M\"untz linear transforms of Brownian motion

We consider a class of linear Volterra transforms of Brownian motion associated to a sequence of M\"untz Gaussian spaces and determine explicitly their kernels; some interesting links with M\"untz-Legendre polynomials are provided. This gives new explicit examples of progressive Gaussian enlargement of the Brownian filtration. By exploiting a link to stationarity, we give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional M\"untz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process.


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 ([4]- [6]) and the references therein. In the meanwhile, Volterra transforms with non square integrable kernels, involving some functional spaces, provide interesting examples of non-canonical decompositions of the Brownian filtration. This motivated many studies on the topic, for instance see ( [3], [8], [13], [15], [29]). Our  [33]) for studies on this topic in more general frameworks.
Applying then the characterization given in [26], a glance at (6) t > 0. Partial answers are given in ( [13], [17], [18]) where the authors established the existence of such transforms. In particular, for an infinite sequence Λ satisfying either sup λ j = +∞ or 0 < λ 1 < λ 2 < ... there exists no such a transform such that (T (B) t , t ≥ 0) is a semimartinagle relative to the filtration of B. We see that a necessary and sufficient condition for the existence of transforms with infinite dimensional orthogonal complement is the Müntz-Szasz condition

Müntz Gaussian spaces and transforms
Throughout this paper, we assume that Λ = {λ 1 , λ 2 , λ 3 , · · · } is a sequence of distinct real numbers such that which ensures that the generalized Müntz polynomials f i (x) := x λ i , for i = 1, 2, · · · , lie in L 2 loc (R + ). For t > 0, let us introduce which are called Müntz spaces. An associated orthogonal system, known as Müntz-Legendre polynomials, is specified by L 1 (x) = x λ 1 and L 2 , L 3 · · · , described by see [5] and [6]; note that we use slightly different notations since we start the sequence Λ with λ 1 instead of λ 0 . Recall that L k (1) = 1 for k = 1, 2, · · · . The celebrated Müntz-Szasz theorem, see e.g. [5], states that f 1 , f 2 , · · · is complete in L 2 ((0, 1]) if and only if condition (7) does not hold. Next, to the linear spaces M n,t and M ∞,t we associate, respectively, the families of Müntz Gaussian spaces defined by (2) and (3). Recall that It follows that the orthogonal complements of G t (λ 1 , · · · , λ n ; B) and G t (λ 1 , λ 2 · · · ; B), in H t (B), are respectively given by One of our aims in this paper is to compute explicitly the kernel of a transform of the form (1) such that H t (T n (B)) = G ⊥ t (λ 1 , · · · , λ n ; B), holds for t > 0, and examine what happens as n → ∞. In the following result, we treat the case when n is finite.
Theorem 2.1. Assuming that n < ∞ then the following assertions hold true.
1) The kernel is a Goursat-Volterra kernel of order n. Furthermore, writing T n for the Goursat-Volterra transform associated to k n (., .), the orthogonal complement of H t (T n (B)) in Proof. 1) T n (B) is a Brownian motion if and only if k n (., .) satisfies the self-reproduction property The latter is obtained by writing E[T n (B) t T n (B) s ] = s ∧ t, differentiating and rearranging terms. But, if we set k n (t, s) = t −1 n j=1 a j,n (s/t) λ j , then (13) is equivalent to saying that (a i,n , i = 1, 2, · · · , n) solves the linear system (5). To study the system, consider the n-degree polynomial which, of course, has at most n roots. But p n (x) = 0 is equivalent to n i=1 a i,n x+λ i +1 = 1. This fact, when combined with lim x→∞ p n (x)/x n = 1, implies that p n (x) = n j=1 (x − λ j ). Now, let us choose m ∈ {1, · · · , n} and substitute the latter product formula in the expression of p n (x). Dividing then both sides by j =m (x + λ j + 1), rearranging terms and letting x → −(λ m + 1) we obtain the expressions of a 1,n ,a 2,n , · · · , a n,n . Next, k n (., .) is a Volterra kernel because it is continuous on {(u, v) ∈ (0, +∞) × (0, +∞) : u > v} and satisfies the following integrability condition which is enough for (4) to be well defined. We have where we used the homogeneity and the self-reproduction properties of k n (., .). Finally, we need to identify H t (B) ⊖ H t (T n (B)) for an arbitrarily t > 0. But, this amounts to solving the integral equation This is easily seen, by differentiation, to be equivalent to an ordinary linear differential equation of degree n. The functions s → s λ j , j = 1, · · · , n, being n linearly independent solutions, we conclude that G t (λ 1 , · · · , λ n ; B) is the orthogonal complement of H t (T n (B)) in H t (B) as required. 6 2) By using the homogeneity property of k n (., .) and the stochastic Fubini theorem, we can write The process (T n (B) t , t ≥ 0) is a semimartingale relative to the filtration of B since we can write ρ n (.) = c n + The covariance matrix of the Gaussian process t 0 f * (s)dB s , where f := (f 1 , · · · , f n ) * is the transpose of (f 1 , · · · , f n ), has an inverse matrix which we denote by α n t . m n 1 is a Cauchy matrix and an explicit formula for its inverse can be found in ( [14], [32]). Note also that the Goursat form of k n (., .) given below is given in a semi-explicit form in [16]. Here we propose another method to compute the entries of α n t and φ.
Remark 2.1. In terms of filtration, for n < ∞ and 0 < T < ∞, we have In fact, F on (0, T ); see [1] for more details on these processes.
The objective of the next proposition is to show that we can express K n (.) in terms of Müntz-Legendre polynomials, given in (10), which form an orthogonal basis of M n,1 . An integro-difference equation is discovered in [8] for the kernels ρ n (., .), n = 1, 2, · · · , for the example where λ i = i; assertion 2) of the following result proves to be useful for finding the analogue of Chiu's result for the kernels k n (., .), n = 1, 2, · · · , in the Müntz framework.
2) The sequence K n (.), n = 1, 2, · · · , satisfies the integro-difference equation Proof. 1) We have (1 + 2λ n )c n,n = a n,n and (1 + 2λ n )c j,n = a j,n − a j,n−1 . Thus, we can write Iterating, with the convention that K 0 (.) ≡ 0, and summing up the equations we get the first formula; K n (1) is obtained by setting x = 1 an using L j (1) = 1 for j = 1, 2, · · · , n. As a by-product formula, we note that (λ j + λ n + 1)c j,n = a j,n for j ≤ n. The second assertion is easily obtained by integration.
2) We quote, from [6], the recurrence formula Combining this with the first assertion and simplifying yields Differentiating, we find −λ n K n (x) + xK ′ n (x) = (λ n + 1)K n−1 + xK ′ n−1 . This is nothing but a differential form of the integro-difference equation. It remains to use 1) on the form K n (x) = K n−1 (x) + (1 + 2λ n )L n (x) and the fact that L n (1) = 1 to conclude.
Our aim now is to outline a connection between self-reproducing kernels and the classical kernel systems. Proposition 2.3. For each fixed t > 0, the kernel system associated to M n,t is given Proof. The kernel system is given by g n,t (u, v) = n k=1 q n k,t (u)q n k,t (v) where (q n k,t (.), n = 1, · · · , n) is an orthonormal sequence that generates M n,t . This is a reproducing kernel in the sense that, for any Q t ∈ M n,t , we Exploiting homogeneity, we easily check that the sequence (q n j,t (x), x ∈ [0, t]; j = 1, 2, · · · , n) defined by q n m,t (u) := m k=1 c k,m (t)u λ k = (1 + 2λ m )/tL m (u/t) satisfies the requirements. We conclude using continuity and the fact that L n (1) = 1.
In the proof of the following result, we use the well known fact that the set of covariance functions is closed under limits and some classical techniques from the mean squared calculus, see for example [31].
Proof. By using the self-reproduction property (13), we see that Γ n (., .) is the covariance function of the Gaussian process (X n (t), t ≥ 0) which gives the first statement.
(i)⇒(ii) is classical from the mean square calculus, see e.g. ( [31], P. 135). That is, Using Hölder inequality, we can get all these terms tend to zero. (ii)⇒ (iv) is seen from the convergence of Γ n (t, t) = 1 t n 1 (1 + λ i ). Now, (ii)⇔ (iii) is obtained from the expression of Γ n (., .) and the fact that K n (.) is a Cauchy sequence in L 2 ((0, 1]) if and only if Γ(t, t) converges for some, and hence for all, 0 < t ≤ 1. The proof is now complete.
Remark 2.2. Using the first assertion of Proposition 2.4, Cauchy-Schwarz inequality implies that |K n (r)| ≤ n 1 (1 + 2λ i )/ √ r which remains bounded, under condition (iv), as long as r > 0. It would be interesting to see whether the derivative of K n (r) remain as well uniformly bounded for r > 0; this proves to be useful in the proof of Theorem 3.1 which is given below. If that is the case then Arzelà-Ascoli theorem would imply that the convergence of K n (.), under condition (iv), is uniform on (ε, 1) for any ε > 0. Thus, it is not clear how to approach directly the problem of convergence of Volterra transforms as n → ∞.

Connection to stationary Ornstein-Uhlenbeck processes
We discuss here a question tackled in [18] which consists on determining a necessary and sufficient condition for the existence of an infinite dimensional kernel associated to Λ. Let us recall some excerpts from [25] and [27] on linear transforms of Brownian motions and stationarity. For that, we define for some η ∈ L 2 (R + ), where β is a Brownian motion indexed by R. Assume that β is characterized in terms of B by ∞ 0 ϕ(s)dB s = R V ϕ(r)dβ r for ϕ ∈ L 2 (R + ) where V is the isometry which maps L 2 (R + ) to L 2 (R) which is defined by V ϕ(r) = e r/2 ϕ(e r ).
We clearly have S(β) = U • T (B) where Uφ(r) = e −r/2 φ(e r ) and η(.) = Uρ(.). 11 We set T n = U • T n . Because T n (B) is a Brownian motion, T n (B) is a stationary Ornstein-Uhlenbeck process. The focus in the next result is on the m.a.r. of the latter when n < ∞.
Proof. From the last part of the proof of Theorem 2. (14) holds with η = η n . We havê a j,n λ j + 1/2 − iξ where we used Fubini theorem for the third equality and the fact that 1 + 2λ j > 0, j = 1, 2, · · · , for the last equality. The last term is now evaluated by using the obvious decomposition Note that the latter decomposition allows as well to resolve the system (5).
Recall that η ∈ C 2 (R + ) if and only ifη ∈ H 2 + whereη be the Fourier transform of η. It is known thatη ∈ H 2 + has a unique, up to some multiplicative constant of modulus 1, decomposition of the formη(ξ) = cI(ξ)O(ξ) where c is a constant, O and I = ΠG are, respectively, the outer and the inner parts. The latter are further specified by where h ≥ 0, ε i = 1 if |a i | ≤ 1 and ε i = |a i |/a i and F is a nondecreasing singular function subject to (u 2 + 1) −1 dF (u) < ∞.
Theorem 3.1. There exists a transform T ∞ of the form (1) associated to Λ such , for all t > 0, if and only if (7) holds.
We call n the order of the transform. The corresponding kernel (k n (., .) or ρ n (., .)) is called a Müntz kernel of order n.  Furthermore, since we are in the homogeneous case, we can show that the decomposition F B t = ∞ k=1 σ t 0 u λ j dT (k) n (B) u , 1 ≤ j ≤ n holds true. Here, by F ⊗ G, for two σ-algebras F and G, we mean F ∨ G with independence between F and G. It follows that Müntz transforms are strongly 14 mixing and ergodic. We also refer to [25] for a proof of this, in a more general framework, which uses the connection to stationarity.

Example of infinite order transforms
Kernels of infinite orders take complicated forms which could sometimes involve special functions. To illustrate that, we shall now give a family of examples. as n → ∞. We haveη ∞ (ξ) = 1 1/2 − iξ ∞ k=1 k r + i/ξ k r − i/ξ which can be evaluated using (18) to get (16) as desired.