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 which is much easier to solve. However, an optimal solution μ of the latter solves the former if and only if the measure μ is supported on a ''tr...

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Дата:	2015
Автор:	Lasserre, J.B.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2015
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/147149
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Moments and Legendre-Fourier Series for Measures Supported on Curves / J.B. Lasserre // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 15 назв. — англ.

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spelling	irk-123456789-1471492019-02-14T01:26:24Z Moments and Legendre-Fourier Series for Measures Supported on Curves Lasserre, J.B. Some important problems (e.g., in optimal transport and optimal control) have a relaxed (or weak) formulation in a space of appropriate measures which is much easier to solve. However, an optimal solution μ of the latter solves the former if and only if the measure μ is supported on a ''trajectory'' {(t,x(t)):t∈[0,T]} for some measurable function x(t). We provide necessary and sufficient conditions on moments (γij) of a measure dμ(x,t) on [0,1]² to ensure that μ is supported on a trajectory {(t,x(t)):t∈[0,1]}. Those conditions are stated in terms of Legendre-Fourier coefficients fj=(fj(i)) associated with some functions fj:[0,1]→R, j=1,…, where each fj is obtained from the moments γji, i=0,1,…, of μ. 2015 Article Moments and Legendre-Fourier Series for Measures Supported on Curves / J.B. Lasserre // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 42C05; 42C10; 42A16; 44A60 DOI:10.3842/SIGMA.2015.077 http://dspace.nbuv.gov.ua/handle/123456789/147149 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Some important problems (e.g., in optimal transport and optimal control) have a relaxed (or weak) formulation in a space of appropriate measures which is much easier to solve. However, an optimal solution μ of the latter solves the former if and only if the measure μ is supported on a ''trajectory'' {(t,x(t)):t∈[0,T]} for some measurable function x(t). We provide necessary and sufficient conditions on moments (γij) of a measure dμ(x,t) on [0,1]² to ensure that μ is supported on a trajectory {(t,x(t)):t∈[0,1]}. Those conditions are stated in terms of Legendre-Fourier coefficients fj=(fj(i)) associated with some functions fj:[0,1]→R, j=1,…, where each fj is obtained from the moments γji, i=0,1,…, of μ.
format	Article
author	Lasserre, J.B.
spellingShingle	Lasserre, J.B. Moments and Legendre-Fourier Series for Measures Supported on Curves Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Lasserre, J.B.
author_sort	Lasserre, J.B.
title	Moments and Legendre-Fourier Series for Measures Supported on Curves
title_short	Moments and Legendre-Fourier Series for Measures Supported on Curves
title_full	Moments and Legendre-Fourier Series for Measures Supported on Curves
title_fullStr	Moments and Legendre-Fourier Series for Measures Supported on Curves
title_full_unstemmed	Moments and Legendre-Fourier Series for Measures Supported on Curves
title_sort	moments and legendre-fourier series for measures supported on curves
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/147149
citation_txt	Moments and Legendre-Fourier Series for Measures Supported on Curves / J.B. Lasserre // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 15 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT lasserrejb momentsandlegendrefourierseriesformeasuressupportedoncurves
first_indexed	2025-07-11T01:27:59Z
last_indexed	2025-07-11T01:27:59Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 077, 10 pages Moments and Legendre–Fourier Series for Measures Supported on Curves? Jean B. LASSERRE LAAS-CNRS and Institute of Mathematics, University of Toulouse, 7 Avenue du Colonel Roche, BP 54 200, 31031 Toulouse Cédex 4, France E-mail: lasserre@laas.fr URL: http://homepages.laas.fr/lasserre/ Received August 28, 2015, in final form September 26, 2015; Published online September 29, 2015 http://dx.doi.org/10.3842/SIGMA.2015.077 Abstract. Some important problems (e.g., in optimal transport and optimal control) have a relaxed (or weak) formulation in a space of appropriate measures which is much easier to solve. However, an optimal solution µ of the latter solves the former if and only if the measure µ is supported on a “trajectory” {(t, x(t)) : t ∈ [0, T ]} for some measurable function x(t). We provide necessary and sufficient conditions on moments (γij) of a measure dµ(x, t) on [0, 1]2 to ensure that µ is supported on a trajectory {(t, x(t)) : t ∈ [0, 1]}. Those conditions are stated in terms of Legendre–Fourier coefficients fj = (fj(i)) associated with some functions fj : [0, 1] → R, j = 1, . . ., where each fj is obtained from the moments γji, i = 0, 1, . . ., of µ. Key words: moment problem; Legendre polynomials; Legendre–Fourier series 2010 Mathematics Subject Classification: 42C05; 42C10; 42A16; 44A60 1 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 γi(j) = ∫ xitj 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 (γi(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 x : [0, 1]→ [0, 1]. ?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica- tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html mailto:lasserre@laas.fr http://homepages.laas.fr/lasserre/ http://dx.doi.org/10.3842/SIGMA.2015.077 http://www.emis.de/journals/SIGMA/OPSFA2015.html 2 J.B. Lasserre 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 νx and νt, and which minimizes some cost func- tional ∫ 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) = δx(t) dνt(t) for some measurable function t 7→ x(t) (the transportation plan) and a crucial issue is to provide conditions for this to happen1. 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 formu- lation 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 ⊂ Rn, U ⊂ Rm. 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 solu- tion µ∗ (or every element of a minimizing sequence) of the relaxed problem is such that every marginal µ∗j of µ∗ with respect to (xj , t), j = 1, . . . , n, and every marginal µ∗` of µ∗ with respect to (u`, t), ` = 1, . . . ,m, is supported on a trajectory {(t, xj(t)) : t ∈ [0, 1]} and on a trajectory {(t, u`(t)) : t ∈ [0, 1]} for some measurable functions t 7→ xj(t) and t 7→ 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 Ms (where Ms[(i, j), (k, `)] = γi+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 6= y and so even in this case additional conditions are needed to ensure existence of a trajectory {(t, x(t)) : t ∈ [0, 1]}. We provide a set of explicit necessary and sufficient conditions on the moments γi = (γi(j)) which state that for every fixed i, the moments γi(j), j = 0, 1, . . ., are limits of certain i-powers of the moments γ1. More precisely, an explicit linear transformation ∆γ1 of the infinite vector γ1 is the vector of (shifted) Legendre–Fourier coefficients associated with the function t 7→ x(t). Then the con- ditions state that for each fixed i = 2, 3, . . ., the vector ∆γi should be the vector of (shifted) Legendre–Fourier coefficients associated with the function t 7→ x(t)i, which in turn are express- ible in terms of limits of “i-powers” of coefficients of ∆γ1. 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 1Here δz denote the Dirac measure at the point z. Moments and Legendre–Fourier Series for Measures Supported on Curves 3 {(t,x(t)) : t ∈ [0, 1]} ⊂ [0, 1]n+1 for some measurable mapping x : [0, 1] → [0, 1]n. Indeed by proceeding coordinate-wise for each function t 7→ xi(t), i = 1, . . . , n, one is reduced to the case [0, 1]2 investigated here. 2 Notation, definitions and preliminary results 2.1 Notation and definitions Given a Borel probability measure µ on [0, 1]2, define γi(j) = ∫ [0,1]2 tjxi dµ(x, t), i, j = 0, 1, . . . , (2.1) and for every fixed i ∈ N, denote by γi the vector of moments (γi(j)), j = 0, 1, . . .. Let (Lj), j = 0, 1, . . ., be the family of orthonormal polynomials with respect to the Lebesgue measure on [0, 1]. They can be deduced from the Legendre polynomials2 via the change of variable t′ = (2t−1); the (Lj) are called the shifted Legendre polynomials. The polynomials (Lj) can be also computed exactly from the moments γ0 of the Lebesgue measure dt on [0, 1] by computing some determinants of modified Hankel moment matrices. For instance, L0 = 1, and L1(t) = a det ( 1 1/2 1 t ) = a(t− 1/2) with a2(1/3− 1/2 + 1/4) = 1, i.e., a = √ 12 and L1(t) = 2 √ 3t− √ 3, and L2(t) = bdet  1 1/2 1/3 1/2 1/3 1/4 1 t t2  = b [ t2/12− t/12 + 1/72 ] , with b > 0 such that ∫ 1 0 L2(t) 2dt = 1. See, e.g., [4] and [6]. The Lebesgue space L2([0, 1]) := {f : [0, 1] → R, ∫ 1 0 f 2 dx < ∞}, equipped with the scalar product 〈f, g〉 = ∫ 1 0 fg dx and the associated norm ‖ · ‖, is a Hilbert space and the polynomials are dense in L2([0, 1]). In particular the family (Lj), j = 0, 1, . . ., form an orthonormal basis of L2([0, 1]). Let `2 denotes the space of square-summable sequences with norm also denoted by ‖ · ‖. Finally, let ‖f‖∞ := ess sup x∈[0,1] \|f(x)\|, and similarly, for p ∈ R[x] let ‖p‖∞ := sup x∈[0,1] \|p(x)\|. Then L∞([0, 1]) := {f : ‖f‖∞ <∞}. 2.2 Some preliminary results We next state some useful auxiliary results, some of them being standard in Real Analysis. Proposition 2.1. Let t 7→ f(t) be an element of L2([0, 1]) and define f = (f(j)) by f(j) := ∫ 1 0 Lj(t)f(t) dt, j = 0, 1, . . . . Then one has ∞∑ j=0 f(j)Lj := lim n→∞ n∑ j=0 f(j)Lj  = f in L2([0, 1]), (2.2) and this decomposition is unique. Moreover f ∈ `2 and ‖f‖ = ‖f‖. 2The Legendre polynomials are orthonormal w.r.t. to the Lebesgue measure on [−1, 1]. 4 J.B. Lasserre 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 fk stands for the function t 7→ f(t)k, k ∈ N. If fk ∈ L2([0, 1]) we denote by fk = (fk(j)) ∈ `2 its (shifted) Legendre–Fourier coefficients so that ‖fk‖ = ‖fk‖ (where again the latter norm is that of `2). Notice that we also have: Proposition 2.2. Let f, fk, g ∈ L2([0, 1]) with k ∈ N. Then g = fk if and only if ĝ = f̂k. This follows from the uniqueness of the decomposition in the basis (Lj). We also have the following helpful results: Lemma 2.3. Let f ∈ L2([0, 1]) with Legendre–Fourier coefficients f = (f(j)), and let (pn) ⊂ R[t] be a sequence of polynomials such that ‖pn − f‖ → 0 as n→∞. If p̂n = (p̂n(j)) denotes the (shifted) Legendre–Fourier coefficients of pn, for all n = 1, 2, . . ., then ‖p̂n − f‖ → 0 in `2 as n→∞. Proof. As ‖pn − f‖2 = ∫ 1 0 (pn − f)2 dt and with dn = deg(pn), ∫ 1 0 (pn − f)2 dt = ∫ 1 0  dn∑ j=0 p̂n(j)Lj − f 2 dt = dn∑ j=0 (p̂n(j))2 ∫ 1 0 L 2 j dt︸︷︷︸ =1 +2 ∑ k<j p̂n(j)p̂n(k) ∫ 1 0 LjLk dt︸︷︷︸ =0 − 2 dn∑ j=0 p̂n(j) ∫ 1 0 Ljf dt︸︷︷︸ f(j) + ‖f‖2︸︷︷︸ =‖f‖= ∑ j f(j) 2 ≥ dn∑ j=0 p̂n(j)2 − 2p̂n(j)f(j) + f(j)2 = dn∑ j=0 (p̂n(j)− f(j))2, and the result follows because ‖pn − f‖ → 0 as n→∞. � Lemma 2.4. Let f ∈ L∞([0, 1]) (hence fk ∈ L2([0, 1]) for every k ∈ N). If a sequence (pn) ⊂ R[x] is such that supn ‖pn‖∞ <∞ and ‖pn − f‖ → 0 as n→∞, then for every k ∈ N, lim n→∞ ∥∥pkn − fk∥∥ = 0. In addition, if p̂kn denotes the (shifted) Legendre–Fourier coefficients of pkn, n = 1, . . ., then ‖p̂kn − fk‖ → 0 in `2 as n→∞. Proof. Let M > max[‖f‖∞, supn ‖pn‖∞] and fix k ∈ N, ∥∥pkn − fk∥∥2 = ∫ 1 0 ( pkn − fk )2 dx = ∫ 1 0 (pn − f)2 ( k−1∑ `=0 pk−1−`n f ` )2 dx ≤ ∫ 1 0 (pn − f)2 ( k∑ `=1 ∣∣pk−`n f `−1 ∣∣)2 dx ≤ ( kMk−1)2 ∫ 1 0 (pn − f)2 dx = ( kMk−1)2‖pn − f‖2 (→ 0 as n→∞.) Then the last statement follows from Lemma 2.3. � Moments and Legendre–Fourier Series for Measures Supported on Curves 5 Definition 2.5. Let f ∈ L2([0, 1]) with Legendre–Fourier coefficients f . For every k, n ∈ N, define the polynomial f (k) n ∈ R[x] and the vector f (k) n ∈ Rkn+1 by t 7→ f (k)n (t) :=  n∑ j=0 f(j)Lj(t) k = nk∑ j=0 f (k)n (j)Lj(t). Observe that each entry f (k) n (j), j = 0, . . . , nk, is a degree-k form of the first n + 1 Legendre– Fourier coefficients of f̂ . 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 f (k) = f (k−1)?f since f (k) n (t) = f (k−1) n (t)f (1) n (t) for all t ∈ [0, 1], and f (k−1) n → fk−1 as n→∞, as well as f (1) n → f . Lemma 2.6. Let f ∈ L∞([0, 1]), hence fk ∈ L2([0, 1) with (shifted) Legendre–Fourier coeffi- cients fk ∈ `2 for every k ∈ N, and assume that sup n ‖f (1)n ‖∞ = sup n ‖ n∑ j=0 f(j)Lj‖∞  <∞. Then fk = f (k) = f ? · · · ? f (k times) for every k = 1, 2, . . ., meaning that for every fixed k ∈ N, lim n→∞ ∥∥∥∥∥∥∥  n∑ j=0 f(j)Lj k − fk ∥∥∥∥∥∥∥ = lim n→∞ ∥∥∥∥∥∥ kn∑ j=0 f (k)n (j)Lj − fk ∥∥∥∥∥∥  = lim n→∞ ∥∥∥∥∥∥ n∑ j=0 fk(j)Lj − fk ∥∥∥∥∥∥ = 0. Equivalently, fk = fk−1 ? f for every k = 2, 3, . . .. Proof. The result is a direct consequence of Lemmas 2.3 and 2.4 with pn = f (1) n (and the definition of the limit “?-product” in Definition 2.5). � 3 Main result Assume that we are given all moments of a nonnegative measure dµ(x, t) on a box [a, b]× [c, d] ⊂ R2. 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 γi(j) = ∫ [0,1]2 xitj dµ(x, t), i, j = 0, 1, . . . . We further assume that the marginal measure µt with respect to the variable t, is the Lebesgue measure on [0, 1], that is, γ0(j) = 1/(j + 1), j = 0, 1, . . .. A standard disintegration of the measure µ yields γi(j) = ∫ [0,1]2 tjxi dµ(x, t) = ∫ 1 0 tj (∫ [0,1] xiψ(dx\|t)︸︷︷︸ =:fi(t) ) dt, i, j = 0, 1, . . . , (3.1) 6 J.B. Lasserre where the stochastic kernel ψ(·\| t) is the conditional probability on [0, 1] given t ∈ [0, 1]. Observe that the measurable function fi in (3.1) is nonnegative and uniformly bounded by 1 because \|xi\| ≤ 1 on [0, 1] for every i, and so fi ∈ L∞([0, 1]) for every i = 1, . . .. The vector γi = (γi(j)), j = 0, 1, . . ., is the vector of moments of the measure dµi(t) = fi(t)dt on [0, 1], for every i = 1, 2, . . .. The (shifted) Legendre–Fourier vector of coefficients q̂i of fi are obtained easily from the (infinite) vector γi via a triangular linear system. Indeed write Lj(t) = j∑ k=0 ∆jkt k, ∀ t ∈ [0, 1], j = 0, 1, . . . , where ∆jj > 0, or in compact matrix form L0 L1 · Ln ·  = ∆  1 t · tn ·  , (3.2) for some infinite lower triangular matrix ∆ with all diagonal elements being strictly positive. Therefore ∆γi = ∆ ∫ 1 0  1 t · tn ·  fi(t) dt = ∫ 1 0  L0 L1 · Ln ·  fi(t) dt = q̂i. Suppose that the measure µ is supported on a trajectory {(t, x(t)) : t ∈ [0, 1]} ⊂ [0, 1]2 for some measurable (density) function x : [0, 1] → [0, 1]. The measurable function t 7→ x(t) is an element of L∞([0, 1]) because ‖x‖∞ ≤ 1. Then by Proposition 2.1, x = ∞∑ j=0 x̂(j)Lj in L2([0, 1]), (3.3) where x̂ = (x̂(j)) ∈ `2 is its vector of (shifted) Legendre–Fourier coefficients (with ‖x̂‖ = ‖x‖). Similarly, for every k = 2, 3, . . ., the function t 7→ x(t)k is in L∞([0, 1]) and xk = ∞∑ j=0 x̂k(j)Lj in L2([0, 1]), with vector of (shifted) Legendre–Fourier coefficients x̂k ∈ `2 such that ‖xk‖ = ‖x̂k‖. We also recall the notation x̂(k) n ∈ Rkn+1 for the vector of coefficients in the basis (Lj) of the polynomial t 7→ ( n∑ j=0 x̂(j)Lj(t) )k , and when considered as an element of `2 (by completing with zeros) denote by x̂(k) ∈ `2 its limit when it exists. Theorem 3.1. Let µ be a Borel probability measure on [0, 1]2 and let γi(j), i, j = 0, 1, . . ., be the moments of µ in (2.1). Moments and Legendre–Fourier Series for Measures Supported on Curves 7 (a) If µ is supported on a trajectory {(t, x(t)) : t ∈ [0, 1]} for some nonnegative measurable function t 7→ x(t) on [0, 1] and if sup n ∥∥∥∥∥ n∑ j=0 x̂(j)Lj ∥∥∥∥∥ ∞ <∞, then x̂i = x̂(i) = x̂ ? · · · ? x̂︸︷︷︸ i times = x̂i−1 ? x̂, ∀ i = 2, 3, . . . , (3.4) Equivalently ∆γi = (∆γ1) (i) = ( ∆γi−1 ?∆γ1 )(i) , ∀ i = 2, 3, . . . , (3.5) where ∆ is the non singular triangular matrix defined in (3.2). (b) Conversely, if (3.5) holds then µ is supported on a trajectory {(t, x(t)) : t ∈ [0, 1]} for some measurable function t 7→ x(t) on [0, 1], and (3.4) also holds. Proof. The (a) part. As µ is supported on [0, 1]2 one has ‖x‖∞ ≤ 1 and so the function t 7→ x(t)i is in L2([0, 1]) for every i = 1, 2, . . .. So let t 7→ x(t) be written as in (3.3). Consider the function t 7→ x(t)i, for every fixed i ∈ N, so that xi = ∞∑ j=0 x̂i(j)Lj in L2([0, 1]), where the (shifted) Legendre–Fourier vector of coefficients x̂i is obtained by x̂i = ∆−1γi. But by Lemma 2.6, we also have xi =  ∞∑ j=0 x̂(j)Lj i = lim n→∞  n∑ j=0 x̂(j)Lj i in L2([0, 1]), with ‖xi‖ = ‖x̂i‖ and x̂i = ∆γi. In other words, x̂(i) = x̂i or equivalently, x̂(i) = ∆γi = (∆γ1) (i), which is (3.4). We next prove the (b) part. By the disintegration (3.1) of the measure µ, γi(j) = ∫ 1 0 tjfi(t) dt, i, j = 0, 1, . . . for some nonnegative measurable functions fi ∈ L∞([0, 1]), i = 1, 2, . . .. As (3.5) holds one may conclude that q̂i = q̂ (i) 1 where q̂i is the (shifted) Legendre–Fourier vector of coefficients associated with fi ∈ L2([0, 1]), i = 1, . . .. Hence by Proposition 2.2, fi(t) = f1(t) i a.e. on [0, 1], for every i = 1, 2, . . .. That is, for every i = 1, 2, . . ., there exists a Borel set Bi ⊂ [0, 1] with Lebesgue measure zero such that fi(t) = f1(t) i for all t ∈ [0, 1]\Bi. Therefore the Borel set B = ∪∞i=1Bi has Lebesgue measure zero and for all i = 1, 2, . . ., fi(t) = f1(t) i, ∀ t ∈ [0, 1]\B. Hence for every t ∈ [0, 1]\B,∫ [0,1] xiψ(dx\|t) = f1(t) i = ∫ [0,1] xi dδf1(t), ∀ i = 1, 2, . . . . where δf1(t) is the Dirac measure at the point f1(t) ∈ [0, 1]. As measures on compact sets are moment determinate, one must have ψ(dx\|t) = δf1(t), for all t ∈ [0, 1]\B. Therefore dµ(x, t) = δf1(t) dt, i.e., the measure µ is supported on the trajectory {(t, x(t)) : t ∈ [0, 1]}, where x(t) = f1(t) for almost all t ∈ [0, 1]. � 8 J.B. Lasserre Remark 3.2. If the trajectory t 7→ x(t) is a polynomial of degree say d, then the vector of Legendre–Fourier coefficients x̂ ∈ `2 has at most d + 1 non-zero elements. Therefore for every j = 2, . . ., x̂j ∈ `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=0 x̂(j)Lj ∥∥∥∥∥ ∞ < ∞ which is much weaker than, e.g., assuming the uniform convergence ∥∥∥∥∥ n∑ j=0 x̂(j)Lj − 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 L2 function in [−1, 1] furnishes enough information such that an exponential convergent approximation can be constructed in any subinterval in which f is analytic”. Example 3.3. To illustrate Theorem 3.1 consider the following toy example with µ on [0, 1]2 and with marginal w.r.t. “t” being the uniform distribution on [0, 1] and conditional ψ(dx\|t) = δexp(−t) for all t ∈ [0, 1]. That is, t 7→ x(t) = exp(−t). Then the first 11 Legendre–Fourier coefficients x̂(j), j = 0, . . . , 10 of x read x̂ = [0.63212055 −0.1795068 0.0230105 −0.0019370 0.0001217 −0.0000061 0.0000002 −0.00000001 −0.00000004 −0.0000015 −0.0000625]. Similarly the first 11 Legendre–Fourier coefficients of t 7→ x(t)2 = exp(−2t) read x̂2 = [0.4323323 −0.2344075 0.0588678 −0.0097965 0.0012219 −0.0001219 0.0000101 −0.0000007 0.00000004 −0.000000004 −0.00000004]. With n = 5 the polynomial t 7→ x (2) 5 (t) := ( 5∑ k=0 x̂(j)Lj(t) )2 reads x (2) 5 (t) = 10∑ k=0 x̂ (2) 5 (j) Lj(t), t ∈ R, with x̂ (2) 5 = [0.4323336 −0.2344129 0.0588626 −0.0097976 0.0012219 −0.0001218 0.0000098 −0.0000006 0.00000003 −0.000000001 0.0000000], and we can observe that x̂ (2) 5 − x̂2 ≈ O ( 10−5 ) . In fact the curves t 7→ x (2) 5 (t) and t 7→ exp(−2t) are almost indistinguishable on the inter- val [0, 1]. 3.1 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 7→ fi(t) of L2([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 ∈ L1([0, 1]) with all moments finite. The Moments and Legendre–Fourier Series for Measures Supported on Curves 9 only change is that now we have to consider the orthonormal polynomials t 7→ Hj(t), j = 0, 1, . . ., with respect to ν. Recall that all the Hj ’s can be computed from the moments γ0(j) = ∫ tjdµ(x, t) = ∫ 1 0 tj dν(t) = ∫ 1 0 tjh(t) dt, j = 0, 1, . . . . Then proceeding as before, for every i = 1, 2, . . .,∫ tjxi dµ(x, t) = ∫ 1 0 tj (∫ [0,1] xiψ(dx\|t)︸︷︷︸ =fi(t)∈L2([0,1],ν) ) h(t) dt, j = 0, 1, . . . , and we now consider the vector of coefficients f̂hi = (f̂hi(j)) defined by f̂hi(j) = ∫ 1 0 Hj(t)fi(t)h(t) dt, j = 0, 1, . . . the analogues for the measure dν = h(t)dt and the function fi ∈ L2([0, 1], ν), of the (shifted) Legendre–Fourier coefficients f̂i(j) of fi in (3.1) for the Lebesgue measure on [0, 1]. Then the conditions in Theorem 3.1 would be exactly the same as before, excepted that now, x̂ = (x̂(j)) with x̂(j) = ∫ 1 0 Hj(t)x(t)h(t) dt, j = 0, 1, . . . . 3.2 Discussion Theorem 3.1 may have some practical implications. For instance consider the weak formula- tion 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 (Pk), k = 1, 2, . . ., of P. Each optimal solution of Pk provides with a finite sequence zk = (zkj,α,β) such that when k →∞, zk → z∗ where z∗ is the infinite sequence of some measure dµ(t,x,u) on [0, 1] ×X ×U, where X ⊂ Rn, U ⊂ Rm, 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 zk 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, (x1(t), . . . , xn(t), u1(t), . . . , um(t)) : t ∈ [0, 1]}, one considers each coordinate xi(t) or uj(t) separately. For instance, for xi(t) one considers the subset of moments γk(j) = (z∗j,α,0) with j = 0, 1, . . ., α = (0, . . . , 0, k, 0, . . . , 0) ∈ Nn, k = 0, 1, . . ., with k in position i. If (3.4) holds then indeed the marginal µt,xi of µ on (t, xi), with moments (γk(j)) is supported on a trajectory {(t, xi(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 numer- ically a truncation Pk of P, and (b) the convergence zk → 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. 10 J.B. Lasserre Acknowledgements Research funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement ERC-ADG 666981 TAMING). References [1] Beiglböck M., Griessler C., A land of monotone plenty, Technical report, Department of Mathematics, University of Vienna, 2015. [2] Charina M., Lasserre J.B., Putinar M., Stöckler J., Structured function systems and applications, Oberwol- fach Rep. 10 (2013), 579–655. [3] Diaconis P., Freedman D., The Markov moment problem and de Finetti’s theorem. I, Math. 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Comp. 81 (2012), 861–877. http://dx.doi.org/10.4171/OWR/2013/11 http://dx.doi.org/10.4171/OWR/2013/11 http://dx.doi.org/10.1007/s00209-003-0636-6 http://dx.doi.org/10.1017/CBO9780511565717 http://dx.doi.org/10.1017/CBO9780511565717 http://dx.doi.org/10.1137/0733015 http://dx.doi.org/10.1214/07-AOP365 http://dx.doi.org/10.1214/07-AOP365 http://arxiv.org/abs/math.PR/0702314 http://dx.doi.org/10.1007/s00013-013-0557-5 http://arxiv.org/abs/1304.1716 http://dx.doi.org/10.1137/070685051 http://arxiv.org/abs/math.OC/0703377 http://arxiv.org/abs/1011.2911 http://dx.doi.org/10.1006/jfan.1996.0033 http://dx.doi.org/10.1006/jath.1997.3117 http://arxiv.org/abs/math.CA/9512222 http://dx.doi.org/10.1007/b12016 http://dx.doi.org/10.1137/0331024 http://dx.doi.org/10.1090/S0025-5718-2011-02549-4 1 Introduction 2 Notation, definitions and preliminary results 2.1 Notation and definitions 2.2 Some preliminary results 3 Main result 3.1 A more general case 3.2 Discussion References

Moments and Legendre-Fourier Series for Measures Supported on Curves

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