Müntz linear transforms of Brownian motion Larbi Alili

El e c t ro nic J

ou o

f Pr

ob a bi l i t y

Electron. J. Probab.19(2014), no. 36, 1–15.

ISSN:1083-6489 DOI:10.1214/EJP.v19-2424

Müntz linear transforms of Brownian motion

Larbi Alili

Ching-Tang Wu

Abstract

We consider a class of Volterra linear transforms of Brownian motion associated to a sequence of Müntz Gaussian spaces and determine explicitly their kernels; the ker- nels take a simple form when expressed in terms of Müntz-Legendre polynomials.

These are new explicit examples of progressive Gaussian enlargement of a Brownian filtration. We give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional Müntz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the fil- tration of the original process. This completes some already obtained partial answers to the aforementioned problems in the infinite dimensional case.

Keywords: Enlargement of filtration ; Gaussian process ; Müntz polynomials ; noncanonical representation ; self-reproducing kernel ; Volterra representation.

AMS MSC 2010:Primary 45D05 ; 60G15 , Secondary 26C05 ; 46E22.

Submitted to EJP on November 8, 2012, final version accepted on January 24, 2014.

1 Introduction

There has been a renewed interest in Müntz spaces which is particularly motivated by topics related to Markov inequalities and approximation theory, see for example ([5]–

[7]) and the references therein. In the meanwhile, Volterra transforms with non square- integrable kernels, involving some functional spaces, provide interesting examples of noncanonical decompositions of the Brownian filtration. This motivated many studies on the topic, for instance see ([4], [10], [17], [19], [33]). Our aim in this paper is to study the class of Volterra transforms involving Gaussian spaces which are generated from sequences of Müntz polynomials. This gives new explicit examples of progressive enlargement of filtrations and interesting links with Müntz-Legendre polynomials; see ([26]–[29], [37]) for studies on this topic in more general frameworks.

To be more precise, let us fix our mathematical setting. Let B := (Bt, t ≥ 0) be a standard Brownian motion defined on a complete probability space (Ω,F,P), and denote by{F_t^B, t ≥ 0} the filtration it generates. We encountered, in literature, two

∗Department of Statistics, University of Warwick, Coventry, UK. E-mail:[email protected]

†Department of Mathematics, National Taitung University, Taiwan. E-mail:[email protected]

types of linear transforms ofB which are of our interest in this paper. The first type consists of transforms of the form

T(B)_t= Z t

0

ρ(t/s)dB_s (1.1)

for allt >0and someρ∈ M, with

M={ρ: [1,∞)→Rmeasurable function s.t.

Z 1 0

ρ²(1/v)dv <∞}.

These transforms were intensively studied in [29]. In particular, we found in Proposi- tion 15 therein a variant of Theorem 6.5 of [31] which states that (T(B)_t, t ≥ 0) is a semimartingale relative to the filtration ofBif and only if there existsc∈R^andg∈ M such that

ρ(.) =c+ Z .

1

1

yg(y)dy.

The second type consists of Volterra transforms with non-square-integrable kernels which are of the form

T(B)_t=B_t− Z t

0

ds Z s

0

l(s, v)dB_v (1.2)

for allt >0, where the kernell :R²+ →R, which satisfiesl(s, v) = 0fors < v, is such that the symmetrized kernel

˜l(t, s) :=

l(t, s) if s≤t;

l(s, t) if s≥t,

is continuous on R²+. These transforms were studied for example in ([2], [14], [20]).

Note that, in the semimartingale case withc= 1, the transform (1.1) becomes a Volterra transform of the form (1.2) with kernel l(t, s) = t⁻¹g(t/s) for s ≤ t and l(t, s) = 0 otherwise. Conversely, all transforms of the form (1.2) which we will consider in this paper are of the form (1.1). Uniqueness when defining T(B) by either (1.1) or (1.2) holds only up to a stochastic modification and we work with the continuous one.

Let f_j(x) := x^λj, j = 1,2,· · ·, be a sequence of Müntz polynomials where Λ = {λ₁, λ₂,· · · }is a sequence of reals satisfying

λ_j>−1/2, j= 1,2,· · ·. (1.3) These generalized polynomials are defined on[0,∞)and the value offj atx= 0is the limit off_j(x)asx→0from(0,∞)forj= 1,2,· · ·. For each fixedt >0, let us define the Müntz Gaussian spaces

Gt(λ1,· · ·λn;B) =Span Z t

0

s^λjdBs, j= 1,2,· · ·n

(1.4) and

Gt(λ1, λ2,· · ·;B) =Span Z t

0

s^λjdBs, j= 1,2,· · ·

(1.5) and let Ht(B)be the closed linear span of {Bs, s≤ t}. A Müntz transform of ordern associated toλ1,· · ·,λn, is a linear transformTnof the form (1.1) such that the following two properties hold true:

(i) (T_n(B)_t, t≥0)is a Brownian motion;

(ii) we have the orthogonal decomposition

Ht(B) =Ht(Tn(B))⊕Gt(λ1,· · ·, λn;B), t >0. (1.6)

Following [2] and [20], if n < ∞ then such transforms exist. As we shall see, the transform T_n of the form (1.2) with l(t, s) = k_n(t, s) := t⁻¹K_n(s/t) for s ≤ t where Kn(x) := Pn

j=1aj,nx^λj for0 ≤x≤1 anda1,n, a2,n, ..., an,n are uniquely determined by the system

n

X

j=1

aj,n

λ_j+λ_k+ 1 = 1, k= 1,· · ·, n, (1.7) is a Müntz transform of ordern. The latter system, whenλ_j =j forj = 1,2,· · ·, was first discovered by P. Lévy, see ([32], [34]) and was further studied in [10]. Solving (1.7), we found that the sequence of Müntz polynomials(Kn, n= 1,2,· · ·)can be simply expressed in terms of Müntz-Legendre polynomials which allows to simplify the study of some of their properties. Note thatTntakes the form (1.1) with

ρ_n(x) = 1− Z x

1

K_n(1/r)dr

r , x≥1. (1.8)

Thus, we are in the semimartingale case withc= 1andg(.) =−K_n(1/.).

The kernels described above are homogeneous of degree−1in the sense that k_n(αt, αs) =α⁻¹k_n(t, s), 0< s≤t <∞, α >0.

As a consequence, the associated Volterra transforms have a close connection to a class of stationary processes. That is, the process

(e^−u/2Tn(B)e^u, u∈R)

is an Ornstein-Uhlenbeck process; the conventional value−1/2of the parameter will be dropped from our notations. It has the moving average representation, m.a.r. for short,

S_n(W)_u:=

Z u

−∞

η_n(u−r)dW_r

foru∈R^{, where}W is a Brownian motion indexed byR^andη_n ∈L¹(R+)∩L²(R+)has the Fourier transform

ˆ ηn(ξ) :=

Z

e^iξxηn(x)dx= 1 1/2−iξ

n

Y

j=1

ξ−ipj

ξ+ip_j, ξ∈R, (1.9) wherepj =λj+¹₂ forj = 1,2,· · ·. Applying then the characterization given in [30], the presence of the inner part, given by the product in (1.9), implies that the latter m.a.r. is not canonical with respect toW.

A natural question is to know whether there exists a transform T of the form (1.1) such that

Ht(B) =Ht(T(B))⊕Gt(λ1, λ2,· · · ;B)

for allt >0. Partial answers are given in ([17], [21], [22]) where the authors established the existence of such transforms. In particular, for an infinite sequence Λ satisfying eithersupλ_j = +∞ or0 < λ₁ < λ₂ < ...there exists no such a transform such that (T(B)_t, t≥0)is a semimartingale relative to the filtration ofB. We see that a necessary and sufficient condition for the existence of transforms of the form (1.1) with infinite dimensional orthogonal complement is

∞

X

j=1

pj

p²_j+ 1 <∞. (1.10)

This is the well-known Müntz-Szasz condition which is necessary and sufficient forf₁, f2,· · ·, to be incomplete inL²[0,1], see for example [6]. Furthermore,(T(B)t, t≥0)is a semimartingale relative to the filtration ofBif and only if(λk)is bounded and satisfies (1.10). Plainly, the latter happens if and only ifP∞

j=1pj <∞.

2 Müntz Gaussian spaces and transforms

Throughout this paper, we assume thatΛ ={λ1, λ2,· · · }is a sequence of distinct real numbers satisfying condition (1.3). Thus, the generalized Müntz polynomialsf_j(x) :=

x^λj, forj= 1,2,· · ·, lie in

L²_loc(R+) :={f :R+→R;f ∈L²[0, t]for all0< t <∞}.

Fort >0, let us introduce

Mn,t=Span{x^λ1, x^λ2,· · ·, x^λn;x∈[0, t]}

and

M_∞,t=Span{x^λ1, x^λ2,· · · ;x∈[0, t]}

which are called Müntz spaces. An associated orthogonal system, known as Müntz- Legendre polynomials, is specified byL1(x) =x^λ1 andL2,L3· · ·, described by

L_k(x) =

k

X

j=1

c_j,kx^λj, c_j,k= Qk−1

l=1(λl+λj+ 1) Qk

l=1,l6=j(λj−λl)

, k= 2,3,· · ·, (2.1) see [6] and [7]; note that we use slightly different notations since we start the sequence Λ with λ1 instead of λ0. Recall that Lk(1) = 1 fork = 1,2,· · ·. Next, to the linear spacesMn,tandM_∞,twe associate, respectively, the families of Müntz Gaussian spaces defined by (1.4) and (1.5). Recall that the closed linear span of{Bs, s≤t}, or the first Wiener chaos ofB, is given by

H_t(B) = Z t

0

f(u)dB_u;f ∈L²[0, t]

. (2.2)

It follows that the orthogonal complements ofG_t(λ₁,· · · , λ_n;B)andG_t(λ₁, λ₂· · ·;B), in Ht(B), are respectively given by

G^⊥_t(λ₁,· · · , λ_n;B) = Z t

0

f(u)dB_u;f ∈L²[0, t], Z t

0

f(s)p(s)ds= 0, p∈M_n,t

and

G^⊥_t(λ₁, λ₂,· · · ;B) = Z t

0

f(u)dB_u;f ∈L²[0, t], Z t

0

f(s)p(s)ds= 0, p∈M_∞,t

. Following [2], the transformT(resp.kernel) defined by (1.2) is called a Goursat-Volterra transform (resp.kernel) of ordernif(T(B)t, t≥0)is a Brownian motion and there exists nlinearly independent functionsgj ∈L²_loc(R⁺)such that

Ht(B) =Ht(T(B))⊕Span{ Z t

0

gj(s)dBs, j= 1,2,· · ·n}

for allt >0. We are now ready to determine a Goursat-Volterra transformT_nassociated to the Müntz polynomialsf1,f2,· · ·,fn, in the case whennis finite.

Theorem 2.1. Assuming thatn <∞then kn(t, s) :=

t⁻¹K_n(s/t) if s≤t;

0 otherwise,

where

Kn(s) =

n

X

j=1

aj,ns^λj, aj,n= Qn

l=1(λj+λl+ 1) Qn

l=1,l6=j(λ_j−λ_l), j= 1, ..., n, (2.3)

is a Goursat-Volterra kernel of order n. Furthermore, writing T_n for the Goursat- Volterra transform associated tok_n, the orthogonal complement ofH_t(T_n(B))inH_t(B) isGt(λ1,· · ·, λn;B)for allt≥0. Note thatTn is of the form (1.1) withρprescribed by (1.8).

Proof of Theorem 2.1. Tn(B) is a Brownian motion if and only if kn satisfies the self- reproduction property

k_n(t, s) = Z s

0

k_n(t, u)k_n(s, u)du (2.4)

for a.e. s ≤ t, which is found in Theorem 6.1 in [14]. This is obtained by writing E[Tn(B)tTn(B)s] =s∧t, differentiating and rearranging terms. But, if we setkn(t, s) = t⁻¹Pn

j=1a_j,n(s/t)^λj, then (2.4) is equivalent to saying that(a_j,n, j = 1,2,· · · , n)solves the linear system (1.7). To study the system, consider then-degree polynomial

p_n(x) =

n

Y

j=1

(x+λ_j+ 1)−

n

X

k=1

a_k,n

n

Y

j=1,j6=k

(x+λ_j+ 1)

which, of course, has at mostnroots. Butpn(x) = 0is equivalent toPn k=1

a_k,n x+λ_k+1 = 1. This fact, when combined withlimx→∞pn(x)/xⁿ= 1, implies thatpn(x) =Qn

j=1(x−λj). Now, let us choose m ∈ {1,· · ·, n} and substitute the latter product formula in the expression ofp_n(x). Dividing both sides byQ

j6=m(x+λ_j+ 1), rearranging terms and letting x → −(λm+ 1) we obtain the expressions of a1,n,a2,n, · · ·, an,n. Next, kn is a Volterra kernel because it is continuous on {(u, v) ∈ R²+;u > v} and satisfies the following integrability condition which is enough for (1.2) to be well defined. We have

Z t 0

Z u 0

k_n²(u, v)dv ^1/2

du = Z t

0

Z 1 0

k²_n(u, ur)u dr ^1/2

du

= 2√ t

Z 1 0

K_n²(r)dr ^1/2

= 2√

tK_n^1/2(1)<+∞,

where we used the homogeneity and the self-reproduction properties of kn. Finally, we need to identify H_t(B) H_t(T_n(B)) for an arbitrarily fixed t > 0. The condition Rt

0f(u)dB_u⊥T(B)_sfor alls≤tis equivalent to Z s

0

f(r)dr= Z s

0

du Z u

0

k_n(u, v)f(v)dv. (2.5)

If we write k_n(u, v) = Pn

j=1ϕ_j(u)f_j(v) then by differentiating the latter equation we obtain the integral equation

f(s) = Z s

0

kn(s, v)f(v)dv

=

n

X

j=1

ϕ_j(s) Z s

0

f_j(v)f(v)dv

for a.e.t >0, this can also be found in [28]. Clearly, iff solves it thenf(t)/ϕ₁(t)must be absolutely continuous with respect to the Lebesgue measure. Repeating this argument, we see that (2.5) is equivalent to an ordinary linear differential equation of degreen which should hold for a.e.s∈[0, t]. The functionsu→u^λj,j= 1,· · · , n, beingnlinearly

independent solutions, we conclude that G_t(λ₁,· · · , λ_n;B) is the orthogonal comple- ment ofH_t(T_n(B))in H_t(B)as required. Next, by using the homogeneity property of the kernelknand the stochastic Fubini theorem, we can write

Tn(B)t = Bt− Z t

0

Z u 0

kn(u, v)dBvdu

= Z t

0

1−

Z t v

kn(u, v)du

dBv

= Z t

0

1− Z t/v

1

kn(vr, v)v dr

! dBv

= Z t

0

ρn(t/v)dBv, t≥0, where

ρn(x) = 1− Z x

1

kn(r,1)dr= 1− Z x

1

Kn(1/r)dr r .

Remark 2.2. Since when n < ∞ we have Kn ∈ L²[0,1], by using the homogeneity property of kn we obtain thatR1

0dsR1

0k_n²(s, v)dv = +∞i.e. kn ∈/ L²([0,1]×[0,1]). The representation (1.2) is the semimartingale decomposition ofT_n(B)with respect toF^B; it is noncanonical relative to the filtration of B since F_t^T(B) F_t^B for all t > 0. The Volterra representation ofT_n(B)asT_n(B)_t=X_t−Rt

0dsRs

0l(s, v)dX_v, wherel:R²+→R is such thatl(s, v) = 0fors < vandX is a Brownian motion, is not unique. Indeed, one representation is given withX =Bandl=knand another one is given withX =Tn(B) andl≡0. But if we add the conditionl∈L²([0,1]×[0,1])then the representation above is unique, see [25].

The covariance matrix

(mⁿ_t)lj = t^λl+λj+1

λ_l+λ_j+ 1, l, j= 1,2,· · · , n, of the Gaussian process(Rt

0f^∗(s)dBs), wheref := (f1,· · ·, fn)^∗ is the transpose of the row vector(f1,· · · , fn), has an inverse matrix which we denote by αⁿ_t. In fact, mⁿ₁ is a Cauchy matrix and an explicit formula for its inverse can be found in ([18], [36]). Note also that the Goursat form of k_n given below is given in a semi-explicit form in [20].

Here we propose another method to compute the entries ofαⁿ_t andϕ. Proposition 2.3. The kernelk_nof Theorem 2.1 satisfies

kn(t, s) =

ϕ^∗(t)·f(s) if s≤t;

0 otherwise,

whereϕ(.) =α(.)·f(.),ϕ_l(t) =a_l,nt^−λl−1,l= 1,2,· · · , nand the entries ofαⁿ_t are given by(αⁿ_t)l,j =al,naj,n(λl+λj+ 1)⁻¹t^{−λl−λj−1}.

Proof of Proposition 2.3. Assume thatknis of the given Goursat form whereϕ1, ϕ2,· · ·, ϕn

are unknown. By using the self-reproduction property (2.4) a little algebra gives that ϕ(.) = α(.)·f(.). The entries ofϕ(t)are identified from the expression ofkn given in Theorem 2.1. Next, from Theorem 2.2 in [2] , we know that(αⁿ_t, t >0)is given, in terms ofϕ, by

αⁿ_t = Z ∞

t

ϕ(u)·ϕ^∗(u)du+α_∞, ϕ(t) =αt·f(t), t >0.

But, hereαⁿ_∞ ≡0becausef1, f2,· · ·,fn are not square-integrable over(0,+∞). Plug- ging in the vectorϕwe obtain the matrixαⁿ_t.

Remark 2.4. In terms of filtrations, forn <∞and0< T <∞, we haveF_T^B=F_T^Tn(B)⊗ σ(GT(λ1,· · · , λn;B)). In fact, F_T^Tn(B) coincides, up to null sets, withσ{Bu^(br), u ≤ T}, where(Bu^(br), u≤T)is thef-generalized bridge over the interval[0, T]. A realization of this is given byBu^(br)=Bu−ψ^∗_T(u)·RT

0 f(s)dBswhereψT(u) =αⁿ_T·Ru

0f(r)dr, foru < T. This is called a generalized bridge because RT

0 f_j(s)dBs^(br) = 0 forj = 1,· · · , n. Note thatTn(B^(br)) =Tn(B)on(0, T); we refer to [1] for more details on these processes.

The objective of the next proposition is to show that we can expressKnin terms of Müntz-Legendre polynomials given by formula (2.1) which form an orthogonal basis of M_n,1. In the special case whenλ_j =jfor allj, an integro-difference equation satisfied byρ_n,n= 1,2,· · ·, was discovered in [10]. The second assertion of the following result proves useful for finding the analogue of Chiu’s result in the general Müntz framework.

Proposition 2.5. Recall that the functions Ln and Kn are given by (2.1) and (2.3), respectively. The following assertions hold true.

1) We have

Kn(x) =

n

X

j=1

(1 + 2λj)Lj(x), x≤1. (2.6)

In particular,K_n(1) =Pn

j=1(1 + 2λ_j). Consequently, we have Kn(x) =x^−λn ∂

∂x x^λn+1Ln(x) and, equivalently,

Ln(x) =x^−λn−1 Z x

0

s^λnKn(s)ds.

Note that unlike Müntz-Legendre polynomials, the Müntz polynomialK_n does not de- pend on the order ofλ1, λ2,· · ·, λn.

2) The sequenceKn,n= 1,2,· · ·, satisfies the integro-difference equation

Kn(x) =Kn−1(x) + (2λn+ 1)x^λn

1− Z 1

x

u^−λn−1Kn−1(u)du

.

Proof of Proposition 2.5. 1) We have(1 + 2λ_n)c_n,n=a_n,nand(1 + 2λ_n)c_j,n=a_j,n−aj,n−1

forj= 1,2,· · ·, n−1. Thus, we can write

Kn(x)−Kn−1(x) = an,nx^λn+

n−1

X

j=1

(aj,n−aj,n−1)x^λj

= (1 + 2λn)cn,nx^λn+ (1 + 2λn)

n−1

X

j=1

cj,nx^λj

= (1 + 2λn)Ln(x).

Iterating, with the convention thatK0 ≡0, and summing up the equations we get the first formula;K_n(1)is obtained by settingx= 1an usingL_j(1) = 1forj = 1,2,· · ·, n. As a by-product formula, we note that(λ_j+λ_n+ 1)c_j,n = a_j,n forj ≤n. The second assertion is easily obtained by integration.

2) We quote from [7] the recurrence formula x^{λn+λn−1+1} x^−λnLn(x)⁰

= x^λn−1+1Ln−1(x)⁰ .

Combining this with the first assertion and simplifying yields x^−λnLn(x)0

= x^−λn−1Kn(x)−(2λn+ 1)x^−2λn−2 Z x

0

s^λnKn(s)ds.

= x^−λn−1K_n−1(x).

Differentiating, we find−λ_nK_n(x) +xK_n⁰(x) = (λ_n+ 1)K_n−1+xK_n−1⁰ . This is nothing but a differential form of the integro-difference equation. It remains to use the first assertion on the formKn(x) =K_n−1(x) + (1 + 2λn)Ln(x)and the fact thatLn(1) = 1to conclude.

Our aim now is to outline a connection between self-reproducing kernels and the classical kernel systems.

Proposition 2.6. For each fixed t > 0, the kernel system associated toMn,t is given by gn,t(u, v) = ¹_tPn

l=1(1 + 2λl)Ll(^u_t)Ll(^v_t)for0 < u, v ≤ t. Letting u → t we get that k_n(t, s) =g_n,t(t, s) =¹_tPn

l=1(1 + 2λ_l)L_l(^s_t)for0< s≤t <∞.

Proof of Proposition 2.6. The kernel system is given by g_n,t(u, v) = Pn

k=1q_k,tⁿ (u)q_k,tⁿ (v) where(qⁿ_k,t, n = 1,· · · , n) is an orthonormal sequence that generates Mn,t. This is a reproducing kernel in the sense that, for anyQt∈Mn,t, we have

Q_t(u) = Z t

0

g_n,t(u, v)Q_t(v)dv.

Exploiting homogeneity, we easily check that the sequence(q_j,tⁿ (x), x∈[0, t];j= 1,2,· · ·, n) defined by q_m,tⁿ (u) := Pm

k=1ck,m(t)u^λk = p

(1 + 2λm)/tLm(u/t) satisfies the require- ments. We conclude using continuity and the fact thatLn(1) = 1.

3 Connection to stationary Ornstein-Uhlenbeck processes

We discuss here a question tackled in [22]; this consists of determining a necessary and sufficient condition for the existence of transforms of the form (1.1) or (1.2) with an infinite dimensional orthogonal complement associated to Λ. Let us recall some excerpts from [29] and [31] on linear transforms of Brownian motions and stationarity.

If two semimartingalesW andB are related by

Wu=

( B1+Re^u 1

dB√_s

s if u≥0;

B₁−R1 e^u

dB√_s

s if u≤0, ^(3.1)

and, equivalently, by

B_t= Z logt

−∞

e^r/2dW_r, t >0, (3.2)

then it is easily checked that B is standard Brownian motion if and only if W is a Brownian motion indexed byR^i.e. W is a centered continuous Gaussian process with independent increments such thatE

(Wu−Wv)²

=|u−v| for all uandv ∈ R^{. Fur-} thermore, we have R∞

0 ϕ(s)dB_s = R

RV ϕ(r)dW_r for ϕ ∈ L²(R+) where the isometry V : L²(R+)→ L²(R)is defined by V ϕ(u) =e^u/2ϕ(e^u). We need to introduce the map- ping U : C(R⁺,R) → C(R,R) which is specified by U ϕ(u) = e^−u/2ϕ(e^u), for u ∈ R^, and denote byU⁻¹its inverse operator. Keeping in mind thatT is defined by (1.1) and settingΘ =U◦T, we clearly have thatΘ(B) =S(W)where the transformS is defined by

S(W)_u= Z u

−∞

η(u−v)dW_v, u∈R, (3.3)

with η(u) = 1_u>0U ρ(u). Plainly, T(B) is a Brownian motion if and only if Θ(B) is a stationary Ornstein-Uhlenbeck process. Moreover, for somef ∈L²_loc(R+)we have

Ht(T(B))⊥ Z t

0

f(s)dBs, t >0,

if and only if

H(S(W))u⊥ Z u

−∞

V(f)(r)dWr, u∈R,

whereH(S(W))u stand for the closed linear span of{S(W)r, r ≤ u}. The focus in the next result is on the m.a.r. ofΘn(B) := (U◦Tn(B)u, u∈R)in case whenn <∞.

Proposition 3.1. Assume thatn <∞andT_nis the Goursat-Volterra transform of The- orem 2.1. The processΘ_n(B)has the m.a.r. (3.3) whereW is given by (3.1) andη :=η_n has the Fourier transform given by

ˆ

η_n(ξ) = (1/2−iξ)⁻¹Π_n(ξ), ξ∈R, (3.4) where

Π_n(ξ) :=

n

Y

j=1

ξ−ip_j ξ+ipj

andpj= ¹₂+λjforj= 1,· · ·, n.

Proof of Proposition 3.1. Using Theorem 2.1 and the recalls above, we see that formula (3.3) holds withη_n(t) = 1_{t>0}U ◦ρ_n(t)andη =η_n. Note that η_n ∈ L¹(R+)∩L²(R+). Now, forξ∈R^{, we have}

ˆ

η_n(ξ) = Z ∞

0

e^iξte^−t/2ρ_n(e^t)dt

= Z ∞

0

e^{−(1/2−iξ)t} 1− Z e^t

1

Kn(1/r)(1/r)dr

! dt

= (1/2−iξ)⁻¹− Z ∞

1

Z ∞ lnr

e^{−(1/2−iξ)t}dt

K_n(1/r)(1/r)dr

= 1

1/2−iξ



1−

n

X

j=1

aj,n

p_j−iξ





where we used Fubini theorem for the third equality and condition (1.3) to justify the last equality. The last term is now evaluated by using the obvious decomposition

n

Y

j=1

x−λj

x+λ_j+ 1 = 1−

n

X

j=1

aj,n

x+λ_j+ 1, x6=−λj−1, j= 1,2,· · ·, n.

Note that the latter decomposition allows as well to resolve the system (1.7).

Our aim now is to look for the analogue of Proposition 3.1 whenn= +∞. Observe that for the transform (3.3) to be well defined we merely needη ∈L²(R⁺)and we can even take η ∈ L²_C(R+). Of course, we need then to work with the Fourier-Plancherel transform instead of the Fourier transform. We recall that this is connected to the Hardy classH₊² of holomorphic functionsH in the upper half-planeC+ ={z ∈C,Im(z)>0}

such thatsup_b>0R

R|H(a+ib)|da <∞, see [12]. We gather in the following result some well known results which are mostly taken from [21] and [29]; for completeness a full proof will be given.

Theorem 3.2. Assuming thatW is a Brownian motion indexed byR^andBis a standard Brownian motion satisfying (3.1) and (3.2) then the following assertions are equivalent.

(1) Λsatisfies (1.10).

(2) There exists a transformSof the form (3.3) associated toΛsuch that(S(W)_u, u∈ R)is an Ornstein-Uhlenbeck process and

Hu(W) =Hu(S(W))⊕Span{ Z u

−∞

e^pjudWu, j= 1,2,· · · } (3.5) for allu∈R^whereH(S(W))ustand for the closed linear span of{S(W)r, r≤u}. (3) There exists a transformT of the form (1.1) such that(T(B)t, t≥0)is a standard

Brownian motion and

Ht(B) =Ht(T(B))⊕Span{ Z t

0

s^λjdBs, j= 1,2,· · · } (3.6) for allt >0.

Proof of Theorem 3.2. (1) ⇔ (2) Assuming that equation (1.10) is not satisfied then by Müntz-Szasz theorem, see e.g. [6], the sequence (fk) is complete in L²[0, t] for all t >0. It follows that the sequence(e^pjx, j= 1,2,· · ·)is total inL²(−∞, a]for allareal.

Hence Hu(W) = Span{Ru

−∞e^pjsdWs, j = 1,2,· · · } which shows that it is not possible to construct a transform S satisfying (3.5) such thatS(W) is an Ornstein-Uhlenbeck process. Conversely, condition (1.10) ensures the convergence of the infinite product (3.7). It is seen in Theorem 2 of [21], see also ([29], p. 60), that under the condition (1.10) the functionH :R→C^{defined by}

H(ξ) = 1 1/2−iξ

∞

Y

j=1

ξ−ipj

ξ+ipj

|1−pj| 1−pj

(3.7)

is the Fourier-Plancherel transform of a function η_∞ ∈ L²_C(R⁺, dx), where dx is the Lebesgue measure; this follows from the fact thatH ∈H₊². Note thatη∞is real-valued sinceH(ξ) = H(−ξ)forξ ∈R^{. Let}S_∞ be defined by (3.3) withη =η_∞. The process (S_∞(W)u, u ∈ R) is a continuous stationary Gaussian process with spectral measure (2π)⁻¹|H(ξ)|²= (2π)⁻¹(ξ²+ 1/4)⁻¹and covariance function

E[S_∞(W)uS_∞(W)v] =

Z u∧v

−∞

η_∞(u−r)η_∞(v−r)dr

= Z ∞

0

η_∞(r)η_∞(u∨v−u∧v−r)dr

= 1

2π Z +∞

−∞

|ˆη∞(ξ)|²ei(u∨v−u∧v)ξdξ

= 2

π Z +∞

−∞

ei(u∨v−u∧v)ξ dξ 1 + 4ξ²

= e^−12|u−v|

foruandv ∈R^{. Thus,}S∞(W)is a stationary Ornstein-Uhlenbeck process. Now, for all u < vandj= 1,2,· · ·,S_∞(W)_uis independent ofRv

−∞e^pjrdW_rsince E

S∞(W)u

Z v

−∞

e^pjrdWr

= Z u

−∞

e^(λj+1/2)rη∞(u−r)dr

= e^pju Z ∞

0

e^−pjsη∞(s)ds= 0

where the last equality is obtained from the factip_j is a zero point ofH.

(2)⇔(3)Assuming (2), we can setρ_∞ =U⁻¹η_∞, whereη_∞ is as above, and defineB by (3.2). Sinceη_∞ ∈L²(R⁺), we clearly have thatρ_∞ ∈ M. Let us now defineT_∞ by (1.1) whereρ_∞andB are as prescribed above. Clearly,T_∞(B)is a standard Brownian motion. Furthermore, we have

Ht(T_∞(B))⊥ Z t

0

u^qdBu

for allt >0, withq >−1/2, if and only if H(S_∞(W))u⊥

Z u

−∞

e^prdWr

for allu∈R^{, with}p=q+ 1/2. Conversely, by reversing the steps we see that(3)implies (2).

Remark 3.3. When we outlined the connection with stationary processes, we could have consideredW^(α) satisfyingR∞

0 ϕ(s)dB_s = R

RV^(α)ϕ(r)dWr^(α) forϕ ∈ L²(R+), for some α > 0, where the isometry V^(α) : L²(R⁺) → L²(R) is defined by V ϕ(u) =

√αe^αu/2ϕ(e^αu), i.e. dW^(α) = α^−1/2e^−αu/2dB(e^αu) with W₀^(α) = B1. But, we need to useU^(α)(φ)(u) =α^−1/2e^αu/2φ(e^αu)instead ofU. The authors used this transformation withα= 2in [21] and [22]. Of course, the conclusions are the same up to working with p_j = 2λ_j+ 1instead ofp_j =λ_j+ 1/2.

Theorem 3.4. There exists a transform of the form (1.1) such that(T_∞(B)t, t ≥0) is a Brownian motion satisfying (3.6) and is a semimartingale inF^B if and only if(λk)is bounded and satisfies the Müntz-Szasz condition (1.10) i.e.P∞

j=1pj<∞.

Proof of Theorem 3.4. By Theorem 3.2 there exists a transform of the form (1.1) such that (3.6) holds and(T(B), t ≥ 0) is a Brownian motion if and only if condition (1.10) is satisfied. Assuming that (1.10) is satisfied, let us check that the semimartingale property cannot hold if there exists a subsequence(nk)k∈N such that λn_k → ∞. For this, let us quote an argument, from [17] and [22], to show that we necessarily have R∞

1 ρ⁰(u)²du= +∞in this case. Using the fact thatT(B)₁is independent ofR1

0u^λnkdB_u and a change of variables, we obtainR1

0u^λnkρ(1/u)du=R∞

1 u^−(λnk+2)ρ(u)du = 0. This, when combined with integration by parts, yields

Z ∞ 1

u^−(λnk+1)ρ⁰(u)du = Z ∞

1

u^−(λnk+1)ρ⁰(u)du−(1 +λn_k) Z ∞

1

u^−(λnk+2)ρ(u)du

= h

ρ(u)u^−(λnk+1)i∞ 1

=−ρ(1).

By using the Cauchy-Schwartz inequality, we obtain (1 + 2λn_k)|ρ(1)|² ≤ R∞

1 ρ⁰(u)²du which implies thatR∞

1 ρ⁰(u)²du = R1

0ρ⁰(1/v)²v⁻²dv = +∞. IfT(B) were anF_t^B semi- martingale then, by applying Theorem 6.5 of [31], we would have the existence ofg such thatρ(t) =c+R.

1y⁻¹g(y)dywithg∈ Mandc6= 0. But thenρ⁰(y) =g(y)/y,y >1, and we should haveR1

0g²(1/v)dv=R1

0ρ⁰(1/v)²v⁻²dv <∞. This contradicts the fact that R∞

1 ρ⁰(v)²dv= +∞.

Let us now examine the case where(λk)satisfies (1.10) and is bounded. i.e.P∞ 1 (1 + 2λj)<∞. We shall first show thatKnconverges asn→ ∞inL²[0,1]. Using Proposition

2.5, for any positive integersn > m, we can write

Z 1 0

(K_n(u)−K_m(u))² du = Z 1

0





n

X

j=m+1

(1 + 2λ_j)L_j(u)





2

du

=

n

X

j=m

(1 + 2λj)

n

X

k=m+1

(1 + 2λk) Z 1

0

Lj(u)Lk(u)du

=

n

X

j=m+1

(1 + 2λj)→0, asm, n→ ∞,

where we have used the fact thatL₁,L₂,· · ·, are orthogonal andR1

0L²_j(r)dr= 1/(1+2λ_j) forj = 1,· · ·, n. This shows that(Kn)is a Cauchy sequence inL²[0,1]. Hence, it must converge to a limit which we denote byK. Withρn(.) = 1−R.

1Kn(1/r)r⁻¹dr,n= 1,2· · ·, let us show thatρn(1/.)→ρ(1/.)inL²[0,1]where

ρ= 1− Z ∞

1

K(1/r)r⁻¹dr. (3.8)

To this end, we quote from [11] the following variant of Hardy inequality. For any g∈L²[0,1], we have

Z 1 0

Z 1 u

g(r)r⁻¹dr ²

du≤4 Z 1

0

g²(u)du.

Now, we can write Z 1

0

(ρn(1/v)−ρ(1/v))²dv = Z 1

0

Z 1/v 1

(Kn(1/z)−K(1/z))z⁻¹dz

!2

dv

= Z 1

0

Z 1 v

(Kn(r)−K(r))r⁻¹dr ²

dv

≤ 4 Z 1

0

(Kn(u)−K(u))²du→0 an→ ∞, which is our claim. It follows that

Tn(B)t= Z t

0

ρn(t/s)dBs→ Z t

0

ρ(t/s)dBs=:T(B)t inL²(Ω,F,P).

Similar arguments show thatηn→ηinL²(R⁺), withη(t) =1t>0U◦ρ(t), andSn(W)u→ Ru

−∞η(u−v)dWv =:S(W)uinL²(Ω,F,P). We need to show thatη=η∞whereη∞is the Fourier inverse of the functionH defined by (3.7). Applying Plancherel Theorem, with ˆ

η_n prescribed by (3.4), we see thatηˆ_n→ηˆinL²_C(R). But by Theorem 13.12 in [35], we know that

Πn(ξ)→Π∞(ξ) :=

∞

Y

j=1

ξ−ip_j ξ+ipj

as n→ ∞,

uniformly on compact subsets ofR\ {0}. It follows thatηˆ_n →H, asn→ ∞, uniformly on compact subsets ofR\ {0}. We conclude that necessarilyηˆ=H a.e. which implies thatη =η_∞ a.e.. It follows from the proof of Theorem 3.2 thatS_∞(W)is an Ornstein- Uhlenbeck process which implies thatT_∞(B) is a standard Brownian motion. Due to the fact thatρ is given by (3.8), Proposition 15 on p. 69 of [29] shows that T(B)is a semimartingale with respect toF^B.

Definition 3.5.A Müntz transform is a transformTof the form (1.1) such that(T(B)_t, t≥ 0)is a standard Brownian motion for any Brownian motion(B_t, t ≥0)and the orthog- onal decompositionHt(B) =Ht(T(B))⊕Gt(λ1,· · · , λn;B), for some sequence of reals

−1/2 < λ1,· · ·, λn and n ∈ {1,2,· · · }, holds for all t > 0. We call n the order of the transform. The corresponding kernelρ(./.)(orkn in the semimartingale case) is called a Müntz kernel of ordern.

Remark 3.6. As a by-product of the discussion for the order to be infinite we mention the following result. Let ϕ be a C^∞([0,1]) function satisfying |ϕ^(m)| ≤ M for all m, whereM is some positive constant. Thenϕis a solution to the integral equationϕ(u) = R1

0ϕ(uv)ϕ(v)dv, defined on [0,1], if and only if ϕ(.) = kn(1, .) =Kn(.)whereλj =j for j≥0andnis some finite positive integer.

Remark 3.7. Forn∈ {1,2,· · · },kn andTnas above, introduce the notationsTn⁽⁰⁾ =Id, Tn⁽¹⁾ =Tn andTn^(m) =Tn^(m−1)◦Tn, form≥2, where◦stands for the composition rule for the iterated transforms. We clearly have formpositive integer

· · · F_t^Tn(m+1)(B) F_t^Tn(m)(B) · · · F_t^Tn(B) F_t(B).

Furthermore, since we are in the homogeneous case, we can show that the decomposi- tion

F_t^B =

∞

O

k=1

σ Z t

0

u^λjdT_n^(k)(B)_u,1≤j≤n

holds true. Here, by F ⊗ G, for two σ-algebras F and G, we mean F ∨ G with inde- pendence betweenF andG. It follows that Müntz transforms are strongly mixing and ergodic. We also refer to [29] for a proof of this, in a more general framework, which uses the connection to stationarity.

Example 3.8. Forr >0, let us setλj= (j^−r−1)/2and sopj=j^−r/2,j= 1,2,· · ·. For npositive integer, we obtain

ak,n= 2 k^r

n

Y

j=1,j6=k

j^r+k^r

j^r−k^r, k= 1,2,· · ·n.

The hyperharmonic seriesP∞

1 (1 + 2λj) =P∞

1 j^−rconverges if and only ifr >1which, by Theorem 3.4, is the necessary and sufficient condition for the existence of an associ- ated Müntz transform of infinite order. Ifr >1then

H(ξ) = 1 1/2−iξ

∞

Y

k=1

k^r−i/(2ξ)

k^r+i/(2ξ), ξ∈R.

If furthermoreris a an integer then H(ξ) =−(1/2−iξ)⁻¹

2r

Y

j=1

Γ(−(i/(2ξ))^1/rω_2r^j )^(−1)j+1

where we have used the relationship Y

j≥1

j^r−z^r j^r+z^r =−

2r

Y

j=1

Γ(−zω_2r^j )^(−1)j+1,

withω2r = exp(πi/r), which is valid for z /∈ {0,1,2,· · · } and is found in ([5], pp. 6-7).

Since the residue ofΓ(z)atz=−kis(−1)^k/k!, we have (k^r−z^r)Γ(−z)→(−1)^k

k! rk^r−1 as z→k.

It follows that

Y

j=1,j6=k

j^r−k^r

j^r+k^r = (−1)^k+12k(k!) r

2r−1

Y

j=1

Γ(−kω_2r^j )^(−1)j+1

which leads to

ak,n → 2 k^r







(−1)^k+12k(k!) r

2r−1

Y

j=1

Γ(−kω^j_2r)^(−1)j+1







−1

= (−1)^k+1 r k^r+1k!

2r−1

Y

j=1

Γ(−kω_2r^j )^(−1)j

asn→ ∞.

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Acknowledgments. The authors would like to thank Y. Hibino for pointing out a mis- take in a previous version of Theorem 3.1. We are grateful to the anonymous referee for a careful reading of the manuscript and for providing interesting comments which lead to an improvement of this paper. The first author is indebted to l’Agence Nationale de la Recherche for the research grant ANR-09-Blan-0084-01.

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