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Ten Lectures on the Moment Problem Konrad Schmüdgen Ten Lectures on the Moment Problem August 31, 2020 arXiv:2008.12698v1 [math.FA] 28 Aug 2020 Preface If µ is a positive Borel measure on the line and k is a nonnegativeinteger, the number k sk sk(µ)= x dµ(x) (0.1) ≡ ZR is called the k-th moment of µ. For instance, if the measure µ comes from the dis- tribution function F of a random variable X, then the expectation value of X k is just the k-the moment, +∞ k k E[X ]= sk = x dF(x), Z ∞ − 2 2 2 2 and the variance of X is Var(X)= E[(X E[X]) ]= E[X ] E[X] = s2 s1 (of course, provided the corresponding numbers− are finite). − − The moment problem is the inverse problem of “finding" the measure when the moments are given. In order to apply functional-analytic methods it is convenient to rewrite the mo- ment problem in terms of linear functionals. For a real sequence s = (sn)n N0 let Ls ∈ n denote the linear functional on the polynomial algebra R[x] defined by Ls(x )= sn, n N0. Then, by the linearity of the integral, (0.1) holds for all k if and only if ∈ +∞ Ls(p)= p(x)dµ(x) for p R[x]. (0.2) Z ∞ ∈ − That is, the moment problem asks whether a linear functional on R[x] admits an integral representation (0.2) with some positive measure µ on R. This is the simplest version of the moment problem, the so-called Hamburger moment problem. Moments and the moment problem have natural d-dimensional versions. For a Rd Nd positive measure µ on and a multi-index α = (α1,...,αd ) 0 , the α-the mo- ments is defined by ∈ α1 αd sα (µ)= x x dµ(x). ZRd 1 ··· d 3 4 Let be a closed subset of Rd. An important variant of the moment problem, called theK -moment problem, requires that the representing µ is supported on . If only a finiteK number of moments are given, we have the truncated moment problemK . Moments were applied to the study of functions by the Russian mathematicians P.L. Chebychev (1874) and A.A. Markov (1884). However, the moment problem first appeared in a famous memoir (1894) of the Dutch mathematician T.J. Stieltjes. He formulated and solved this problem for the half-line and gave the first explicit example of an indeterminate problem. The cases of the real line and of bounded intervals were studied only later by H. Hamburger (1920) and F. Hausdorff (1920). Because of the simplicity of its formulation it is surprising that the moment prob- lem has deep interactions with many branches of mathematics (functional analysis, function theory, real algebraic geometry, spectral theory, optimization, numerical analysis, convex analysis, harmonic analysis and others) and a broad range of ap- plications. The AMS volume “Moments in Mathematics", edited by H.J. Landau (1987), and the book “Moments, Positive Polynomials and Their Applications" by J.P. Lasserre (2015) illustrate this in a convincing manner. The following notes grew out from a series of lectures given at the “School on Sums of Squares, Moment Problems, and Polynomial Optimization", which was organized from 1.1.2019 till 31.3.2019 at the Vietnam Institute for Advanced Studies in Mathematics (VIASM) in Hanoi. I would like to thank my Vietnamese colleague, Prof. Trinh Le Cong, for the kind invitation and his warm hospitality during this visit. The audience consisted of researchers, young scientists, and graduate students. The first three lectures deal with one-dimensional full and truncated moment problems and contain well-known classical material. The topics of the other lec- tures were chosen according to the themes of the VIASM school. In Lectures 4–7, we investigate the multi-dimensional full moment problem with particular empha- sis on the interactions with real algebraic geometry. Lecture 7 gives a digression into applications of moment methods and real algebraic geometry to polynomial op- timization. The final Lectures 8–10 are devoted to the multidimensional truncated moment problem. In Lectures 4–10, I have covered selected advanced results as well as recent developments. The following are smoothed and slightly extended versions of the handouts I had given to the audience. I have tried to maintain the casual style of the lectures and to illustrate the theory by many examples. The notes are essentially based on my book “The Moment Problem”, Graduate Text in Mathematics, Springer, 2017, which is quoted as [MP] in what follows. For some results complete proofs are given, while for most of them I referred to [MP]. These notes have no bibliography. The book [MP] contains additional material and also detailed references for the 5 main results. Finally, it should be emphasized that these notes are not a book on the moment problem. It is more appropriate to think of them as an entrance guide into some developments of the beautiful subject. Contents 1 Integral representations of linear functionals ...................... 9 1.1 Positive linear functionalson adapted spaces..... ... ............ 10 1.2 Positivepolynomialsonintervals .................. ........... 13 1.3 Momentproblemsonintervals ....................... ........ 14 2 One-dimensional moment problem: determinacy .................. 17 2.1 An indeterminate measure: lognormal distribution . ........... 17 2.2 Carleman’s condition for the Hamburger moment problem . ....... 18 2.3 Krein’s condition for the Hamburger moment problem . ....... 19 2.4 Carleman and Krein conditions for the Stieltjes moment problem .. 20 3 The one-dimensional truncated moment problem on a bounded interval ................................................... .... 23 3.1 Existenceofasolution ............................ .......... 23 3.2 Themomentconeanditsboundarypoints............... ....... 24 3.3 Interiorpointsandprincipalmeasures .............. ........... 26 4 The moment problem on compact semi-algebraic sets .............. 29 4.1 Basic notionsfrom real algebraicgeometry ........... .......... 29 4.2 Strict Positivstellensatz and solution of the moment problem ...... 31 4.3 Localizing functionalsand Hankel matrices...... .... ........... 33 4.4 Momentproblemcriteriabased on semirings ........... ........ 34 4.5 Two technical results and the proof of Theorem4.11 ... .. ........ 35 5 The moment problem on closed semi-algebraic sets: the fibre theorem 39 5.1 Positive functionals which are not moment functionals . .......... 39 5.2 Properties(MP) and (SMP) and the fibre theorem......... ....... 40 5.3 First application: cylinder sets with compact base. ............ 43 5.4 (SMP)forsubsetsoftherealline .................... ......... 44 5.5 Second application: the two-sided complex moment problem...... 45 7 8 Contents 6 The moment problem on closed sets: determinacy and Carleman condition ................................................... ... 47 6.1 Variousnotionsofdeterminacy ..................... .......... 47 6.2 Determinacyviamarginalmeasures .................. ......... 48 6.3 The multivariate Carleman condition and Nussbaum’s theorem .... 49 6.4 Some operator-theoreticreformulations .... ..... ... ............ 51 7 Polynomial optimization and semidefinite programming ............ 53 7.1 Semidefiniteprogramming.......................... ......... 53 7.2 Lasserre relaxations of polynomial optimization with constraints ... 55 7.3 Polynomialoptimizationwith constraints ........... ........... 57 8 Truncated multidimensional moment problem: existence via positivity 59 8.1 TheRichter-TchakaloffTheorem .................... ......... 60 8.2 Positive semidefinite 2n-sequences............................ 61 8.3 The truncated moment problem on projectivespace . ........ 62 8.4 Stochel’stheorem ................................ .......... 64 9 Truncated multidimensional moment problem: existence via flat extensions ................................................... .. 67 9.1 Hankelmatrices.................................. .......... 67 9.2 The full moment problem with finite rank Hankel matrix.. ....... 69 9.3 Flat extensionsand the flat extensiontheorem ......... .......... 70 9.4 Hankel matrices of functionals with finitely atomic measures...... 71 10 Truncated multidimensional moment problem: core variety and moment cone .................................................. 75 10.1 Strictly positive linear functionals .... ..... ..... ............... 75 10.2 Thecorevariety................................. ........... 76 10.3 Themomentcone .................................. ........ 79 Lecture 1 Integral representations of linear functionals Abstract: An integral representation theorem of positive functionals on Choquet’s adapted spaces is obtained. As applications, Haviland’s theorem is derived and existence results for moment problems on intervals are developed. All variants of the moment problem deal with integral representations of certain linear functionals. We begin with a rather general setup. Suppose is a locally compact topological Hausdorff space. The Borel algebra B( ) is theXσ-algebra generated by the open subsets of . X X Definition 1.1. A Radon measure on is a measure µ : B( ) [0,+∞] such that µ(K) < ∞ for each compact subset KXof and X → X µ(M)= sup µ(K) : K M, K compact for all M B( ). (1.1) { ⊆ } ∈ X Thus, in our terminology Radon measures are always nonnegative! Closed subsets of Rd are locally compact in the induced topology of Rd. Further, suppose E is a vector space of continuous real-valued functions on . In a very general and modern form, the moment problem is the following problem:X Given a linear functional L : E R and a closed subset of , when does there exist a Radon measure µ supported→ on such that K X K L( f )= f (x)dµ(x) for f E? (1.2) ZX ∈ In this case, L is called a -moment functional
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