Moments and Legendre-Fourier Series for Measures Supported on Curves

Some important problems (e.g., in optimal transport and optimal control) have a relaxed (or weak) formulation in a space of appropriate measures whichis much easier to solve. However, an optimal solution $\mu$ of the latter solves the former if and only if the measure $\mu$ is supported on a"trajectory"$\{(t,x(t))\colon t\in [0,T]\}$ for some measurable function $x(t)$. We provide necessary and sufficient conditions on moments $(\gamma\_{ij})$ of a measure $d\mu(x,t)$ on $[0,1]^2$ to ensure that $\mu$ is supported on a trajectory $\{(t,x(t))\colon t\in [0,1]\}$. Those conditions are stated in terms of Legendre-Fourier coefficients ${\mathbf f}\_j=({\mathbf f}\_j(i))$ associated with some functions $f\_j\colon [0,1]\to {\mathbb R}$, $j=1,\ldots$, where each ${\mathbf f}\_j$ is obtained from the moments $\gamma\_{ji}$, $i=0,1,\ldots$, of $\mu$.


Introduction
This paper is in the line of research concerned with the following issue: which type and how much of information on the support of a measure can be extracted from its moments (a research issue outlined in a Problem session at the 2013 Oberwolfach meeting on Structured Function Systems and Applications [2]). In particular, a highly desirable result is to obtain necessary and/or sufficient conditions on moments of a given measure to ensure that its support has certain geometric properties. For instance there is a vast literature on the old and classical L-moment problem, which asks for moment conditions to ensure that the underlying measure µ is absolutely continuous with respect to some reference measure ν, and with a density in L ∞ (ν). See, for instance, [3,10,11], more recently [7], and the many references therein.
Here we are interested in a problem that is somehow "orthogonal" to the L-moment problem. Namely, we consider the following generic problem: Let dµ(x, t) be a probability measure on [0, 1] × [0, 1]. Provide necessary and/or sufficient conditions on the moments of µ to ensure that µ is singular with respect to the Lebesgue measure d(x, t) on [0, 1] 2 . In fact, and more precisely, suppose that: • one knows all moments γ i (j) = x i t j dµ(x, t), i, j = 0, 1, . . ., of the measure µ, and • the marginal of µ with respect to the "t" variable is the Lebesgue measure dt on [0, 1].
Then provide necessary and/or sufficient conditions on the moments (γ i (j)) of µ to ensure that µ is supported on a trajectory {(t, x(t)) : t ∈ [0, 1]} ⊂ [0, 1] 2 , for some measurable function In contrast to the L-moment problem, and to the best of our knowledge, the above problem stated in this form has not received a lot of attention in the past even though it is crucial in some important applications (two of them having motivated our interest).
Motivation. In addition of being of independent interest, this investigation is motivated by at least two important applications: -The mass transfer (or optimal transport) problem. In the weak (or relaxed) Monge-Kantorovich formulation of the mass transport problem originally stated by Monge, one searches for a measure dµ(x, t) with prescribed marginals ν x and ν t , and which minimizes some cost functional c(x, t) dµ(x, t). However in the original Monge formulation, ultimately one would like to obtain an optimal solution µ * of the form dµ * (x, t) = δ x(t) dν t (t) for some measurable function t → x(t) (the transportation plan) and a crucial issue is to provide conditions for this to happen 1 . For more details the interested reader is referred, e.g., to [13, pp. 1-5] and [9]. There exist some characterizations of the support of an optimal measure for the weak formulation. For instance, c-cyclical monotonicity relates optimality with the support of solutions, and more recently [1] have shown in the (more general) context of the generalized moment problem that under some weak conditions the support of optimal solutions is finitely minimal / c-monotone. (As defined in [1] a set Γ is called finitely minimal / c-monotone if each finite measure α concentrated on finitely many atoms of Γ is cost minimizing among its competitors; in the optimal transport context, a competitor of α is any finite measure α with same marginals as α.) For more details the interested reader is referred to [1] and the references therein. But such a characterization does not say when this support is a trajectory.
-Deterministic optimal control. Using the concept of occupation measures, a weak formulation of deterministic optimal control problems replaces the original control problem with an infinite-dimensional optimization problem P on a space of appropriate (occupation) measures on a Borel space X × U × [0, 1] with X ⊂ R n , U ⊂ R m . For more details the interested reader is referred, e.g., to [8,14], and the many references therein. An important issue is to provide conditions on the problem data under which the optimal value of the relaxed problem P is the same as that of the original problem; see, e.g., [14]. Again this is the case if some optimal solution µ * (or every element of a minimizing sequence) of the relaxed problem is such that every marginal µ * j of µ * with respect to (x j , t), j = 1, . . . , n, and every marginal µ * of µ * with respect to (u , t), = 1, . . . , m, is supported on a trajectory {(t, x j (t)) : t ∈ [0, 1]} and on a trajectory {(t, u (t)) : t ∈ [0, 1]} for some measurable functions t → x j (t) and t → u (t) on [0, 1].
Contribution. Of course there is a particular case where one may conclude that µ is singular with respect to the Lebesgue measure on [0, 1] 2 . If there is a polynomial p ∈ R[x, t] of degree say d, such that its vector of coefficients p is in the kernel of the moment matrix M s (where M s [(i, j), (k, )] = γ i+k,j+ , i + j, k + ≤ s, with d ≤ s), then µ is supported on the variety {(x, t) ∈ [0, 1] 2 : p(x, t) = 0} and therefore is singular with respect to the Lebesgue measure on [0, 1] 2 . But it may happen that p(x, t) = p(y, t) = 0 for some t and some x = y and so even in this case additional conditions are needed to ensure existence of a trajectory {(t, We provide a set of explicit necessary and sufficient conditions on the moments γ i = (γ i (j)) which state that for every fixed i, the moments γ i (j), j = 0, 1, . . ., are limits of certain i-powers of the moments γ 1 .
More precisely, an explicit linear transformation ∆γ 1 of the infinite vector γ 1 is the vector of (shifted) Legendre-Fourier coefficients associated with the function t → x(t). Then the conditions state that for each fixed i = 2, 3, . . ., the vector ∆γ i should be the vector of (shifted) Legendre-Fourier coefficients associated with the function t → x(t) i , which in turn are expressible in terms of limits of "i-powers" of coefficients of ∆γ 1 .
At last but not least, it should be noted that all results of this paper are easily transposed to the multi-dimensional case of a measure dµ(x, t) on [0, 1] n × [0, 1] and supported on a trajectory Indeed by proceeding coordinate-wise for each function t → x i (t), i = 1, . . . , n, one is reduced to the case [0, 1] 2 investigated here.

Some preliminary results
We next state some useful auxiliary results, some of them being standard in Real Analysis.

2)
and this decomposition is unique. Moreover f ∈ 2 and f = f .
When the interval is [−1, 1] (instead of [0, 1] here) (2.2) is called the Legendre (or Legendre-Fourier) series expansion of the function f and f = (f (j)) is called the vector of Legendre-Fourier coefficients.
The notation f k stands for the function we denote by f k = (f k (j)) ∈ 2 its (shifted) Legendre-Fourier coefficients so that f k = f k (where again the latter norm is that of 2 ). Notice that we also have: This follows from the uniqueness of the decomposition in the basis (L j ). We also have the following helpful results: be a sequence of polynomials such that p n − f → 0 as n → ∞.
is such that sup n p n ∞ < ∞ and p n − f → 0 as n → ∞, then for every k ∈ N, In addition, ifp k n denotes the (shifted) Legendre-Fourier coefficients of p k n , n = 1, . . ., then Then the last statement follows from Lemma 2.3.
Observe that each entry f (k) n (j), j = 0, . . . , nk, is a degree-k form of the first n + 1 Legendre-Fourier coefficients off . Completing with zeros, consider f (k) n to be an element of 2 and if f (k) n converges in 2 as n → ∞, call f (k) ∈ 2 its limit.
The limit f (k) can also be denoted f · · · f , the limit of the k times " -product" in 2 of the vector f ∈ 2 by itself. Equivalently one may write Legendre-Fourier coefficients f k ∈ 2 for every k ∈ N, and assume that Then f k = f (k) = f · · · f (k times) for every k = 1, 2, . . ., meaning that for every fixed k ∈ N, Equivalently, f k = f k−1 f for every k = 2, 3, . . ..

Proof . The result is a direct consequence of Lemmas 2.3 and 2.4 with p n = f
(1) n (and the definition of the limit " -product" in Definition 2.5).

Main result
Assume that we are given all moments of a nonnegative measure dµ(x, t) on a box [a, b] × [c, d] ⊂ R 2 . After a re-scaling of its moments we may and will assume that µ is a probability measure supported on [0, 1] 2 with associated moments We further assume that the marginal measure µ t with respect to the variable t, is the Lebesgue measure on [0, 1], that is, γ 0 (j) = 1/(j + 1), j = 0, 1, . . .. A standard disintegration of the measure µ yields dt, i, j = 0, 1, . . . , where ∆ jj > 0, or in compact matrix form for some infinite lower triangular matrix ∆ with all diagonal elements being strictly positive. Therefore wherex = (x(j)) ∈ 2 is its vector of (shifted) Legendre-Fourier coefficients (with x = x ). Similarly, for every k = 2, 3, . . ., the function t → x(t) k is in L ∞ ([0, 1]) and with vector of (shifted) Legendre-Fourier coefficientsx k ∈ 2 such that x k = x k . We also recall the notationx (k) n ∈ R kn+1 for the vector of coefficients in the basis (L j ) of , and when considered as an element of 2 (by completing with zeros) denote byx (k) ∈ 2 its limit when it exists. (j)L j ∞ < ∞, then

4)
Equivalently where ∆ is the non singular triangular matrix defined in (3.2).
Proof . The (a) part. As µ is supported on [0, 1] 2 one has x ∞ ≤ 1 and so the function t → x(t) i is in L 2 ([0, 1]) for every i = 1, 2, . . .. So let t → x(t) be written as in (3.3). Consider the function t → x(t) i , for every fixed i ∈ N, so that where the (shifted) Legendre-Fourier vector of coefficientsx i is obtained byx i = ∆ −1 γ i . But by Lemma 2.6, we also have In other words,x (i) =x i or equivalently,x (i) = ∆γ i = (∆γ 1 ) (i) , which is (3.4). We next prove the (b) part. By the disintegration (3.1) of the measure µ, Hence for every t ∈ [0, 1]\B, 1] x i dδ f 1 (t) , ∀ i = 1, 2, . . . . Remark 3.2. If the trajectory t → x(t) is a polynomial of degree say d, then the vector of Legendre-Fourier coefficientsx ∈ 2 has at most d + 1 non-zero elements. Therefore for every j = 2, . . .,x j ∈ 2 also has at most jd + 1 non-zero elements and the condition (3.5) can be checked easily.
In Theorem 3.1(a) one assume that sup n n j=0x (j)L j ∞ < ∞ which is much weaker than, e.g., assuming the uniform convergence n j=0x (j)L j − x ∞ → 0 as n → ∞. The latter (which is also much stronger than the a.e. pointwise convergence) can be obtained if the function x(t) has some smoothness properties. For instance if x belongs to some Lipschitz class of order larger then or equal to 1/2, then uniform convergence takes place and one may even obtain rates of convergence; see, e.g., [12] and also [15] for a comparison (in terms of convergence) of Legendre and Chebyshev expansions. In fact, quoting the authors of [5], ". . . knowledge of the partial spectral sum of an L 2 function in [−1, 1] furnishes enough information such that an exponential convergent approximation can be constructed in any subinterval in which f is analytic". With n = 5 the polynomial t → x

A more general case
We have considered a measure µ on [0, 1] 2 whose marginal with respect to t ∈ [0, 1] is the Lebesgue measure. The conditions of Theorem 3.1 are naturally stated in terms of the (shifted) Legendre-Fourier coefficients associated with the functions t → f i (t) of L 2 ([0, 1]) defined in (3.1).
However, the same conclusions also hold if the marginal of µ with respect to t ∈ [0, 1] is some measure dν = h(t)dt for some nonnegative function h ∈ L 1 ([0, 1]) with all moments finite. The only change is that now we have to consider the orthonormal polynomials t → H j (t), j = 0, 1, . . ., with respect to ν. Recall that all the H j 's can be computed from the moments Then proceeding as before, for every i = 1, 2, . . ., and we now consider the vector of coefficientsf hi = (f hi (j)) defined bŷ

Discussion
Theorem 3.1 may have some practical implications. For instance consider the weak formulation P of an optimal control problem P as an infinite-dimensional optimization problem on an appropriate space of (occupation) measures, as described, e.g., in [14]. In [8] the authors propose to solve a hierarchy of semidefinite relaxations (P k ), k = 1, 2, . . ., of P. Each optimal solution of P k provides with a finite sequence z k = (z k j,α,β ) such that when k → ∞, z k → z * where z * is the infinite sequence of some measure dµ(t, x, u) on [0, 1] × X × U, where X ⊂ R n , U ⊂ R m , are compact sets.
Under some conditions both problems P and its relaxation P have same optimal value. If µ is supported on feasible trajectories {(t, x(t), u(t)) : t ∈ [0, 1]} then these trajectories are optimal for the initial optimal control problem P. So it is highly desirable to check whether indeed µ is supported on trajectories from the only knowledge of its moments z * = (z * j,α,β ). By construction of the moment sequences z k one already knows that the marginal of µ with respect to the variable "t" is the Lebesgue measure on [0, 1]. Therefore we are typically in the situation described in the present paper. Indeed to check whether µ is supported on trajectories {(t, (x 1 (t), . . . , x n (t), u 1 (t), . . . , u m (t)) : t ∈ [0, 1]}, one considers each coordinate x i (t) or u j (t) separately. For instance, for x i (t) one considers the subset of moments γ k (j) = (z * j,α,0 ) with j = 0, 1, . . ., α = (0, . . . , 0, k, 0, . . . , 0) ∈ N n , k = 0, 1, . . ., with k in position i. If (3.4) holds then indeed the marginal µ t,x i of µ on (t, x i ), with moments (γ k (j)) is supported on a trajectory {(t, x i (t)) : t ∈ [0, 1]}.
Of course, in (3.4) there are countably many conditions to check whereas in principle only finitely many moments of z * are available (and with some inaccuracy due to (a) solving numerically a truncation P k of P, and (b) the convergence z k → z * has not taken place yet). So an issue of future investigation is to provide necessary (or sufficient?) conditions based only on finitely many (approximate) moments of µ.