Convergence of Random Fractional Wavelet Series and some of its Properties

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Convergence of Random Fractional Wavelet Series and some of its Properties

Juan Miguel Medina, Fernando Dobarro, Bruno Cernuschi-Frias

To cite this version:

Juan Miguel Medina, Fernando Dobarro, Bruno Cernuschi-Frias. Convergence of Random Fractional Wavelet Series and some of its Properties. 2018. �hal-01946303�

Convergence of Random Fractional Wavelet Series and some of its Properties

BY

Juan M. Medina, Fernando R. Dobarro

and

Bruno Cernuschi-Frías

Abstract. For appropriate orthonormal wavelet basis {ψ_{j k}^e }_j∈

Zk∈Z^de∈{0,1}^d, con- stants p and γ, if I_γ denotes the Riesz fractional integral operator of order γ and (ηj k e)_j∈Zk∈Z^d_e∈{0,1}^d a sequence of independent identically distributed sym- metric p-stable random variables, we investigate the convergence of the series

P

j k e

η_{j k e}I_γψ_{j k}^e . Similar results are also studied for modified fractional integral operators.

Keywords and phrases: Fractional Processes, Wavelets

2010 Mathematics Subject Classifications:60G22 (Primary), 42C40, 60G20 (Secondary)

1 Introduction

It is of practical interest to obtain uncoupled representations of random processes which, on the other hand, may posses some interesting statisti- cal properties for applications. A classical example for Gaussian processes is the Karhunen-Lóeve (KL) representation. Motivated in part by applications in signal and image processing [2, 18], a usual requirement for a random pro- cess defined on R^d is to be self similar (see section 2.2) in some previously specified sense, since there exists several related notions in the literature. For the finite variance case, several KL like representations for the family of 1 f of self-similar processes and related were proposed e.g. [2, 4, 13, 18] among others. In this case, these representations have in general the form:

X_γ =^X

I

η_II_γψ_I, (1)

whereI_γis some fractional integration operator,{ψ_I}_I is an orthonormal ba- sis ofL²(R^d)or other Hilbert space of functions and the{η_I}_Iis a sequence of finite variance identically distributed random variables, in most cases Gaus- sian. The parameterγis usually linearly related to the self-similarity orHurst parameter H of the process [3]. Apart from applications, series like (1) and its geometric properties were extensively studied in the case of Fourier Gaus- sian random series, see for example [9]. Considering this sum as ageneralized random process in the sense of Gelfand and Vilenkin [5, Chapter 3, p. 237], if the η_I’s are Gaussian and I_γ is the Riesz fractional integration operator (definition 3) then this sum converges a.s. in the sense of distributions, i.e.

in D⁰(R^d) to a self-similar process as defined here in Section 2.2 in terms of equality in probability law betweenX_γand a re-scaled version of it: a^δX_γ(a .) for some δ ∈ R. Indeed, in this particular case X_γ is a fractional Gaussian noise (See Theorem 3.2). This type of representations have received some interest for its simplicity for modelling certain random signals (see e.g. [18]) since one only needs to know the probability distribution of the coefficients η_I and the parameter γ or similar. On the other hand, the finite variance requirement may be a constraint in some applications. A first and almost obvious attempt to overcome this limitation, retaining at the same time some of the properties of interest of X_γ, is to substitute theη_I’s by non Gaussian p-stable random variables, p ∈ (0,2) [16, Chapter 0]. However, it may be- come a no trivial task to check which of these properties are preserved. For example, besides self similarity, in [15] is proved that it is not possible to represent a p-stable stationary random process by a series like (1). In this work we prove that for appropriate parameters γ ≤ d

2 and p, if we consider {ψ_I}_I a suitable wavelet basis, the series (1) stills converges a.s. in D⁰(R^d) and changing I_γ by a modified operator, it converges to an ordinary process for the case d

2 < γ ≤ d

2 + 1. If p= 2 the limit of the series (1) is self similar of parameter d

2+γ and in the case p6= 2 although its limit is not necessarily self similar we can prove that the distribution function of the re-scaled pro- cess a^d2+γX_γ(a .)is, in some sense, properly stochastically dominated. In the Gaussian case of p = 2 the series of equation (1) converges to a fractional Gaussian noise, which an integrated version of it gives the well known frac- tional Brownian motion and its d-dimensional analogues with their known

“fractal” properties. We shall see, for appropriate parameters p and γ, that integrated versions of the process Xγ have a graph with Hausdorff dimension greater than d, justifying the possible use of the process defined by (1) as a model of fractal process still for p6= 2.

2 Auxiliary results and definitions.

2.1 Function spaces, Fourier transforms and Wavelets.

On the following, if p∈ [1,∞] and µis a Borel measure over R^d, the corre- sponding Lebesgue spaces of functions (indeed equivalence classes) associated to it will be denoted by L^p(R^d, dµ), and if µ is the usual Lebesgue measure, we will write shortly L^p(R^d). When p = 2 it becomes a Hilbert space and the L²(R^d)inner product will be denoted byh. , .i. Ifx∈C^d(d∈N)we will denote its usual norm by |x| and the support of a function f is defined by supp(f) ={x :f(x)6= 0}. The Schwartz class of functions S(R^d) is defined as the linear space of smooth functions rapidly decreasing at infinity, together with its derivatives, this means that φ∈ S(R^d)whenever φ ∈C^∞ÄR^d

ä and sup

(x1,...xd)∈R^d d

Y

i=1

|x_i|^αi

∂

∂x^β₁¹... ∂

∂x^β_d^dφ(x₁, ...x_d)

<∞ ∀α_jβ_j ∈ N,

endowed with its usual topology. We will denoteD(R^d)the space of functions which are in C^∞ÄR^d

ä and have compact support. Both spaces are topolog- ical vector spaces (For more details see [7, Chapter 2, p. 109], and their duals are denoted as: S⁰(R^d) (Tempered distributions) andD⁰(R^d)(distribu- tions) respectively. Clearly: D(R^d)⊂ S(R^d)and thenS⁰(R^d)⊂ D⁰(R^d). The Fourier Transform f^boff ∈ S(R^d)is defined asf^b(λ) = ^R

R^d

f(x)e^−2πiλ.xdx .It is a known fact that f^balso belongs to the space S(R^d). The Fourier trans- form can be defined, as usual as a linear map over L¹(R^d), as an isometry on L²(R^d)or over the class of tempered distributions. The inverse Fourier trans- form

∨

f is defined in an analogous way. For more references about Fourier transforms and series we refer the reader, for example, to [7].

Later, we will need a variant of the classic sampling theorem of Shannon, Nyquist and Kotelnikov.

Theorem 2.1. If f ∈L²(R^d) is such thatsupp(f)⊂[−x_o, x_o]^d withx_o < 1 2. Then there exists φ∈ S(R^d) such that

fb(λ) = ^X

k∈Z^d

fb(k)φ(λ−k) (2)

Proof. Let f^e(x) = ^P

k∈Z^d

f(x+k) be the periodization of f. f^e verifies f^e ∈

L²

ñ

−1 2,1

2

ôd!

⊂L¹

ñ

−1 2,1

2

ôd!

and therefore f^ehas Fourier series given by

X

k∈Z^d

a_ke^−2πix.k,

and then lim

R→∞

P

k∈DR

a_ke^−2πix.k = f^e a.e. and in L¹

ñ

−1 2,1

2

ôd!

(and in L²) norm for a suitable domainD_R∈R^d. Now, we can takeφ∈ S(R^d)such that

∨

φ(x) =

® 1,|x_i|< x₀

0,|x_i| ≥1−x₀ .

Defining S_R(x) =

∨

φ(x)

Ç P

k∈D_R

a_ke^−2πix.k

å

then is easy to check thatf =f^e

∨

φ and that lim

R→∞kS_R−fk_L1(R^d) = 0. This implies

R→∞lim sup

λ∈R^d

S”_R(λ)−f(λ)^b = 0, but (see e.g. [7, Excersise 3.6.4, p.236]): a_k =f(k), then^b

S”R(λ) = ^X

k∈D_R

fb(k)φ(λ−k).

Then (2) follows immediately from this.

In our developments we will sometimes use some fractional integral oper- ators, let us review some of their properties.

We begin with a definition ([8, Chapter 6, p. 2] or [17, Chapter 5, p. 117]):

Definition 2.2. Let 0 < α < d. For f ∈ S(R^d) we can define its Riesz Potential:

(I_γf)(x) = 1 Cγ

Z

R^d

f(y)

|x−y|^d−γdy (3)

where Cγ =

π^d/22^αΓ

Åγ 2

ã

Γ

Çd 2− γ

2

å .

Riesz potentials have the following scaling property: for every a 6= 0:

I_γ(f(a .)) = |a|^−γ(I_γf)(a .)A crucial result for this integral operators is the following [8, Chapter 6, p.3] :

Theorem 2.3. (Hardy, Littlewood and Sobolev) Let 0< γ < d, 1≤p < q <

∞ and 1 q = 1

p − γ d then:

(a) For all f ∈L^p(R^d), the integral that defines I_γf converges a.e.

(b)If p > 1 then

kI_γfk_Lq(R^d) ≤C_pqkfk_Lp(R^d) . (4) Note that, in the appropriate sense, the Fourier Transform ofI_γf is given by:

I‘_γf(λ) = (2π)^−γ|λ|^−γf(λ)^b (5) and that it is easy to check that forf ∈ S(R^d)andα+β < dthenI_α(I_βf) = I_α+β(f). Furthermore, if ∆f = ^Pd

j=1

∂²f

∂x_j² is the Laplacian of f , ∆(I_γf) = Iγ−2f. Finally, Iγ can be thought as defined by the convolution with the locally integrable function k_γ(x) = 1

C_γ 1

|x|^d−γ and is formally self adjoint in the sense that for every f, g ∈ S(R^d):

hI_γf, gi=hf,I_γgi. (6)

Considering again k_γ, we can define a fractional integral operator for f ∈ L^p(R^d), in the following way:

K_γf(x) =

Z

R^d

(k_γ(x−y)−k_γ(y))f(y)dy =

Z

R^d

K_γ(x, y)f(y)dy

The modified kernel K_γ(x, y) =k_γ(x−y)−k_γ(y)is easier to control, and we can sketch the proof of the following lemma:

Lemma 2.4. If 1< p <∞ and 0< d

Ç

1− 1 p

å

< γ < d

Ç

1− 1 p

å

+ 1, then K_γ(x, .)∈L^p(R^d) and moreover:

(i) There exists a positive constant C_{p β d} such that for each x∈R^d: kK_γ(x, .)k_Lp(R^d)=C_{p γ d}|x|^γ−(¹⁻¹p)^d.

(ii) For everyx, x⁰ ∈R^d: kK_γ(x, .)−K_γ(x⁰, .)k_Lp(R^d) =kK_γ(x−x⁰, .)k_Lp(R^d). Proof. (sketch) Since

kK_γ(x, .)k^p_Lp(R^d)=

Z

{|y|<2|x|}

|K_γ(x, y)|^pdy+

Z

{|y|≥2|x|}

|K_γ(x, y)|^pdy .

The condition d

Ç

1−1 p

å

< γ gives the appropriate exponent for the bound- edness of the first integral. In addition, sinceγ < d

Ç

1− 1 p

å

+ 1and consid- ering that for some positive constant C

|K_γ(x, y)| ≤C|x−y|^γ−d−1|x|,

if |y| > 2|x|, then the second integral is also finite. From all this, the map x 7→ kK_γ(x, .)k_Lp(R^d) is well defined and by a change of variable is easy to check that it is also an homogeneous function depending only on |x| from which assertion (i) follows. (ii) is also obtained by a change of variable.

For fixed x ∈R^d we note that in the Fourier domain K_γ can be charac- terized, in an appropriate sense [2, Chapter 3, p.45] by:

Kγf(x) = 1 (2π)^γ

Z

R^d

e^−2πiλx−1

|λ|^γ

!

f(λ)dλ .b (7)

Some formal manipulations shows that combing equations (5) and (7), for suitable parameters β and γ:

(I¤_γK_β(x, .))(λ) = K^¤_β+γ(x, .)(λ) = 1 (2π)^γ+β

e^−2πiλx−1

|λ|^β

! 1

|λ|^γ. (8) and

K_γ(I_βf)(x) =K_β+γf(x) =

Z

R^d

K_β+γ(x, y)f(y)dy . (9) For s ∈ R another related operator J_sf is defined, formally, by its Fourier transform as:

J‘_sf(λ) = (1 +|λ|²)^s/2fˆ(λ). (10) Theorem 2.5. [8, Chapter 6, p.8] If s < 0 and p ≥ 1, J_s : L^p(R^d) −→

L^p(R^d) defines a continuous linear operator i.e. there existsC_p >0such that kJ_sfk_Lp(R^d)≤C_pkfk_Lp(R^d) .

For1< p <∞, and s∈R, we introduce the Sobolev spaces H_s^p(R^d):

H_s^p(R^d) =^¶f ∈ S⁰(R^d) : J_sf ∈ L^p(R^d)^© .

These are Banach spaces of tempered distributions with the norm defined by kfk_H^p

s(R^d) = kJ_sfk_Lp(R^d). Moreover ([14] p.168), if s ≥ 0, this norm is

equivalent to kfk_Lp(R^d)+(|.|^sf)ˆ^∨

L^p(R^d). Recalling again equation (7) the equivalence of norms for K_γ(x, .) takes the following form which will be useful in the sequel:

kK_γ(x, .)k_H^p

s(R^d) ∼ kK_γ(x, .)k_Lp(R^d)+k(Kγ−s(x, .)k_Lp(R^d) . (11) In the particular case s =−d, only when p= 2 the H_s^p(R^d) spaces coincide with the followingFL^p_w spaces, which are introduced for auxiliary purposes.

Proposition 2.6. For 1≤p≤2, the space

FL^p_w =ⁿf ∈ S⁰(R^d) : f(1 +ˆ |.|²)^−d∈ L^p(R^d)^o

is a Banach space with the norm defined by kfk_FLp

w =f(1 +ˆ |.|²)^−d

L^p(R^d). Moreover convergence in FL^pw implies convergence in S⁰(R^d).

Proof. Observe that defining w(λ) = (1 +|λ|²)^−d then f ∈ FL^p_w if and only if fˆ ∈ L^p(R^d, wdλ). Let (f_n)n∈N be a Cauchy sequence en FL^p_w which is equivalent to ( ˆfn)n∈N being a Cauchy sequence in L^p(R^d, wdλ), and then there exists a unique g ∈ L^p(R^d, wdλ) such that fˆ_n−g

L^p(R^d,wdλ) −→ 0, when n −→ ∞. We shall verify that g ∈ S⁰(R^d) and therefore taking f :=

g^∨ ∈ S⁰(R^d)we are done. For this take 1 p+1

q = 1 and m > d

Ç

1 + 2q p

å

then by Hölder’s inequality:

Z

R^d

|g(λ)|

(1 +|λ|)^mdλ=

Z

R^d

|g(λ)|

(1 +|λ|)^m

(1 +|λ|²)^dp (1 +|λ|²)^dp

dλ

≤

Ö Z

R^d

|g(λ)|^p(1 +|λ|²)^−ddλ

è¹_pÖ Z

R^d

(1 +|λ|²)^dqp (1 +|λ|)^m dλ

è¹_q

<∞,

thus (See e.g. [7, Exercise 2.3.1, p.122]) g ∈ S⁰(R^d) and thereforef ∈ FL^p_w. Finally, f_n −→f

n−→∞ in FL^p_w if and only if fˆ_n−→fˆ

n−→∞ in L^p(R^d, wdλ). Let ϕ ∈ S(R^d), then, if 1

p+1

q = 1, by the definition of Fourier Transform of a tempered distribution and Hölder’s inequality one gets:

|hfn, ϕi − hf, ϕi|=|hfˆn−f , ϕˆ ^∨i|=

Z

R^d

( ˆfn(λ)−f(λ))ϕˆ ^∨(λ)dλ

=

Z

R^d

( ˆfn(λ)−fˆ(λ))ϕ^∨(λ)(1 +|λ|²)^dp (1 +|λ|²)^dp

dλ

≤

Ö Z

R^d

|fˆ_n(λ)−fˆ(λ)|^p 1

(1 +|λ|²)^ddλ

è¹_p Ö Z

R^d

|ϕ^∨(λ)|^q(1 +|λ|²)^dqpdλ

è¹_q

,

which proves the last assertion of Proposition 2.6.

The following estimate for theFL^pw norm will be useful in the sequel.

Lemma 2.7. Let 1 ≤ p ≤ 2, then L²(R^d) ⊂ FL^p_w and moreover, if Q =

ñ

−1 4,1

4

åd

, there exits a positive constant Cp d such that for everyf ∈L²(R^d), f = 0 a.e. in Q^c, the following inequality holds:

kfk^p_FLp

w ≤C_{p d} ^X

k∈Z^d

|fˆ(k)|^p(1 +|k|²)^−d. (12) Proof. If p = 2 is immediate. To prove the first assertion for p 6= 2, by Hölder’s inequality one has the following estimate

kfk^p_FLp

w ≤ kfk^p_L2(R²)

Ö Z

R^d

dλ (1 +|λ|²)^d/(1−p2)

è1−^p₂

.

For the second assertion, under these conditions we can write f(λ) =ˆ ^X

k∈Z^d

f(k)φ(λˆ −k),

as in Theorem 2.1 and therefore:

kfk_FLp w =

Z

R^d

|f(λ)|ˆ ^p(1 +|λ|²)^−ddλ

≤

Z

R^d

Ñ X

k∈Z^d

|fˆ(k)||φ(λ−k)|(1 +|λ|²)^−d/p

ép

dλ

≤

Z

R^d

Ñ X

k∈Z^d

|f(k)||φ(λˆ −k)|2^d/p(1 +|k|²)^−d/p(1 +|λ−k|²)^d/p

ép

dλ (13) since (1 +|λ|²)^−d ≤2^d(1 +|k|²)^−d(1 +|λ−k|²)^d by Peetre’s inequality. Now, if 1

p +1

q = 1, take a_k(λ) =|φ(λ−k)|^1q and

b_k(λ) =|fˆ(k)||φ(λ−k)|2^dp(1 +|k|²)^−dp(1 +|λ−k|²)^dp|φ(λ−k)|^1p,

by Hölder’s inequality we get:

kfk_FLp w ≤

Z

R^d

X

k∈Z^d

|b_k(λ)|^p

Ñ X

k∈Z^d

|a_k(λ)|^q

é^p_q

dλ , (14)

finally, taking into account that for some positive constant C:

X

k∈Z^d

|a_k(λ)|^q = ^X

k∈Z^d

|φ(λ−k)| ≤C,

equation (14) becomes

≤2^dC

Z

R^d

X

k∈Z^d

|f(k)|ˆ ^p(1 +|k|²)^−d(1 +|λ−k|²)^d|φ(λ−k)|dλ

= 2^dC

Z

R^d

(1 +|λ|²)^d|φ(λ)|dλ ^X

k∈Z^d

|f(k)|ˆ ^p(1 +|k|²)^−d.

2.2 Some probability, stable laws and generalized ran- dom processes.

Let (Ω,F,P) be a probability space and X a random variable variable defined on it. The distribution function of X is defined, for x ∈ R, as F_X(x) = P(X ≤ x). If ϕ is any Borel measurable real function, we will denote the expectation of ϕ(X) with E(ϕ(X)). The characteristic function of X is Φ_X(ξ) =E(e^iξX). For p∈ (0,2], we say that a random variable η is symmetric p-stable of parameter σ >0if Φ_η(ξ) =e^−σp|ξ|p and this situation will be denoted by η ∼ SpS. When we write F_ηp we shall be referring to the distribution function of such a random variable with σ = 1. Note that p = 2 corresponds to the Gaussian case and therefore η ∼ N(0, σ). Let us list some basic properties of stable distributions, for more references see [16, Chapter 1, p.10] and [10, Chapter 0, p.5].

1. If η₁, . . . , η_n are independent and η_i ∼ SpS, with parameter σ_i then

n

P

i=1ηi ∼SpS, with σ⁰ =k(σ_ηi)ik_lp.

2. Let p < 2. If η ∼ SpS and 0 < r < p then (E|η|^r)^1/r = C_rσ_η, where C_r^r=E|η_p|^r, and E|η|^r =∞ forr ≥p.

Now, let µ be a non negative Borel measure on R^d. We shall need a result on the a.s. convergence of random elements in L^r(R^d, dµ). This Theorem is a particular case of a more general one in [10, Chapter 2].

Theorem 2.8. Let 0< r < p <2, {f_j}j∈N⊂L^r(R^d, dµ), and let {η_j}j∈N∼ SpS be a sequence of independent and identically distributed random vari- ables. Then the series ^P∞

i=1ηifi converges in L^r(R^d, dµ) a.s. if and only if

∞

X

i=1

|fi|^p

!1/p _Lr(

R^d,dµ)

<∞.

Our results, are aimed at the construction of certain random variables taking values in D⁰(R^d). In this case, everyD⁰(R^d)- valued random variable, say X, takes the form of a random linear functional defined on D(R^d). Pre- viously, we will also need to define the class of generalized random processes, of which these D⁰(R^d)- valued random variables are particular cases. Fol- lowing [5, Chapter 3, p.237] and [18, Chapter 4, p.57], we will say that a generalized random functional is defined on D(R^d) if for every ϕ ∈ D(R^d) there is associated a real valued random variable X(ϕ) = hX, ϕi. In accor- dance with the way that one usually specifies the probability distributions of a countable set of real random variables, given n ∈ N, ϕ₁, . . . , ϕ_n ∈ D(R^d) one gives the probability of the events, {ak ≤ hX, ϕki < bk}, k = 1, . . . , n , and these probability distributions are compatible in the usual sense. On the other hand, the linearity means that for any a, b ∈ R, ϕ, ψ ∈ D(R^d):

hX, aϕ+bψi=ahX, ϕi+bhX, ψi a.s for a comprehensive reference on this topic see [5]. In an analogous way to real valued random variables, for each ϕ ∈ D(R^d) we can calculate the characteristic function of the real random variable hX, ϕi, ΦhX,ϕi(ξ) =E(e^iξhX,ϕi). In fact ifξ= 1 and consideringϕas a variable, this gives the characteristic functional of X,Φ_X(ϕ) = E(e^ihX,ϕi), which completely determines its distributions as in the case of ordinary ran- dom processes. Finally, self-similarity, for generalized random processes can be defined in the following analogous way to [18, p.178]: If there exists a constant δ >0 such that

Φ_X(ϕ) = Φ_X(a^δϕ(a .)), (15) in distribution, for every dilation factor a > 0 and ϕ∈ D(R^d). This means that X is equivalent, in probability law, to a^rX(. /a), for some appropri- ate constant r. In this context, it would be useful to recall the Hausdorff dimension [3, Chapter 2, p.21] of a subset A of R^d denoted by dim_H(A).

Although self similarity is associated to the notion of “fractality”, the last

one has not a precise meaning. However subsets of R^d with a non integer Hausdorff dimension are regarded to display a fractal behaviour. A common task is to study the fractal behaviour of the graph of a function calculating its Hausdorff dimension. Usually, the estimation of a lower bound for this value is calculated by potential methods ([3, Chapter 2, p.26][9, Chapter 10, p.132]). A direct application of it takes form in the following:

Lemma 2.9. If B is a compact subset of R^d and G ⊂ R^d+1 denotes the graph of a measurable function f : B −→ R and ^R

B

R

B

(|x− x⁰|² + |f(x)− f(x⁰)|²)^−ρ/2dxdx⁰ <∞ then dim_H(G)> ρ.

Other related results, if needed, will be introduced within the text and used in the final section to estimate the Hausdorff dimension of certain pro- cesses arising from the construction introduced in equation (1).

2.3 Wavelets.

Let{ψ_{j k}^e }_j∈Zk∈Z^d_e∈E (withE ={0,1}^d) be an orthonormal wavelet basis [14, Chapter 2] of L²(R). Obviously, in this case one has Parseval’s identity:

kfk²_L2(R²) = ^X

e∈E

X

j∈Z

X

k∈Z^d

|hf, ψ^e_{j k}i|², (16) therefore the norm kfk²_L2(R²) can be estimated from the wavelet coefficients hf, ψ_{j k}^e i. Under some additional conditions, for example if the wavelet basis arises from ar-regular wavelet multirresolution approximation ofL²(R^d)and if {I_{j k}}_j∈Z,k∈Z^d denotes the family of dyadic cubes of R^d, for some positive constants c_p, c_{p s}, C_p, C_{p s} we have the following estimations for the L^p(R^d) and H_s^p(R^d)norms respectively [14, Chapter 6]:

c_pkfk_Lp(R^d)≤

Ñ X

j k e

|hf, ψ^e_{j k}i|²2^dj1_{Ij k}

é¹_{2 Lp(}

R^d)

≤C_pkfk_Lp(R^d) , (17)

and for 0≤s≤r, c_{p s}kfk_H^p

s(R^d) ≤

Ñ X

j k e

|hf, ψ_{j k}^e i|²(1 + 4^sj)2^dj1_{Ij k}

é¹_{2 L}p(R^d)

≤C_{p s}kfk_H^p

s(R^d). (18) In order to simplify the notation involving wavelet expansions we will some- times omit the summation limits as in equations (17) and (18).

3 Main Results.

3.1 Convergence.

First, we shall prove an inequality involving the l^p norm of the wavelet coef- ficients of a function. This inequality implies as a by side product one case of Sobolev’s embeddings (See e.g. Theorem 7.57 of [1] and further).

Theorem 3.1. Let {ψ_{j k}^e }j k e be anr-regular orthonormal wavelet basis, 1<

p <2 and d

Ç1 p − 1

2

å

< s < r then there exists a positive constant C_{p s} such that:

kfk_L2(R^d) ≤

Ñ X

j k e

|hf, ψ_{j k}^e i|^p

é¹_p

≤C_{p s}kfk_H^p

s(R^d) , (19)

for all f ∈H_s^p(R^d). If p= 2, the inequality (19) holds for s≥0.

Proof. The case p= 2is immediate since kfk_L2(R^d) ≤H_s²(R^d). If 1< p≤2, the lower bound holds, since

X

j k e

|hf, ψ_{j k}^e i|^p ≥

Ñ X

j k e

|hf, ψ^e_{j k}i|²

é^p₂

=kfk^p_L2(R^d) .

The upper bound is obtained splitting the sum:

X

j k, e

|hf, ψ_{j k}^e i|^p = ^X

j<1k e

|hf, ψ^e_{j k}i|^p+ ^X

j≥1k e

|hf, ψ^e_{j k}i|^p.

Then for each e ∈E:

X

j≥1k

|hf, ψ^e_{j k}i|^p =

Z

R^d

X

j≥1

2^jd(^1−p2)4^−jsp24^jsp2

Ñ X

k∈Z^d

|hf, ψ_{j k}^e i|²1_{Ij k}(x)2^jd

é^p₂

dx ,

since I_{j k}^TI_{j k}⁰ =∅ if k 6=k⁰. The inner integrand can be rewritten as

X

j≥1

2^jd(^(1−p₂^)−sp_d)4^−jsp2

Ñ X

k∈Z^d

|hf, ψ_{j k}^e i|²1_{Ij k}2^jd

é^p₂

≤

Ñ X

j≥1

Å

2^jd(^(1−p₂^)−sp_d)^ã

2 2−p

é^2−p₂ Ñ X

j≥1

4^{js X}

k∈Z^d

|hf, ψ_{j k}^e i|²1_{Ij k}2^jd

é^p₂

,

by Hölder’s inequality with exponents 2

p and 2

2−p and sinces > d

Ç1 p − 1

2

å

. Hence

X

j≥1k

|hf, ψ^e_{j k}i|^p ≤C_{d p s}

Z

R^d

Ñ X

j≥1

4^{js X}

k∈Z^d

|hf, ψ^e_{j k}i|²1_{Ij k}(x)2^jd

é^p₂

dx (20)

X

j≥1k

|hf, ψ^e_{j k}i|^p ≤C_{d p s}⁰ kfk^p_H^p

s(R^d) .

Now we give a bound on the other term. Similarly to the previous case:

X

j<1k

|hf, ψ_{j k}^e i|^p =

Z

R^d

X

j<1

2^jd(^1−p₂)

Ñ X

k∈Z^d

|hf, ψ^e_{j k}i|²1I_{j k}(x)2^jd

é^p₂

dx .

Therefore by by Hölder’s inequality with exponents 2

p and 2 2−p, if

C_{d p}⁰⁰ =

Ñ X

j<1

2^jd(^1−p2)2−p²

é^2−p_p

,

we get

X

j<1k

|hf, ψ_{j k}^e i|^p ≤C_{d p}⁰⁰

Z

R^d

Ñ X

j<1

X

k∈Z^d

|hf, ψ_{j k}^e i|²1_{Ij k}(x)2^jd

é^p

2

dx

≤C_{d p}⁰⁰ kfk_Lp(R^d) ≤C_{d p}⁰⁰ kfk_H^p

s(R^d) (21)

Combining equations (20) and (21) and sinceE is finite we get the result.

Now, we can prove one of the main results of this work.

Theorem 3.2. Let {ψ_{j k}^e }_{j k e} be an r-regular orthonormal wavelet series, d

Ç1 p − 1

2

å

< γ ≤ d

Ç

1−1 p

å

, 3

4 ≤p≤2, γ < r and (η_{j k e})_{j k e} a sequence of independent identically distributed random variables such that η_{j k e} ∼ SpS. Then the series defined by

X_γ =^X

j k e

η_{j k e}I_γψ^e_{j k}

converges a.s. in D⁰(R^d). If p= 2, the result remains true for 0≤γ ≤ d 2.

Proof. We shall prove the case p < 2, the p = 2 case is very similar us- ing Parseval’s identity instead of Theorem 3.1. Let Q =

ñ−1 4 ,1

4

åd

, since (I_γψ_{j k}^e )1_Q ∈ L²(R^d), then by lemma 2.7,

(I_γψ_{j k}^e )1_Q

FL^pw

≤C_{p d} ^X

n∈Z^d

|(I^¤_γψ_{j k}^e )1_Q(n)|^p(1 +|n|²)^−d,

thus

X

j k e

(Iγψ_{j k}^e )1Q

p

FL^p_w ≤Cp d

X

n∈Z^d

X

j k e

|(I^¤γψ_{j k}^e )1Q(n)|^p(1 +|n|²)^−d

=C_{p d} ^X

n∈Z^d

(1 +|n|²)^−dX

j k e

|(I^¤_γψ^e_{j k})1_Q(n)|^p, (22) but, if e_n(x) =1_Q(x)e^i2πnx, a density argument applied to equation (6) gives:

(I¤_γψ^e_{j k})1_Q(n) = h(I_γψ^e_{j k})1_Q, e_ni=hψ_{j k}^e ,I_γe_ni=hI_γe_n, ψ_{j k}^e i, therefore, by Theorem 3.1, and taking γ =s :

X

j k e

|(I^¤_γψ_{j k}^e )1_Q(n)|^p =^X

j k e

|hI_γe_n, ψ_{j k}^e i|^p ≤C_{p s}kI_γe_nk_H^p

s(R^d) (23)

≤C_{p s}⁰ (kIγenk_Lp(R^d)+kIγ−senk_Lp(R^d))≤C_{p γ}⁰ (kenk_Lr(R^d)+kenk_Lp(R^d)). (24) Where the last inequality holds by the Hardy-Littlewood and Sobolev In- equality with exponents 1

r − 1 p = γ

d, note that the validity of this last step is granted since 4

3 ≤ p ≤ 2 and d

Ç1 p− 1

2

å

≤ γ ≤ d

Ç

1− 1 p

å

. Moreover kenk_Lr(R^d)+kenk_Lp(R^d) is finite and constant in n. Thus from the definition of FL^p_w combined with equations (24), (23) and (22):

Z

R^d

X

j k e

(I¤_γψ_{j k}^e )1_Q(λ)

p

(1 +|λ|²)^−ddλ (25)

= ^X

j k e

(I_γψ_{j k}^e )1_Q^p

FL^pw

≤C_{p d} ^X

n∈Z^d

(1 +|n|²)^−dX

j k e

|(I^¤_γψ_{j k}^e )1_Q(n)|^p <∞.

Taking any 1< r < p, by Hölder’s inequality combined with equation (25):

Z

R^d

Ñ X

j k e

(I¤_γψ_{j k}^e )1_Q(λ)

pé^r

p

(1 +|λ|²)^−ddλ

≤

Ö Z

R^d

X

j k e

(I¤γψ^e_{j k})1Q(λ)

p

(1 +|λ|²)^−ddλ

è^r_pÖ Z

R^d

1

(1 +|λ|²)^ddλ

è1−^r_p

<∞

then, by Theorem 2.8,

X

j k e

η_{j k e}(I^¤_γψ_{j k}^e )1_Q

converges a.s. in L^r(R^d, wdλ)and therefore ^P

j k e

η_{j k e}(I_γψ_{j k}^e )1_Q converges a.s.

in FL^rw and in S⁰(R^d). Notice that the same argument, with slight modifi- cations works with any translate of Q. Finally, to verify that ^P

j k e

η_{j k e}I_γψ^e_{j k}

converges a.s. in D⁰(R^d), take Q=

(

Q=

ñ−1 4 ,1

4

åd

+k

2, k ∈Z^d

)

, Ω⁰ such that P(Ω⁰) = 1 defined by

Ω⁰ = ^\

Q∈Q







ω ∈Ω :

X

j k e

η_{j k e}(ω)(I_γψ^e_{j k})1_Q

_F

r

<∞







and ϕ∈ D(R^d). For fixed Q∈ Q, ω∈Ω⁰ and N, M ∈N note that s_{N M Q}(ω) = ^X

|j|≤N|k|≤M

X

e∈E

η_{j k e}(ω)(I_γψ_{j k}^e )1_Q ∈L²(R^d),

and then _∞

X

Q

s_{N M Q}(ω), ϕ

∫

=

l

X

i=1

hs_{N M Qi}(ω), ϕi

for some Q_i such that supp(ϕ) ⊂ ^Sm

i=1

Q_i since ϕ has compact support. Now the result follows from the convergence of hs_{N M Qi}(ω), ϕiwhen N, M −→ ∞ for each i= 1. . . m.

Alternatively, considering γ > d

2 and the operators K_γ instead of I_γ we can prove:

Theorem 3.3. Let {ψ_{j k}^e }_{j k e} be an r-regular orthonormal wavelet series, d

2 < γ ≤ d

Ç

1− 1 p

å

+ 1, 1 ≤ p ≤ 2, γ < r and (η_{j k e})_{j k e} a sequence of independent identically distributed random variables such that ηj k e ∼ SpS. Then, for each x∈R^d the series defined by

Yγ(x) = ^X

j k e

ηj k eKγψ_{j k}^e (x)

converges a.s.. Moreover, {Y_γ(x)}_x∈R^d has a measurable version. If p = 2, the result remains true for d

2 ≤γ ≤ d 2 + 1.

Remark.

Note that the range of validity of the result depends on the dimension d.

Indeed, the restrictions give that 1< 2d

d+ 2 < p≤2 for d≥2.

Proof. Recalling the properties of p stable random variables from Section 2.2. For each x ∈ R^d, we can prove the convergence in r-mean (r < p) of the sum defining Y_γ(x). In fact, by Theorem 3.1 and taking anys such that d

Ç1 p − 1

2

å

< s < γ−d

Ç

1− 1 p

å

, sinceK_γψ_{j k}^e (x) = hK_γ(x, .), ψ_{j k}^e i, for some constant C we obtain:

(E|Yγ(x)|^r)^1r =C

Ñ X

j k e

|hKγ(x, .), ψ^e_{j k}i|^p

é_p¹

≤C⁰kKγ(x, .)k_H^p

s(R^d) <∞, since recalling from Section 2.1 Lemma 2.4 and the equivalence of norms of H_s^p(R^d) of equation (11) one obtains:

kK_γ(x, .)k_Hp

s(R^d) ≤ C(kK_γ−s(x, .)k_Lp(R^d)+kK_γ(x, .)k_Lp(R^d))

≤C⁰(|x|^(γ−s)−(¹⁻¹p)^d+|x|^γ−(¹⁻¹p)^d).

The sum defining Y_γ(x) converges a.s. since convergence in the r-mean of independent random variables implies a.s. convergence. Similarly to the previous bound, if |x−x⁰|<1, by Lemma 2.4 (ii) one gets:

(E|Y_γ(x)−Y_γ(x⁰)|^r)^1r

=C

Ñ X

j k e

|hK_γ(x, .)−K_γ(x⁰), ψ^e_{j k}i|^p

é¹_p

=C

Ñ X

j k e

|hK_γ(x−x⁰, .), ψ^e_{j k}i|^p

é¹_p

≤C⁰|x−x⁰|^(γ−s)−(¹⁻¹p)^d,

From this, applying Chevychev’s inequality it follows the stochastic continu- ity of Yγ(x)and thus there exists a measurable version (Theorem 1, p.157 of [6]) of {Y_γ(x)}_x∈Rd.

3.2 Self similarity analysis

Exact self similarity is broken if p6= 2. However, we can see in the following results that, in some sense, the re scaled versions of X_γ are stochastically dominated and moreover one can expects some kind of fractal behaviour for on an integrated version of X_γ (See for example, the following realizations of Y_γ considering a Daubechies wavelet basis).