moment of every order such that, almost surely

$|\xi_{j,k}|\leq C_{t}\sqrt{\log(3+j+|k-k_{j}(t)|)}.$

As previously, if $s\in(0,1)$ and $\nu\in\mathbb{N}$ is such that

$2^{-\nu}<|t-s|\leq 2^{-\nu+1},$

we first start by considering, for all $j\leq\nu$ ,$|Df_{h,j}(x)|$ for a $x$
between $s$ and $t$. In this case, we have

$\displaystyle|Df_{h,j}(x)|$ $\displaystyle\leq
CC_{t}2^{(1-h)j}\left(\sum_{k\in\kappa_{j}^{t}(j)}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(1+|2^{j}x-k|)^{4}}\right.$
$\displaystyle\left.\quad\quad+\sum_{k\notin\kappa_{j}^{t}(j)}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(3+|2^{j}x-k|)(1+|2^{j}x-k|)^{4}}\right)$

and we bound the second sum by Lemma 2.3, while, for the first sum, we have

$\sum_{k\in\kappa_{j}^{t}(j)}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(1+|2^{j}x-k|)^{4}}\leq
C\sum_{k\in\kappa_{j}^{t}(j)}\frac{\sqrt{\log(j)}}{(1+|2^{j}x-k|)^{4}}\leq
C\sqrt{\log(j)}$

by Lemma 1.1. These inequalities lead to

$|\sum_{j\leq\nu}(f_{h,j}(t)-f_{h,j}(s))|\leq
CC_{t}|t-s|^{h}\sqrt{\log\log|t-s|^{-1}}.$ (15)

To bound $|f_{h,j}(t)|$, for all $j>\nu$, we use the same techniques and get

$|\sum_{j>\nu}f_{h,j}(t)|\leq CC_{t}|t-s|^{h}\sqrt{\log\log|t-s|^{-1}}.$

The bound for $|f_{h,j}(s)|$ is a little bit more tricky. As
$|2^{j}s-k_{j}(t)|\leq 2^{j-\nu+2}$, we first consider the case when
$2^{j-\nu+2}\leq j$, then

$\displaystyle|f_{h,j}(s)|$ $\displaystyle\leq
CC_{t}2^{-hj}\left(\sum_{k\in\kappa_{j}^{t}(2j)}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(1+|2^{j}s-k|)^{4}}\right.$
$\displaystyle\left.\quad\quad+\sum_{k\notin\kappa_{j}^{t}(2j)}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(1+|2^{j}s-k|)^{5}}\right).$

The second sum is this time bounded using Lemma 2.4 while we again use Lemma
1.1 to get

$\sum_{k\in\kappa_{j}^{t}(2j)}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(1+|2^{j}s-k|)^{4}}\leq
C\sqrt{\log(j)}.$

Now, if $j<2^{j-\nu+2}$,

$\displaystyle|f_{h,j}(s)|$ $\displaystyle\leq
CC_{t}2^{-hj}\left(\sum_{k\in\kappa_{j}^{s}(2^{j-\nu+2})}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(1+|2^{j}s-k|)^{4}}\right.$
$\displaystyle\left.\quad\quad+\sum_{k\notin\kappa_{j}^{s}(2^{j-\nu+2})}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(1+|2^{j}s-k|)^{5}}\right)$

and, as $|k_{j}(s)-k_{j}(t)|\leq 2^{j-\nu+1}+2$, for all
$k\in\kappa_{j}^{s}(2^{j-\nu+2})$ we have

$\log(3+j+|k-k_{j}(t)|)\leq\log(3+j+|k_{j}(s)-k_{j}(t)|+|k-k_{j}(s)|)\leq\log(42^{j-\nu+2})$

while, for all $k\notin\kappa_{j}^{s}(2^{j-\nu+2})$,

$\displaystyle\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(1+|2^{j}s-k|)^{5}}$
$\displaystyle\leq\frac{\log(3+j+|k_{j}(s)-k_{j}(t)|+|k-k_{j}(s)|)}{|k-k_{j}(s)|}$
$\displaystyle\leq\frac{\log(4|k-k_{j}(s)|)}{|k-k_{j}(s)|}$ $\displaystyle\leq
C$

which gives us

$|f_{h,j}(s)|\leq CC_{t}2^{-hj}\sqrt{j-\nu}.$ (16)

In total, using (15) and (16), we get

$\displaystyle|\sum_{j>\nu}f_{h,j}(s)|$
$\displaystyle\leq\sum_{j>\nu}CC_{t}2^{-hj}\sqrt{j-\nu}\sqrt{\log(j)}$
$\displaystyle\leq CC_{t}|t-s|^{h}\sqrt{\log\log|t-s|^{-1}}.$

The conclusion comes from Fubini’s Theorem. ∎

###  2.3 Slow points

####  2.3.1 An iterative procedure for slow points

The following procedure is inspired by the one initially described by Kahane
in [22] to identify slow points for the Brownian motion. We generalized it
here by introducing an extra parameter $m\in\mathbb{N}$ in order to use it for
any333Kahane only considered the case $h=\frac{1}{2}$ and so its construction
is made with $m=3$ which is sufficient to fulfill the condition
$h>\frac{1}{m}$ we will need afterward. $h>0$. Some clarifications are also
made.

If $\mu\in\mathbb{N}$, $\xi\sim\mathcal{N}(0,1)$ and $l\in\mathbb{N}_{0}$, we
note

$p_{l}(\mu)=\mathbb{P}(2^{l}\mu<|\xi|\leq 2^{l+1}\mu).$

For all $j,l\in\mathbb{N}_{0}$ and $0\leq k<2^{j}$, we define

$S_{j,l}^{\mu}=\\{k\in\\{0,\ldots,2^{j}-1\\}\,:\,2^{l}\mu<|\xi_{j,k}|\leq
2^{l+1}\mu\\}$

and

$\Lambda_{j,l}(k)=\\{k^{\prime}\in\\{0,\ldots,2^{j}-1\\}\,:\,|k-k^{\prime}|\leq
2^{ml}\\}.$

Note that $\\#\Lambda_{j,l}(k)\leq 2^{ml+1}+1$. For all $j\in\mathbb{N}_{0}$,
we define a closed set from the dyadic intervals $[k2^{-j},(k+1)2^{-j}]$ for
which $k$ belongs to the set

$I_{j}^{\mu}=\\{k\in\\{0,\ldots,2^{j}-1\\}\,:\,\forall
l\in\mathbb{N}_{0},\,\Lambda_{j,l}(k)\cap S_{j,l}^{\mu}=\emptyset\\},$

namely, we consider

$F_{j}^{\mu}=\bigcup_{k\in I_{j}^{\mu}}[k2^{-j},(k+1)2^{-j}].$

We want to show that, almost surely, there exists $\mu\in\mathbb{N}$ such that

$S_{\text{low}}^{\mu}=\bigcap_{j\in\mathbb{N}_{0}}F_{j}^{\mu}\neq\emptyset$

which is equivalent to the fact that, for all $J\in\mathbb{N}_{0}$,

$S_{\text{low},J}^{\mu}=\bigcap_{j\leq J}F_{j}^{\mu}$

is non-empty, as $(S_{\text{low},J}^{\mu})_{J\in\mathbb{N}_{0}}$ is a
decreasing sequence of compact sets. In other words, if we denote by
$N_{J}^{\mu}$ the number of subintervals of $S_{\text{low},J}^{\mu}$, we want
to show that

$\mathbb{P}(\bigcup_{\mu\in\mathbb{N}}\bigcap_{J\in\mathbb{N}_{0}}\\{N_{J}^{\mu}\geq
1\\})=1.$

For this purpose, we will consider sufficiently large $\mu$ such that the
inequality

$\displaystyle\sum_{l=0}^{+\infty}(2^{ml+1}+1)(p_{l}(\mu)+l\sqrt{p_{l}(\mu)(1-p_{l}(\mu))})<\frac{1}{4}$
(17)

holds. Moreover, if $J_{1}$ is fixed, let us remark that the construction of
$S_{\text{low},J_{1}}^{\mu}$ needs to consider at most
$\sum_{j=0}^{J_{1}}2^{j}$ i.i.d. $\mathcal{N}(0,1)$ random variables and, by
increasing $\mu$ if necessary, one can choose to remove the intervals
$[0,2^{-J_{1}}]$ and $[1-2^{-J_{1}},1]$ from $S_{\text{low},J_{1}}^{\mu}$ at
this step if necessary while making
$\mathbb{P}(N_{J_{1}}^{\mu}\geq(\frac{3}{2})^{J_{1}})$ as close to $1$ as we
want. This trick helps us to start our construction with a arbitrary close to
$1$ “initial value” of probability and to make sure that, at the end, the
resulting points will differ from $0$ and $1$.

######  Lemma 2.6.

For all $\mu\in\mathbb{N}$ sufficiently large such that condition (17) holds,
the sequence $(N_{J}^{\mu})_{J\in\mathbb{N}_{0}}$ of random variables
satisfies the formula

$\displaystyle\mathbb{P}(N_{J+1}^{\mu}\geq(\frac{3}{2})^{J+1})\geq(1-(\frac{2}{3})^{J})\mathbb{P}(N_{J}^{\mu}\geq(\frac{3}{2})^{J}),\quad\forall
J\in\mathbb{N}_{0}.$ (18)

###### Proof.

For all $J$, we define444We obviously have
$N_{J}^{\mu}=\\#\mathcal{I}_{J}^{\mu}$.

$\mathcal{I}_{J}^{\mu}=\\{k\in\\{0,\ldots,2^{J}-1\\}\,:\,[k2^{-J},(k+1)2^{-J}]\subseteq
S_{\text{low},J}^{\mu}\\},$

and remark that $\mathcal{I}_{J+1}^{\mu}$ is obtained by removing from
$2\mathcal{I}_{J}^{\mu}\cup 2\mathcal{I}_{J}^{\mu}+1$ the elements
$k\in\\{0,\ldots,2^{J+1}-1\\}$ such that $\Lambda_{J+1,l}(k)\cap
S_{J+1,l}^{\mu}\neq\emptyset$ for a $l\in\mathbb{N}_{0}$. But now, if
$N_{J}^{\mu}=N$, for all such a $l$,

$\\#(S_{J+1,l}^{\mu}\cap(2\mathcal{I}_{J}^{\mu}\cup
2\mathcal{I}_{J}^{\mu}+1))\sim\mathcal{B}(2N,p_{l}(\mu))$

so we have, by Tchebycheff’s inequality, that

$\\#(S_{J+1,l}^{\mu}\cap(2\mathcal{I}_{J}^{\mu}\cup
2\mathcal{I}_{J}^{\mu}+1))\leq
2N(p_{l}(\mu)+(l+1)\sqrt{p_{l}(\mu)(1-p_{l}(\mu))})$

with probability greater than $1-(l+1)^{-2}N^{-1}$. Thus the number of removed
$k$ is bounded by

$2N\sum_{l=0}^{+\infty}(2^{ml+1}+1)(p_{l}(\mu)+l\sqrt{p_{l}(\mu)(1-p_{l}(\mu))})$

with probability greater than $1-N^{-1}$. Now, with condition (17), we get,
for all $J$,

$\mathbb{P}(N_{J+1}^{\mu}\geq\frac{3}{2}N_{J}|N_{J}^{\mu}=N)\geq 1-N^{-1}$

which gives us

$\displaystyle\mathbb{P}(N_{J+1}^{\mu}\geq(\frac{3}{2})^{J+1})$
$\displaystyle\geq\mathbb{P}((N_{J+1}^{\mu}\geq\frac{3}{2}N_{J}^{\mu})\cap(N_{J}^{\mu}\geq(\frac{3}{2})^{J}))$
$\displaystyle=\sum_{N\geq(\frac{3}{2})^{J}}\mathbb{P}(N_{J+1}^{\mu}\geq\frac{3}{2}N_{J}|N_{J}^{\mu}=N)\mathbb{P}(N_{J}^{\mu}=N)$
$\displaystyle\geq\sum_{N\geq(\frac{3}{2})^{J}}(1-N^{-1})\mathbb{P}(N_{J}^{\mu}=N)$
$\displaystyle\geq(1-(\frac{2}{3})^{J})\sum_{N\geq(\frac{3}{2})^{J}}\mathbb{P}(N_{J}^{\mu}=N)$
$\displaystyle=(1-(\frac{2}{3})^{J})\mathbb{P}(N_{J}^{\mu}\geq(\frac{3}{2})^{J}).$

∎

######  Proposition 2.7.

Almost surely, there exists $\mu\in\mathbb{N}$ such that $(0,1)\cap
S_{\text{low}}^{\mu}\neq\emptyset$.

###### Proof.

From formula (18), if $\mu$ is large enough we deduce by induction that for
all $J_{1},J$ with $J_{1}\leq J$,

$\displaystyle\mathbb{P}(N_{J}^{\mu}\geq 1)$
$\displaystyle\geq\mathbb{P}(N_{J}^{\mu}\geq(\frac{3}{2})^{J})$
$\displaystyle\geq\mathbb{P}(N_{J_{1}}^{\mu}\geq(\frac{3}{2})^{J_{1}})\left(\prod_{j=J_{1}}^{J}(1-(\frac{2}{3})^{j})\right).$

Let us remark that, as $\sum_{j=1}^{\infty}(\frac{2}{3})^{j}<\infty$, the
infinite product $\prod_{j=1}^{\infty}(1-(\frac{2}{3})^{j})$ converges to a
non-zero limit and

$\lim_{J_{1}\to+\infty}\prod_{j=J_{1}}^{+\infty}(1-(\frac{2}{3})^{j})=1.$

Now, for all $0<\varepsilon<\frac{1}{2}$, one can choose $J_{1}$ such that

$\prod_{j=J_{1}}^{+\infty}(1-(\frac{2}{3})^{j})>1-\varepsilon$

and, by increasing $\mu$ if necessary, we can choose to remove the intervals
$[0,2^{-J_{1}}]$ and $[1-2^{-J_{1}},1]$ from $S_{\text{low},J_{1}}^{\mu}$, if
necessary and assume

$\mathbb{P}(N_{J_{1}}^{\mu}\geq(\frac{3}{2})^{J_{1}})>1-\varepsilon.$

Therefore, as the sequence of events $(\\{N_{J}^{\mu}\geq
1\\})_{J\in\mathbb{N}_{0}}$ is decreasing, we get

$\displaystyle\mathbb{P}\big{(}\bigcap_{J\in\mathbb{N}_{0}}(N_{J}^{\mu}\geq
1)\big{)}\geq\mathbb{P}(N_{J_{1}}^{\mu}\geq(\frac{3}{2})^{J_{1}})\left(\prod_{j=J_{1}}^{\infty}(1-(\frac{2}{3})^{j})\right)>(1-\varepsilon)^{2}.$

In total, we showed that, for all $0<\varepsilon<\frac{1}{2}$,

$\mathbb{P}\big{(}\bigcup_{\mu\in\mathbb{N}}\bigcap_{J\in\mathbb{N}_{0}}(N_{J}^{\mu}\geq
1)\big{)}>(1-\varepsilon)^{2}$

and the conclusion follows immediately. ∎

####  2.3.2 Existence of slow points

Now, using this iterative procedure, we establish the existence of points
satisfying inequality (12). In this section, it will be convenient to write

$\displaystyle f_{h,j}=\sum_{n\in\mathbb{Z}}f_{h,j}^{[n]}$ (19)

where, for all $n\in\mathbb{Z}$,

$f_{h,j}^{[n]}=\sum_{k=n2^{j}}^{(n+1)2^{j}-1}\xi_{j,k}2^{-hj}\psi(2^{j}\cdot-k)$

is the partial sum involving the $k$ for which the corresponding dyadic
interval $[k2^{-j},(k+1)2^{-j}[$ is included in $[n,n+1[$.

######  Proposition 2.8.

Almost surely, there exists $t\in(0,1)$ such that

$\limsup_{s\to
t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega_{s}^{(h)}(|t-s|)}<+\infty.$

###### Proof.

Let us fix $m\in\mathbb{N}_{0}$ such that $h\geq 1/m$. The iterative procedure
with $m$ gives that almost surely, $(0,1)\cap
S_{\text{low}}^{\mu}\neq\emptyset$. Now let $t\in(0,1)\cap
S_{\text{low}}^{\mu}$. There exists $r>0$ such that $[t-r,t+r]\subset(0,1)$
and so, let us take $s\in[t-r,t+r]$ and $\nu\in\mathbb{N}$ be such that

$2^{-\nu}<|t-s|\leq 2^{-\nu+1}.$

We are going to estimate from above $|f_{h}(t)-f_{h}(s)|$.

For all $j\leq\nu$, once again, by the Mean Value Theorem, there exists $x$
between $s$ and $t$ such that

$|f_{h,j}(t)-f_{h,j}(s)|\leq|s-t||Df_{h,j}(x)|.$

To bound $|Df_{h,j}(x)|$, we will use the decomposition (19). For $n=0$, if
$0\leq l\leq\lceil j/m\rceil$, $0\leq k<2^{j}$ and $|k_{j}(t)-k|\leq 2^{ml}$,
then necessarily $|\xi_{j,k}|\leq 2^{l}\mu$. Thus, if we set

$\Lambda_{j}^{0}(t)=\\{0\leq k<2^{j}\,:\,|k_{j}(t)-k|\leq 1\\}$

and, for all $1\leq l\leq\lceil j/m\rceil$

$\Lambda_{j}^{l}(t)=\\{0\leq k<2^{j}\,:\,2^{m(l-1)}<|k_{j}(t)-k|\leq
2^{ml}\\},$

we have, using the fast decay of $D\psi$,

$\displaystyle|Df_{h,j}^{[0]}(x)|$ $\displaystyle\leq
2^{(1-h)j}\sum_{l=0}^{\lceil
j/m\rceil}\sum_{k\in\Lambda^{l}(t)}|\xi_{j,k}||D\psi(2^{j}x-k)|$ (20)
$\displaystyle\leq C2^{(1-h)j}\sum_{l=0}^{\lceil
j/m\rceil}\sum_{k\in\Lambda_{j}^{l}(t)}2^{l}\mu\frac{1}{(3+|2^{j}x-k|)(1+|2^{j}x-k|)^{4}}.$
(21)

As, for $l\geq 1$ and $k\in\Lambda_{j}^{l}(t)$,
$|2^{j}x-k|\geq|k_{j}(t)-k|-3>2^{m(l-1)}-3$, we obtain

$\displaystyle|Df_{h,j}^{[0]}(x)|$ $\displaystyle\leq C\mu
2^{(1-h)j}\sum_{k=0}^{2^{j}-1}\frac{1}{(1+|2^{j}x-k|)^{4}}.$

Now, there exists555We can make this constant only dependant of $t$. Note
that, as $t$ is random, $C_{t}$ is a random variable as well. $C_{t}>0$, such
that, for all $n\neq 0$ and $n2^{j}\leq k<(n+1)2^{j}-1$, we have
$|2^{j}x-k|\geq C_{t}|n|2^{j}$, which gives

$\displaystyle|Df_{h,j}^{[n]}(x)|$ $\displaystyle\leq
CC_{2}2^{(1-h)j}\sum_{k=n2^{j}}^{(n+1)2^{j}-1}\frac{1}{(1+|2^{j}x-k|)^{4}}\frac{\sqrt{\log(3+j+|k|)}}{C_{t}|n|2^{j}}$
$\displaystyle\leq
CC_{2}2^{(1-h)j}\sum_{k=n2^{j}}^{(n+1)2^{j}-1}\frac{1}{(1+|2^{j}x-k|)^{4}}.$

Let us define the random variable $C_{\mu}=\max(C_{2},\mu)$. By Lemma 1.1, we
obtain, for all $j\leq\nu$,

$\displaystyle|f_{h,j}(t)-f_{h,j}(s)|$ $\displaystyle\leq
C_{\mu}2^{(1-h)j}|s-t|$

and thus

$\displaystyle|\sum_{j\leq\nu}(f_{h,j}(t)-f_{h,j}(s))|$ $\displaystyle\leq
CC_{\mu}2^{(1-h)\nu}|s-t|$ $\displaystyle\leq CC_{\mu}^{*}|s-t|^{h}.$ (22)

Now, we consider the terms for $j>\nu$ and we will bound separately
$|f_{h,j}(t)|$ and $|f_{h,j}(s)|$. For $|f_{h,j}(t)|$, we just have to repeat
the same procedure, using the set $\Lambda_{j}^{l}(t)$ to estimate
$|f_{h,j}^{[0]}(t)|$ and Lemma 1.2 for $|f_{h,j}^{[n]}(t)|$ with $n\neq 0$. We
then conclude that

$\displaystyle|\sum_{j>\nu}f_{h,j}(t)|$ $\displaystyle\leq
C\sum_{j>\nu}C_{\mu}2^{-hj}$ $\displaystyle\leq CC_{\mu}2^{-h\nu}$
$\displaystyle\leq CC_{\mu}|t-s|^{h}$

For $|f_{h,j}(s)|$, the strategy remains the same: nothing changes to bound
$|f_{h,j}^{[n]}(s)|$ with $n\neq 0$ while, for $n=0$, if $l$ is the greatest
integer such that $|s-t|\geq 2^{ml}2^{-j}$, the construction insures that, for
all $1\leq l^{\prime}\leq\lceil j/m\rceil$ and
$k\in\Lambda_{j}^{l^{\prime}}(s)$,

$|\xi_{j,k}|\leq 2^{l}2^{l^{\prime}}\mu$

and we get

$\displaystyle|f^{[0]}_{h,j}(s)|\leq C2^{l}\mu
2^{-hj}\sum_{k=0}^{2^{j}-1}\frac{1}{(1+|2^{j}s-k|)^{4}}.$

But as $|t-s|\leq 2^{-\nu+1}$, we have $l\leq\frac{1}{m}(j+1-\nu)$ and thus

$\displaystyle 2^{l}\mu 2^{-hj}\leq 2^{\frac{1}{m}}\mu
2^{(\frac{1}{m}-h)(j-\nu)}2^{-h\nu}.$

It gives, as we took $\frac{1}{m}<h$, combined with Lemma 1.1,

$\displaystyle|\sum_{j>\nu}f_{h,j}(s)|\leq CC_{\mu}|t-s|^{h}.$ (23)

Finally, combining (2.3.2) with (23) allows to obtain

$\displaystyle|f_{h}(t)-f_{h}(s)|\leq CC_{\mu}|t-s|^{h}$

as desired. ∎

##  3 Asymptotic behavior of wavelet coefficients

In this section we study the asymptotic behavior of the wavelet coefficients
of the Gaussian wavelet series. It will allow to get irregularity properties
for the random wavelets series $f_{h}$, i.e. to prove that the three
corrections obtained in the previous section characterize exactly three
possible pointwise behaviors. Let us start by recalling the following result
from [2, Lemma A.27].

######  Lemma 3.1.

[2] Almost surely, for every $t\in\mathbb{R}$, one has

$\limsup_{j\to+\infty}|\xi_{j,k_{j}(t)}|\geq 2^{-3/2}\sqrt{\pi}.$

Before stating the next lemma, let us recall that if
$\xi\sim\mathcal{N}(0,1)$, then one has

$\lim_{x\to+\infty}\frac{\mathbb{P}\big{(}|\xi|>x\big{)}}{(2\pi^{-1})^{1/2}x^{-1}e^{-x^{2}/2}}=1\,.$
(24)

The following result follows the lines of [4].

######  Lemma 3.2.

Almost surely, for almost every $t\in\mathbb{R}$, one has

$\limsup_{j\to+\infty}\dfrac{|\xi_{j,k_{j}(t)}|}{\sqrt{\log j}}>0\,.$ (25)

###### Proof.

By Fubini theorem, it is enough to prove that for every $t\in\mathbb{R}$, (25)
holds almost surely. Let us fix $t\in\mathbb{R}$. For every $m\in\mathbb{N}$,
we consider the event

$A_{m}(t)=\left\\{\max_{2^{m}\leq
j<2^{m+1}}|\xi_{2^{j},k_{j}(t)}|\geq\sqrt{m\log 2}\right\\}.$

Using the independance of the random variables $\xi_{j,k}$,
$(j,k)\in\mathbb{N}\times\mathbb{Z}$, we have

$\displaystyle\mathbb{P}\left(A_{m}(t)\right)$ $\displaystyle=$ $\displaystyle
1-\prod_{2^{m}\leq
j<2^{m+1}}\mathbb{P}\left(|\xi_{2^{j},k_{j}(t)}|<\sqrt{m\log 2}\right)$
$\displaystyle=$ $\displaystyle 1-\Big{(}1-\mathbb{P}\big{(}|\xi|>\sqrt{m\log
2}\,\big{)}\Big{)}^{2^{m}}$

where $\xi\sim\mathcal{N}(0,1)$. Using (24) and the fact that
$\log(1-x)\leq-x$ if $x\in(0,1)$, we obtain for $m$ large enough

$\displaystyle\mathbb{P}\left(A_{m}(t)\right)\geq
1-\Big{(}1-C\dfrac{2^{-\frac{m}{2}}}{\sqrt{m\log 2}}\Big{)}^{2^{m}}$
$\displaystyle\geq$ $\displaystyle
1-\exp\left(-C\dfrac{2^{\frac{m}{2}}}{\sqrt{m\log 2}}\right)$

where $C=\frac{1}{2}\sqrt{\frac{2}{\pi}}$. Hence, it follows that

$\sum_{m=0}^{+\infty}\mathbb{P}\left(A_{m}(t)\right)=+\infty\,$

and since that the events $A_{m}(t)$, $m\in\mathbb{N}$, are independents, the
Borel-Cantelli lemma implies that

$\mathbb{P}\left(\bigcap_{M\in\mathbb{N}}\bigcup_{m\geq
M}A_{m}(t)\right)=1\,.$

It gives that almost surely, for infinitely many $m\in\mathbb{N}$, there is
$j\in\\{2^{m},\dots,2^{m+1}-1\\}$ such that

$|\xi_{2^{j},k_{j}(t)}|\geq\sqrt{m\log 2}\geq\sqrt{\log j-\log 2}.$

The conclusions follows. ∎

The last lemma we need is proved in [4]. Note that in this paper, the authors
works in a more general context. Indeed, it is only required that the sequence
of standard Gaussian random variables satisfy the following condition: there
is $N\in\mathbb{N}$ such that for every
$(j_{1},k_{1}),(j_{2},k_{2})\in\mathbb{N}\times\mathbb{Z}$ satisfying

$\left(\frac{k_{1}-N}{2^{j_{1}}},\frac{k_{1}+N}{2^{j_{1}}}\right)\cap\left(\frac{k_{2}-N}{2^{j_{2}}},\frac{k_{2}+N}{2^{j_{2}}}\right)=\emptyset\,,$

the random variables $\xi_{j_{1},k_{1}}$ and $\xi_{j_{2},k_{2}}$ are
independant.

######  Lemma 3.3.

[4] Almost surely, for every non-empty open interval $I$ of $\mathbb{R}$,
there is $t\in I$ such that

$\limsup_{j\to+\infty}\left\\{\frac{|\xi_{j,k_{j}(t)}|}{\sqrt{j}}\right\\}>0\,.$

##  4 Proof of the main result

Putting together Propositions 2.2, 2.5, 2.8 and Lemmata 3.1, 3.2 and 3.3, we
can summarize the results obtained in the previous sections as follows.

######  Corollary 4.1.

Almost surely,

  * •

there exists $t\in(0,1)$ such that

$\limsup_{s\to
t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega^{(h)}_{r}(|t-s|)}<+\infty\quad\text{ and
}\quad\limsup_{j\to+\infty}\dfrac{|c_{j,\lfloor
2^{j}t\rfloor}|}{\omega_{r}^{(h)}(2^{-j})}>0,$

  * •

for almost every $t\in(0,1)$,

$\limsup_{s\to
t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega^{(h)}_{o}(|t-s|)}<+\infty\quad\text{ and
}\quad\limsup_{j\to+\infty}\dfrac{|c_{j,\lfloor
2^{j}t\rfloor}|}{\omega_{o}^{(h)}(2^{-j})}>0,$

  * •

there exists $t\in(0,1)$ such that

$\limsup_{s\to
t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega^{(h)}_{s}(|t-s|)}<+\infty\quad\text{ and
}\quad\limsup_{j\to+\infty}\dfrac{|c_{j,\lfloor
2^{j}t\rfloor}|}{\omega_{s}^{(h)}(2^{-j})}>0.$

We are now ready to prove our main result.

###### Proof of Theorem 1.4.

Using Corollary 4.1, it suffices to prove the second part of the result. Let
us consider $t\in(0,1)$ for which there is $v\in\\{r,o,s\\}$ such that

$\limsup_{s\to
t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega_{v}^{(h)}(|t-s|)}<+\infty\quad\text{ and
}\quad\limsup_{j\to+\infty}\dfrac{|c_{j,\lfloor
2^{j}t\rfloor}|}{\omega_{v}^{(h)}(2^{-j})}>0.$ (26)

Let us fix a modulus of continuity $\omega$ such that
$\omega=o(\omega_{v}^{(h)})$. Assume by contradiction that

$\limsup_{s\to t}\dfrac{|f_{h}(s)-f_{h}(t)|}{\omega(|s-t|)}<+\infty.$ (27)

Given an arbitrary fixed $j\in\mathbb{N}$, we set $k=\lfloor 2^{j}t\rfloor$.
The first vanishing moment of the wavelet allows to write

$\displaystyle|c_{j,k}|$ $\displaystyle=$
$\displaystyle\left|2^{j}\int_{\mathbb{R}}\psi(2^{j}x-k)\big{(}f_{h}(x)-f_{h}(t)\big{)}dx\right|$
(28) $\displaystyle\leq$ $\displaystyle
2^{j}\int_{B(t,2^{-j})}|\psi(2^{j}x-k)|\big{|}f_{h}(x)-f_{h}(t)\big{|}dx$
$\displaystyle+2^{j}\sum_{l=0}^{j-1}\int_{B_{l}}|\psi(2^{j}x-k)|\big{|}f_{h}(x)-f_{h}(t)\big{|}dx$
$\displaystyle+2^{j}\int_{\mathbb{R}\setminus
B(t,1)}|\psi(2^{j}x-k)|\big{|}f_{h}(x)-f_{h}(t)\big{|}dx$

where for every $l\in\\{0,\dots,j-1\\}$, $B_{l}$ denotes the set
$B(t,2^{-l})\setminus B(t,2^{-l-1})$ . The first term can be controlled by
using the modulus of continuity $\omega$ in the following way

$\displaystyle
2^{j}\int_{B(t,2^{-j})}|\psi(2^{j}x-k)|\big{|}f_{h}(x)-f_{h}(t)\big{|}dx$
$\displaystyle\leq$ $\displaystyle C\omega(2^{-j})$ (29)

for some positive constant $C$ by assumption (27). In order to deal with the
second term, we use in addition the fast decay of the wavelet to get the
existence of a natural number $N$ and a constant $C$ such that

$\displaystyle
2^{j}\sum_{l=0}^{j-1}\int_{B_{l}}|\psi(2^{j}x-k)|\big{|}f_{h}(x)-f_{h}(t)\big{|}dx$
$\displaystyle\leq$ $\displaystyle
C\sum_{l=0}^{j-1}\omega(2^{-l})\int_{B_{l}}\frac{2^{j}}{(1+|2^{j}x-k|)^{2N}}dx.$

Notice then that if $x\in B_{l}$, then

$1+|2^{j}x-k|\geq 1+2^{j}|x-t|-|2^{j}t-k|\geq 2^{j-l-1}.$

It implies that

$\displaystyle
2^{j}\sum_{l=0}^{j-1}\int_{B_{l}}|\psi(2^{j}x-k)|\big{|}f_{h}(x)-f_{h}(t)\big{|}dx$
$\displaystyle\leq$ $\displaystyle
C\sum_{l=0}^{j-1}\omega(2^{-l})2^{-N(j-l-1)}\int_{B_{l}}\frac{2^{j}}{(1+|2^{j}x-k|)^{N}}dx$
(30) $\displaystyle\leq$ $\displaystyle
C\sum_{l=0}^{j-1}\omega(2^{-l})2^{-N(j-l-1)}\int_{\mathbb{R}}\frac{1}{(1+|y|)^{N}}dy$
$\displaystyle=$ $\displaystyle
C^{\prime}2^{-Nj}\sum_{l=0}^{j-1}\omega(2^{-l})2^{Nl}$ $\displaystyle\leq$
$\displaystyle C^{\prime\prime}\omega(2^{-j})$

for some constants $C^{\prime},C^{\prime\prime}$ and using (7). It remains to
bound the last term. Again, the fast decay of the wavelet together with the
boundedness of the random wavelets series $f_{h}$ leads to

$\displaystyle 2^{j}\int_{\mathbb{R}\setminus
B(t,1)}|\psi(2^{j}x-k)|\big{|}f_{h}(x)-f_{h}(t)\big{|}dx$ $\displaystyle\leq$
$\displaystyle 2\|f_{h}\|_{\infty}\int_{\mathbb{R}\setminus
B(t,1)}\frac{2^{j}}{(1+|2^{j}x-k|)^{2N}}dx$ (31) $\displaystyle\leq$
$\displaystyle
2\|f_{h}\|_{\infty}2^{-Nj}\int_{\mathbb{R}}\frac{2^{j}}{(1+|y|)^{N}}dy$
$\displaystyle\leq$ $\displaystyle C2^{-Nj}$

for some positive constant $C$, since $|2^{j}x-k|\geq 2^{j}$ if $|t-x|\geq 1$.
Putting (28), (29), (30) and (31) together, we finally obtain

$\displaystyle|c_{j,k}|\leq C\big{(}\omega(2^{-j})+2^{-Nj})$

for some positive constant $C$, so that

$\limsup_{j\to+\infty}\frac{|c_{j,k}|}{\omega(2^{-j})}<+\infty$

if $N$ is chosen large enough. This contradicts the second part of (26) since
$\omega=o(\omega_{v}^{(h)})$. ∎

######  Remark 4.2.

As mentionned earlier, let us notice that if the wavelet $\psi$ is compactly
supported, and if $t$ is a rapid, ordinary or slow point, then $\omega_{v}$
gives the exact pointwise behavior of $f$ at $t$, meaning that

$0<\limsup_{s\to t}\dfrac{|f(s)-f(t)|}{\omega_{v}^{(h)}(|t-s|)}<+\infty$

where $v=r$, $v=o$ or $v=s$ respectively. Indeed, in this case, one has
directly

$\displaystyle|c_{j,k}|$ $\displaystyle\leq$ $\displaystyle
2^{j}\int_{B(t,R2^{-j})}|\psi(2^{j}x-k)|\big{|}f_{h}(x)-f_{h}(t)\big{|}dx$
$\displaystyle\leq$ $\displaystyle C\sup_{x:|x-t|<R2^{-j}}|f_{h}(x)-f_{h}(t)|$

where $R$ can be computed via the support of the wavelet and $C$ is a positive
constant.

The last Remak applies in particular to the fractional Brownian motion thanks
to the representation (3). Indeed, as $R$ is a smooth process, it does not
modify the pointwise regularity and irregularity properites.

######  Corollary 4.3.

Almost surely, the fractional Brownian motion satisfies the following property
for every non-empty interval $I$ of $\mathbb{R}$:

  * •

almost every $t\in I$ is ordinary:

$0<\limsup_{s\to
t}\dfrac{|f_{h}(s)-f_{h}(t)|}{|t-s|^{h}\sqrt{\log\log|t-s|^{-1}}}<+\infty,$

  * •

there exists $t\in I$ which is fast:

$0<\limsup_{s\to
t}\dfrac{|f_{h}(s)-f_{h}(t)|}{|t-s|^{h}\sqrt{\log|t-s|^{-1}}}<+\infty,$

  * •

there exists $t\in I$ which is slow:

$0<\limsup_{s\to t}\dfrac{|f_{h}(s)-f_{h}(t)|}{|t-s|^{h}}<+\infty.$

##  5 Extension to the multifractal case

Let us now consider a multifractal version of the previously introduced
Gaussian wavelets series. This is in the spirit of the multifractional
Brownian motion introduced in [36, 8] by replacing the constant Hurt exponent
$h$ in the stochastic integral defining this process by a continuous function
$H(\cdot)$. In this paper, we adopt the strategy of [7] which takes advantage
of wavelets series expansion. It consists in substituting the exponent $h$ at
level $(j,k)$ in (4) by $H(k2^{-j})$. In [3], the authors showed that, under
some Hölderian regularity assumptions for the function $H$, both
generalizations are equivalent in the sense that the multifractional Brownian
Motion $B_{H(\cdot)}$ and the Gaussian wavelets series associated to
$H(\cdot)$ only differs by a smooth process, similarly to (3).

Here, we just need a weaker regularity condition to obtain a generalized
version of Theorem 1.4. Namely, we consider a compact set $K\subseteq(0,1)$
and a function $H\,:\,\mathbb{R}\to K$, for which there exits a constant
$C_{H}>0$ such that

$|H(x)-H(y)|\leq\frac{C_{H}}{|\log|x-y||}$ (32)

for all $x,y$ with $|x-y|<1$. Of course, such a function $H$ is necessarily
continuous and any Hölder-continuous function satisfies (32).

This function allows to define the multifractal random wavelet series

$f_{H}=\sum_{j\in\mathbb{N}}\sum_{k\in\mathbb{Z}}\xi_{j,k}2^{-H(k2^{-j})j}\psi(2^{j}\cdot-k).$
(33)

Our goal is to show that, the function $t\mapsto H(t)$ maps any
$t\in\mathbb{R}$ to its Hölder exponent and that, even in this generalized
setting, slow, ordinary and rapid points can still be highlighted. Note that,
in [14], it is proved that, if $H$ is the function “Hölder exponent” of a
continuous function then there exists a sequence
$(P_{j})_{j\in\mathbb{N}_{0}}$ of polynomials such that

$\left\\{\begin{array}[]{l}H(t)=\liminf_{j\to+\infty}P_{j}(t)\\\\[8.61108pt] \
\|DP_{j}\|_{\infty}\leq j,\quad\forall j\in\mathbb{N}_{0}\\\
\end{array}\right.$ (34)

In the situation of deterministic wavelet series with coefficients
$(2^{-jH(k2^{-j})})_{j,k}$, conditions (34) are also sufficient to recover the
irregularity from the function $H$ [12]. Because of condition (32), our
function $H$ is not as general, but if a function $H$ checks conditions (34)
and if we assume the existence of a constant $C>0$ such that, for all
$t\in\mathbb{R}$ and $j\in\mathbb{N}_{0}$,

$|H(t)-P_{j}(t)|\leq Cj^{-1}$

then, (32) is satisfied.

######  Theorem 5.1.

Let $I$ denote any non-empty interval of $\mathbb{R}$. Almost surely, the
random wavelets series defined in (33) satisfies the following property:

  * •

For almost every $t\in I$,

$\limsup_{s\to
t}\dfrac{|f_{H}(s)-f_{H}(t)|}{\omega_{o}^{(H(t))}(|s-t|)}<+\infty$ (35)

and if $\omega$ is a modulus of continuity such that
$\omega=o(\omega_{o}^{(H(t))})$, then

$\limsup_{s\to t}\dfrac{|f_{H}(s)-f_{H}(t)|}{\omega(|s-t|)}=+\infty,$ (36)

  * •

There exists $t\in I$ such that

$\limsup_{s\to
t}\dfrac{|f_{H}(s)-f_{H}(t)|}{\omega_{r}^{(H(t))}(|t-s|)}<+\infty,$ (37)

and if $\omega$ is a modulus of continuity such that
$\omega=o(\omega_{r}^{(H(t))})$, then

$\limsup_{s\to t}\dfrac{|f_{H}(s)-f_{H}(t)|}{\omega(|s-t|)}=+\infty,$ (38)

  * •

There exists $t\in I$ such that

$\limsup_{s\to t}\dfrac{|f_{H}(s)-f_{H}(t)|}{\omega_{s}^{(H(t))}}<+\infty$
(39)

and if $\omega$ is a modulus of continuity such that
$\omega=o(\omega_{s}^{(H(t))})$, then

$\limsup_{s\to t}\dfrac{|f_{H}(s)-f_{H}(t)|}{\omega(|s-t|)}=+\infty.$ (40)

###### Proof.

The part concerning the irregularity can easily be adapted from the constant
case $H(t)=h$. It suffices to notice that the three superior limits appearing
in Corollary 4.1 and that concerns the wavelet coefficients are still stricly
postive, using (32).

The idea and techniques for the regularity are similar to the ones of
Propositions 2.2, 2.5 and 2.8 except that, in addition to “control” the
randoms coefficients with the help of the fast decay of the wavelets, we also
need to “control” the variation of the function $2^{H(\cdot)j}$, with
$j\in\mathbb{N}$. Again, we will do so using the fast decay of the wavelets.

Let us list how to modify the proofs of Propositions 2.2, 2.5 and 2.8 in order
to find the expected inequalities

Rapid points:

To bound $|Df_{H,j}(x)|$ in (14), the idea consists to split the sum over
$|k|\leq 2^{j+1}$ in two and provides additional information for the sum over
$|k|>2^{j+1}$:

  1. (i)

If $k$ is such that $|k|<2^{j+1}$ and $|2^{j}t-k|\leq 2^{j/2}$ then
$|H(t)-H(k2^{-j})|\leq 2C_{H}j^{-1}$ and thus

$\displaystyle 2^{(1-H(k2^{-j}))j}|\xi_{j,k}||D\psi(2^{j}x-k)|$
$\displaystyle\leq 2^{2C_{H}}2^{(1-H(t))j}|\xi_{j,k}||D\psi(2^{j}x-k)|$
$\displaystyle\leq
C2^{2C_{H}}2^{(1-H(t))j}\frac{\sqrt{\log(3+j+|k|)}}{(1+|2^{j}x-k|)^{4}}.$

  2. (ii)

If $k$ is such that $|k|<2^{j+1}$ and $|2^{j}t-k|>2^{j/2}$, we have

$|2^{j}x-k|>2^{j/2}-2$

and thus, if $\beta>2(\sup K-\inf K)$, we get, using the fast decay of
$D\psi$,

$\displaystyle 2^{(1-H(k2^{-j}))j}|\xi_{j,k}||D\psi(2^{j}x-k)|$
$\displaystyle\leq C\frac{2^{(1-H(t))j}2^{(\sup K-\inf
K)j}|\xi_{j,k}|}{(1+|2^{j}x-k|)^{4}(2+|2^{j}x-k|)^{\beta}}$ $\displaystyle\leq
C2^{(1-H(t))j}\frac{\sqrt{\log(3+j+|k|)}}{(1+|2^{j}x-k|)^{4}}.$

  3. (iii)

For $|k|>2^{j+1}$ of course $|2^{j}x-k|>2^{j}$ and one can use the same trick
to get

$2^{(1-H(k2^{-j}))j}|\xi_{j,k}||D\psi(2^{j}x-k)|\leq
C2^{2C_{H}}2^{(1-H(t))j}\frac{\sqrt{\log(3+j+|k|)}}{(1+|2^{j}x-k|)^{5}}.$

From this, the bound

$|\sum_{j\leq\nu}(f_{H,j}(t)-f_{H,j}(s))|\leq
CC_{2}|t-s|^{H(t)}\sqrt{\log|t-s|^{-1}}$

is obtained.

The same method applied on $|f_{j}(t)|$ ($j>\nu)$ leads to

$|\sum_{j>\nu}f_{H,j}(t)|\leq CC_{2}|t-s|^{H(t)}\sqrt{\log|t-s|^{-1}}$

while, for $|f_{H,j}(s)|$, replacing $t$ by $s$ in the reasoning, we get

$\displaystyle|\sum_{j>\nu}f_{H,j}(s)|$ $\displaystyle\leq
CC_{2}2^{-H(s)\nu}\sqrt{\nu}$ $\displaystyle\leq
CCC_{2}2^{-H(t)\nu}\sqrt{\nu}2^{(H(t)-H(s))\nu}$ $\displaystyle\leq
CC_{2}|t-s|^{H(t)}\sqrt{\log|t-s|^{-1}}\ $

as $|H(t)-H(s)|\leq(\nu-1)^{-1}$.

Ordinary points

In this case, to bound $|Df_{H,j}(x)|$ (with $j\leq\nu$), the sum for
$k\in\kappa_{j}^{t}(j)$ remains untouched as, for all such $k$, we have
$|t-k2^{-j}|\leq 2^{-j}(j+1)$ and thus we get $|H(t)-H(2^{-j})|\leq
CC_{H}j^{-1}$.

This time, this is the sum for $k\notin\kappa_{j}^{t}(j)$ that we need to
split:

  1. (i)

If $k\in\kappa_{j}^{t}(2^{j/2})\setminus\kappa_{j}^{t}(j)$, $|k2^{-j}-t|\leq
22^{-j/2}$ and, similarly to the point (i) in the rapid points, we have

$2^{(1-H(k2^{-j}))j}|\xi_{j,k}||D\psi(2^{j}x-k)|\leq
C2^{2C_{H}}2^{(1-H(t))j}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(3+|2^{j}x-k|)(1+|2^{j}x-k|)^{4}}.$

  2. (ii)

If $k\notin\kappa_{j}^{t}(2^{j/2})$ then $|2^{j}x-k|\geq 2^{j/2}-3$ and we
proceed just like in the point (ii) for the rapid points to get

$2^{(1-H(k2^{-j}))j}|\xi_{j,k}||D\psi(2^{j}x-k)|\leq
C2^{2C_{H}}2^{(1-H(t))j}\frac{\sqrt{\log(3+j+|k-k_{j}(t)|)}}{(3+|2^{j}x-k|)(1+|2^{j}x-k|)^{4}}$

again.

To bound $|f_{H,j}(t)|$ (with $j>\nu$) the strategy remains the same while to
bound $|f_{H,j}(s)|$, like in the proof of Proposition 2.2 we need to consider
separately the case where $j\geq 2^{j-\nu+2}$ and when $j<2^{j-\nu+2}$.

If $j\geq 2^{j-\nu+2}$, we take into account the three sums:

  1. (i)

the sum over $k\in\kappa_{j}^{t}(2j)$ where $|H(t)-H(2^{-j})|\leq
CC_{H}j^{-1}$ and thus which remains untouched,

  2. (ii)

the sum over $k\in\kappa_{j}^{t}(2^{j/2})\setminus\kappa_{j}^{t}(2j)$ where we
use $|H(t)-H(2^{-j})|\leq CC_{H}j^{-1}$ to deal with the exponent,

  3. (iii)

the sum over $k\notin\kappa_{j}^{t}(2^{j/2})$ where we deal with the exponent
just like in (ii) and (iii) for the rapid points.

In total, we get

$|f_{H,j}(s)|\leq CC_{t}2^{-jH(t)}\sqrt{\log(j)}$

Now if $2^{j-\nu+2}<j$ we consider:

  1. (i)

the sum over $\kappa_{j}^{s}(2^{j/2})$ for which we deal similarly to the sum
over $\kappa_{j}^{t}(2^{j/2})$ above

  2. (ii)

the sum over $\kappa_{j}^{s}(2^{j-\nu+2})\setminus\kappa_{j}^{s}(2^{j/2})$
where the exponent is treated like in (ii) and (iii) for the rapid points.

  3. (iii)

the sum for $k\notin\kappa_{j}^{s}(2^{j-\nu+2})$ where, again, we treat the
exponent like in (ii) and (iii) for the rapid points.

In total, in this case, we get

$|f_{H,j}(s)|\leq CC_{t}2^{-jH(s)}\sqrt{j-\nu}.$

The conclusion follows from the same arguments as for the rapid points.

Slow points

First, we note that the natural $m$ in Kahane procedure must be chosen such
that $1/m<\inf K$.

To bound $|Df_{H,j}^{[0]}(x)|$ (with $j\leq\nu$), we split the sum (20) in
two:

  1. (i)

for $0\leq l\leq\lfloor j/2m\rfloor$, then for all $k\in\Lambda_{j}^{l}(t)$,
we have $|x-k2^{-j}|\leq C2^{-j/2}$ and thus one uses $|H(x)-H(k2^{-j})|\leq
CC_{H}j^{-1}$ to deal with the exponent.

  2. (ii)

for $\lfloor j/2m\rfloor<l\leq\lceil j/m\rceil$, then for all
$k\in\Lambda_{j}^{l}(t)$, $|2^{j}x-k|\geq 2^{j/2}-3$ and we treat the exponent
like for the rapid points in (ii).

To bound $|Df_{H,j}^{[n]}(x)|$ with $n\neq 0$, we use $|2^{j}x-k|\geq
C_{t}|n|2^{j}$ to treat the exponent like in the rapid points in (iii).

With this, we obtain

$\displaystyle|\sum_{j\leq\nu}(f_{H,j}(t)-f_{H,j}(s))|$ $\displaystyle\leq
CC^{*}|s-t|^{\alpha}.$

We can use the same method to have

$|\sum_{j>\nu}f_{H,j}(t)|\leq CC_{\mu}2^{-H(t)\nu}$

and, using the same arguments as in the end of Proposition 2.8, with the fact
that $1/m<\inf K\leq H(s)$, we have

$|\sum_{j>\nu}f_{H,j}(t)|\leq CC_{\mu}2^{-H(s)\nu}\leq CC_{\mu}2^{-H(t)\nu}$

just as in the conclusion for the rapid points.

∎

##  6 Genericity of the non-existence of slow points

The aim of this section is to prove that the results obtained in the previous
section are spectific to these Gaussian random wavelet series. On this
purpose, let us define the Fréchet space

$C^{\nearrow h}:=\bigcap_{\alpha<h}C^{\alpha}([0,1]),$

where for every $\alpha\in(0,h)$, $C^{\alpha}([0,1])$ denote the Hölder space
of order $\alpha$. This space and its topology can be equivalently defined
using sequence of wavelet coefficients as follows: If we consider $\alpha>0$,
$\alpha\notin\mathbb{N}$, then for any $f\in C^{\alpha}([0,1])$, one has

$\sup_{j\in\mathbb{N}}\sup_{k\in\\{0,\dots,2^{j}-1\\}}2^{\alpha
j}|c_{j,k}|<+\infty$ (41)

where $(c_{j,k})_{j\in\mathbb{N},k\in\\{0,\dots,2^{j}-1\\}}$ denotes the
sequence of wavelet coefficients of $f$ on $[0,1]$. It allows to identify
(algebraically and topologically) the Hölder space $C^{\alpha}([0,1])$ with
the space of complex sequences
$(c_{j,k})_{j\in\mathbb{N},k\in\\{0,\dots,2^{j}-1\\}}$ satisfying (41), isee
[31]. When $\alpha\in\mathbb{N}$, we will also denote by $C^{\alpha}([0,1])$
the space of functions satisfying the condition (41).

We will prove that in $C^{\nearrow h}$, “most of” the functions do not present
a slow pointwise behavior as the one exhibited for Gaussian wavelet series,
i.e. does not belong to the space $C^{h}(t)$ formed by the functions $f$ such
that

$\limsup_{s\to t}\frac{|f(s)-f(t)|}{|s-t|^{h}}<+\infty.$

We will use in this purpose two notions of genericity: the prevalence and the
Baire category points of view.

The notion of prevalence supplies an extension of the notion of “almost
everywhere” (for the Lebesgue measure) in infinite dimensional spaces. In a
metric infinite dimensional vector space, no measure is both $\sigma$-finite
and translation invariant. However, one can consider a natural extension of
the notion of “almost everywhere” which is translation invariant, see [10,
18].

######  Definition 6.1.

Let $E$ be a complete metric vector space. A Borel set $A\subset E$ is Haar-
null if there exists a compactly supported probability measure $\mu$ such that

$\forall x\in E,\quad\mu(x+A)=0.$ (42)

If this property holds, the measure $\mu$ is said to be transverse to $A$. A
subset of E is called Haar-null if it is contained in a Haar-null Borel set.
The complement of a Haar-null set is called a prevalent set.

In order to prove that a set is Haar-null in a functional space $E$, one can
often use for transverse measure the law of a stochastic process, see [11, 16]
for some applications of this method. If $\mathcal{P}$ is a property that can
be satisfied by points of $E$, one can prove that $\mathcal{P}$ holds only on
a Haar-null set by exhibiting a stochastic process $X$ whose sample paths lies
in a compact subset of $E$ and such that for all $f\in E$ almost surely the
property $\mathcal{P}$ does not hold for $X+f$.

In our setting, the stochastic process that will be used is a random wavelet
series. The following result allows to get that the sample paths of this
series are almost surely in a compact set of $C^{\nearrow h}$. Let us first
describe this subset.

######  Lemma 6.2.

Let $h>0$ and let $(\alpha_{j})_{j\in\mathbb{N}}$ be a non-decreasing sequence
of $(0,h)$ with tends to $h$. The subset

$K=\left\\{f\in C^{\nearrow h}:\max_{k\in\\{0,\dots,2^{j}-1\\}}|c_{j,k}|\leq
2^{-\alpha_{j}j}\,\,\forall j\in\mathbb{N}\right\\}$

is compact in $C^{\nearrow h}$, where
$(c_{j,k})_{j\in\mathbb{N},k\in\\{0,\dots,2^{j}-1\\}}$ denotes the sequence of
wavelet coefficients of $f$.

###### Proof.

Clearly, $K$ is closed in $C^{\nearrow h}$. Since this last space is a Fréchet
space, it suffices now to prove that $K$ is totally bounded. Let us fix
$\varepsilon>0$ and $\alpha<h$. Then there exists $J\in\mathbb{N}$ such that
$2^{-\alpha_{j}j}<\varepsilon 2^{-\alpha j}$ for every $j\geq J$, which
implies that

$\sup_{j\geq J,k\in\\{0,\dots,2^{j}-1\\}}2^{\alpha j}|c_{j,k}|<\varepsilon$

for the sequence of wavelet coefficients
$(c_{j,k})_{j\in\mathbb{N},k\in\\{0,\dots,2^{j}-1\\}}$ of any $f\in K$.
Moreover, since the product

$P=\prod_{j=0}^{J-1}\prod_{k=0}^{2^{j}-1}[-2^{-\alpha_{j,k}j},2^{\alpha_{j,k}j}]$

where we have set $\alpha_{j,k}=\alpha_{j}$ for every
$k\in\\{0,\dots,2^{j}-1\\}$, is compact, one can find a finite number of
sequences $c^{1},\dots,c^{L}$ with support included in $\\{(j,k):j\leq
J-1,k\in\\{0,\dots,2^{j}-1\\}\\}$ such that

$P\subseteq\bigcup_{l=1}^{L}\prod_{j=0}^{J-1}\prod_{k=0}^{2^{j}-1}\big{(}c^{l}_{j,k}-\delta,c^{l}_{j,k}+\delta\big{)},$

where $\delta<2^{-\alpha J}\varepsilon$. For every $l$, we define
$f^{l}=\sum_{j=0}^{J-1}\sum_{k=0}^{2^{j}-1}c^{l}_{j,k}\psi(2^{j}\cdot-k)$. In
order to conclude, let us show that

$K\subseteq\bigcup_{l=1}^{L}\left\\{f\in C^{\nearrow
h}:\|f-f^{l}\|_{\alpha}<\varepsilon\right\\}.$

If $f\in K$, its truncated sequence $(c_{j,k})_{j\leq
J-1,k\in\\{0,\dots,2^{j}-1\\}}$ of wavelet coefficients belongs to $P$. Hence,
for a $l\in\\{1,\dots,L\\}$, one has

$2^{\alpha j}|c_{j,k}-c^{l}_{j,k}|\leq\delta 2^{\alpha j}<\varepsilon$

if $j\leq J-1$, and

$2^{\alpha j}|c_{j,k}-c^{l}_{j,k}|=2^{\alpha j}|c_{j,k}|<\varepsilon$

if $j\geq J$. ∎

Before stating our result, we need to recall the following wavelet “almost
characterization” of the pointwise Hölder regularity. It relies on alternative
quantities, namely the wavelet leaders. In order to define them, we need to
introduce the notation $c_{\lambda}$ to denote the wavelet coefficient
$c_{j,k}$, where $\lambda$ is the dyadic interval

$\lambda=\lambda(j,k)=\left[\frac{k}{2^{j}},\frac{k+1}{2^{j}}\right)$

Then, if $3\lambda$ denote the interval with the same center as $\lambda$ but
three times larger, the wavelet leader $d_{\lambda}$ is defined by

$d_{\lambda}=\sup_{\lambda^{\prime}\subset 3\lambda}|c_{\lambda^{\prime}}|.$
(43)

Note that this supremum is finite as soon as $f$ is locally bounded. In [20],
the author proved that if $f\in C^{h}(t)$ for some $h>0$, then there exists a
constant $C>0$ such that

$d_{\lambda(j,k_{j}(t))}\leq C2^{-\alpha j}.$ (44)

This inequality will allow us to construct wavelet series which do not belong
to $C^{h}(t)$ for every $t\in[0,1]$.

######  Proposition 6.3.

Let $h>0$. The set of functions $f$ such that $f\notin C^{h}(t)$ for every
$t\in[0,1]$ is prevalent in $C^{\nearrow h}$.

###### Proof.

Let us consider a sequence $(j_{n})_{n\in\mathbb{N}}$ satisfying
$j_{n+1}>j_{n}+\lfloor\log_{2}j_{n}^{2}\rfloor+1$ and let us set
$\alpha_{n}=h-\frac{1}{\sqrt{j_{n}}}$ for every $n\in\mathbb{N}$. Let us
define the random wavelets series

$f=\sum_{n\in\mathbb{N}}\sum_{j=j_{n}}^{j_{n+1}-1}\sum_{k=0}^{2^{j}-1}2^{-\alpha_{n}j}\varepsilon_{j,k}\psi_{j,k}$

where $(\varepsilon_{j,k})_{j\in\mathbb{N},k\in\\{0,\dots,2^{j}-1\\}}$ is a
sequence of independent $\mathcal{U}([-1,1])$ random variables. Clearly, for
every $\alpha<h$, one has

$2^{-\alpha_{n}j}\leq 2^{-\alpha j}$

if $j\in\\{j_{n},\dots,j_{n+1}-1\\}$ for $n$ large enough. Using the
characterization given in (41), it follows that $f\in C^{\alpha}([0,1])$.
Moreover, from Lemma 6.2, we know that the process $f$ takes its values in a
compact subset of $C^{\nearrow h}$.

Let us now show that almost surely $f\notin C^{h}(t)$ for every $t\in[0,1]$.
Let us fix $M\in\mathbb{N}$. For every $n\in\mathbb{N}$, if one consider the
subintervals of scale $j^{\prime}=j_{n}+\lfloor\log_{2}j_{n}^{2}\rfloor+1$ in
the supremum appearing in the definition (43) of the wavelet leaders, one has

$\displaystyle\mathbb{P}\left(\inf_{k\in\\{0,\dots,2^{j_{n}}-1\\}}d_{j_{n},k}\leq
M2^{-hj_{n}}\right)$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{2^{j_{n}}-1}\mathbb{P}\left(\sup_{\lambda^{\prime}\subseteq\lambda_{j_{n},k}}|2^{-\alpha_{n}j^{\prime}}\varepsilon_{\lambda^{\prime}}|\leq
M2^{-hj_{n}}\right)$ $\displaystyle\leq$ $\displaystyle
2^{j_{n}}\left(2^{\alpha_{n}j^{\prime}}M2^{-hj_{n}}\right)^{2^{j^{\prime}-j_{n}}}$
$\displaystyle\leq$ $\displaystyle
2^{j_{n}+j_{n}^{2}h}j_{n}^{2hj_{n}^{2}}2^{-j_{n}^{5/2}}M^{j_{n}^{2}}.$

The Borel-Cantelli Lemma implies that almost surely, one has

$d_{j_{n},k}>M2^{-hj_{n}}$

for every $n$ large enough and every $k\in\\{0,\dots,2^{j_{n}}-1\\}$. Since
$M\in\mathbb{N}$ is arbitrary, (44) gives the announced result. In order to
conclude, it suffices to prove that the previous result is still valid if we
replace the wavelet series $f$ by

$\tilde{f}=f+{g}$

for a function ${g}\in C^{\nearrow h}$. In this case, the wavelet coefficients
$2^{-\alpha_{n}j}$ of $f$ are replaced by

$2^{-\alpha_{n}j}\varepsilon_{j,k}+c_{j,k}=2^{-\alpha_{n}j}(\varepsilon_{j,k}+2^{\alpha_{n}j}c_{j,k}).$

It implies that the random variables defining the wavelets series are still
independant but no more centered since they are shifted by a deterministic
quantity. Clearly, the probabilities computed before can only become smaller,
hence the Borel-Cantelli lemma still holds. ∎

To end the paper, we show that the same result holds true if one replaces the
notion of prevalence by the genericity in the sense supplied by the Baire
category theorem. Let us recall that a subset $A$ of a Baire space $X$ is of
first category (or meager) if it is included in a countable union of closed
sets of $X$ with empty interior. The complement of a set of first category is
Baire-residual; it contains a countable union of dense open sets of $X$.

######  Proposition 6.4.

Let $h>0$. The set of functions $f$ such that $f\notin C^{h}(t)$ for every
$t\in[0,1]$ is Baire-residual in $C^{\nearrow h}$.

###### Proof.

Let us consider the non-decreasing sequence $(\alpha_{j})_{j\in\mathbb{N}}$ of
$(0,h)$ with converges to $h$ defined by $\alpha_{j}=h-\frac{1}{\sqrt{j}}$.
For every $J\in\mathbb{N}$, the set $\mathcal{C}_{J}$ is formed by the
functions $f\in C^{\nearrow h}$ whose sequence of wavelet coefficients
$(c_{j,k})_{j\in\mathbb{N},k\in\\{0,\dots,2^{j}-1\\}}$ satisfies

$2^{\alpha_{j}j}|c_{j,k}|\in\mathbb{N}\setminus\\{0\\}\quad\forall j\geq
J\,,\forall k\in\\{0,\dots,2^{j}-1\\}.$

Finally, we define the open sets $U_{J}$ by

$U_{J}=\bigcup_{j\geq J}\left\\{g\in C^{\nearrow h}:\exists
f\in\mathcal{C}_{j}\text{ such that
}\|f-g\|_{\alpha_{j}}<\frac{1}{2}\right\\}.$

Notice that if $g\in U_{J}$ with wavelet coefficients
$(e_{j,k})_{j\in\mathbb{N},k\in\\{0,\dots,2^{j}-1\\}}$, then there is
$f\in\mathcal{C}_{j_{0}}$ for a $j_{0}\geq J$, with wavelet coefficients
$(c_{j,k})_{j\in\mathbb{N},k\in\\{0,\dots,2^{j}-1\\}}$ such that

$|e_{j_{0},k}|\geq|c_{j_{0},k}|-|e_{j_{0},k}-c_{j_{0},k}|\geq
2^{-\alpha_{j_{0}}j_{0}}-\frac{1}{2}2^{-\alpha_{j_{0}}j_{0}}=\frac{1}{2}2^{-\alpha_{j_{0}}j_{0}}$

for every $k\in\\{0,\dots,2^{j_{0}}-1\\}$. Now, assume that $g$ belongs to the
set $\mathcal{R}$ defined by

$\mathcal{R}=\bigcap_{J\in\mathbb{N}}U_{J}.$

If there exists $t\in[0,1]$ such that $g\in C^{h}(t)$, (44) gives the
existence of a constant $C>0$ such that

$d_{j,k_{j}(t)}\leq C2^{-hj}.$

Since $g\in\mathcal{R}$, one gets

$\frac{1}{2}2^{-\alpha_{j}j}=\frac{1}{2}2^{-hj+\sqrt{j}}\leq C2^{-hj}$

for infintely many $j$, which is impossible. Consequently, $g\notin C^{h}(t)$
for every $t\in[0,1]$.

To conclude, it suffices to prove that the open sets $U_{J}$ are dense in
$C^{\nearrow h}$. Let us fix $J\in\mathbb{N}$, $g\in C^{\nearrow h}$,
$\alpha<h$ and $\varepsilon>0$. Let $J_{0}\geq J$ be large enough to ensure
that both $\alpha<\alpha_{J_{0}}$ and
$2^{(\alpha-\alpha_{J_{0}})J_{0}}<\varepsilon$ are satisfied. We construct the
function $f$ via its sequence of wavelet coefficients by setting

$c_{j,k}=\begin{cases}e_{j,k}&\text{ if }j<J_{0}\\\
2^{-\alpha_{j}j}[2^{\alpha_{j}j}e_{j,k}]&\text{ if }j\geq J_{0}\text{ and
}2^{\alpha_{j}j}|e_{j,k}|\geq 2,\\\ 2^{-\alpha_{j}j}&\text{ if }j\geq
J_{0}\text{ and }2^{\alpha_{j}j}|e_{j,k}|<2,\end{cases}$

so that $f\in\mathcal{C}_{J_{0}}\subseteq U_{J}$ and
$|2^{\alpha_{j}j}c_{j,k}-2^{\alpha_{j}j}e_{j,k}|\leq 1$ for every $j\geq
J_{0}$. Hence one has

$2^{\alpha j}|e_{j,k}-c_{j,k}|\leq 2^{(\alpha-\alpha_{j})j}\leq
2^{(\alpha-\alpha_{J_{0}})J_{0}}<\varepsilon$

for every $j\geq J_{0}$, which allows to conclude. ∎

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