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The Truncated Moment Problem The Truncated Moment Problem Philipp J. di Dio To my parents They gave me the freedom to worry only about mathematics. The Truncated Moment Problem Von der Fakult¨atf¨urMathematik und Informatik der Universit¨atLeipzig angenommene DISSERTATION zur Erlangung des akademischen Grades DOCTORRERUMNATURALIUM (Dr. rer. nat.) im Fachgebiet Mathematik vorgelegt von Dipl.-Math. B.Sc. Philipp J. di Dio geboren am 25.02.1986 in Lutherstadt Wittenberg. Einreichungsdatum: 10. Januar 2018. Die Annahme der Dissertation wurde empfohlen von: 1. Prof. Dr. Konrad Schm¨udgen,Universit¨atLeipzig 2. Prof. Dr. Claus Scheiderer, Universit¨atKonstanz Die Verleihung des akademischen Grades erfolgt mit Bestehen der Verteidigung am 28. Mai 2018 mit dem Gesamtpr¨adikat magna cum laude. Contents 1 Introduction 1 2 Preliminaries 5 2.1 Basic Definitions for the Moment Problem and Carath´eodory Numbers . .5 2.2 Three Theorems of Hans Richter . .9 2.3 Apolar Scalar Product and Real Waring Rank . 10 3 Carath´eodory Numbers CA and CA;± 11 3.1 The Moment Cone SA and its Connection to CA ........................ 11 3.2 Continuous Functions . 15 3.3 Differentiable Functions . 18 3.4 Upper Bound Estimates on CA;± and the Real Waring Rank . 27 3.5 One-Dimensional Monomial Case with Gaps . 28 3.6 Multidimensional Monomial Case . 38 4 Set of Atoms, Determinacy, and Core Variety 47 4.1 Set W(s) of Atoms . 47 4.2 Determinacy of Moment Sequences . 54 4.3 Core Variety . 57 5 Weyl-Circles 59 5.1 Preliminaries . 59 5.2 Main Tool . 61 5.3 Hamburger Moment Problem . 62 5.4 Stieltjes Moment Problem on [a; 1).............................. 63 5.5 Truncated Moment Problem on [a; b].............................. 66 5.6 Truncated Moment Problem on R n (a; b)........................... 67 6 Applications 69 6.1 Numerical Integration . 69 6.2 Carath´eodory Number of Convex Hulls of Curves . 71 6.3 Non-negative Functions and Polynomials . 74 6.4 Tangent Planes . 74 6.5 Truncated Moment Problem for Gaussian Measures . 75 7 Conclusions and Open Problems 83 Acknowledgment I Selbst¨andigkeitserkl¨arung III Index and Symbols V Bibliography V i ii CONTENTS Chapter 1 Introduction Let A be a (finite- or infinite-dimensional) vector space of (real valued) functions on a locally compact Hausdorff space X with basis A. The (generalized) moment problem asks: Given a (real) sequence s = (sa)a2A. Does there exists a (positive) Borel measure µ on X such that Z sa = a(x) dµ(x) for all a 2 A; (1.1) X or equivalently is the Riesz functional Ls defined by Ls(a) = sa for all a 2 A represented by a (positive) Borel measure µ: Z Ls(f) = f(x) dµ(x) for all f 2 A? (1.2) X R α The name moment problem stems from the special case of monomials: sα = Rn x dµ(x) is the αth Nn moment of the (probability) measure µ for all α 2 0 , e.g. in probability theory or physics. This is also called classical moment problem and is widely studied together with many versions of this problem. These studies can be found in [Akh65], [KN77], and [Sch17b]. More, including historic remarks and developments, can be found in [SV97], [CF00], [CF05], [FP05], [Mar08], [Lau09], [Las15], and [Fia16]. Several of the present results in this work were published in [SdD17], [dDS18a], and [dDS18b] jointly with K. Schm¨udgen. Others are not jet published. α The generalization is made by going from monomials x to functions ai on X . While the classical moment problem deals with infinitely many moments and is mostly called full moment problem, if finitely many moments are investigated, it is called truncated. The study of the (truncated) general moment problem leads naturally to the study of gaps, i.e., where not all monomials xα are present. This will be one of our main interests in this work and all results in this work are new in the special monominial as well as the general case. Another distinction is between one- (X ⊆ R) and multidimensional (X ⊆ Rn). But this is only a technical distinction, since in the one-dimensional case techniques can be used which do not exist in higher dimensions, e.g., the fundamental theorem of algebra. The (truncated) moment problem appears in several mathematical areas and also techniques from several areas can be applied. Some are depicted in the following scheme: Probability Physics: Numerics: k Optimization Theory: E[Xα] R xαdµ(x) Z X ai(x) dµ(x) ≈ cjai(xj) j=1 Functional Analysis: Real Algebraic Geometry: The Truncated Moment Problem (Mxf)(x) = xf(x) non-negative polynomials Z p(x ; : : : ; x ) si = ai(x) dµ(x) i = 1; : : : ; m 1 n X Non-linear Analysis: Sk;A Linear Algebra: Differential Geometry: Convex Bodies: SA m Convex Analysis: @SA sA(x) = (ai(x))i=1 DSk;A 1 2 CHAPTER 1. INTRODUCTION The (truncated) moment problem connects all these areas: results and techniques transfer trough the moment problem from one to another and give an interesting interplay between these fields. So studying the (truncated) moment problem on its own right is beneficial even in other fields. In the current work we will investigate the (general) truncated moment problem. As shown, its range of application is vast and also often studied, see e.g. [KN77] or [Sch17b] and references therein. We concentrate our effort to two less studies aspects and fundamental questions therein: Carath´eodory numbers and set of atoms. Even for the most important cases where A is the set of polynomials in n variables of degree at most d little is known there. Our treatment is very general and we are therefore able to treat the polynomial cases where all monomials are present, but also the cases with gaps. The starting point of our investigations is a fundamental result by Richter (Theorem 2.23) [Ric57, m Satz 4]: Every truncated moment sequence s = (si)i=1 has a k-atomic representing measure with k ≤ m. So instead of dealing with all possible measures, it is sufficient to deal with k-atomic measures (linear combinations of δx-measures). To efficiently deal with k-atomic measures, we introduce the moment Rm m curve sA : X! ; x 7! (ai(x))i=1, i.e., sA(x) is the moment sequence of δx, and the moment map Pk Pk Sk;A(C; X) = i=1 cisA(xi), i.e., the moment sequence of i=1 ciδxi . From the Richter Theorem and the existence of atomic representing measures fundamental questions arise: Chapter 3: How many atoms are required to represent a fixed/all truncated moment sequence(s)? Chapter 4: What are the possible atom positions appearing in an atomic representing measure? Chapter 5: Does there exist a \good" description of all representing measures of a truncated moment sequence? In Chapter 3 we investigate the Carath´eodory number. Since every moment sequence s has a k- atomic representing measure, there is a smallest k. It is called the Carath´eodory number CA(s) of s. The maximum over all moment sequences is called the Carath´eodory number CA. Interestingly, despite its importance, little is known about CA even for the polynomial case with all monomials, let alone with gaps (some monomials are missing) or even continuous functions. From the map Sk;A we already find the first fundamental result in this work: A general lower bound on the Carath´eodory number CA. Under the assumption of sufficient differentiablity with n being the number of variables we find m ≤ C in Theorem 3.38: n + 1 A The main ingredient in the proof is the application of Sard's Theorem [Sar42]. This lower bound is so general, that it not only holds for polynomials on Rn, but also for polynomials with gaps or Cr-functions n n on subsets X ⊆ R . Even for polynomials on R lower bounds are rarely known, e.g. for A2;2k−1 the polynomials in 2 variables of degree at most 2k − 1 the M¨ollerbound [M¨ol76]is forty years old. The l jA2;2k−1j m 1 2 difference between 3 and M¨ollersbound grows as ∼ 6 k (Lemma 3.45). The implications of this result to numerics is discussed in Section 6.1. In Theorem 3.55 we also use the lower bound on the Carath´eodory number to reprove upper bounds on the real Waring rank of a homogeneous polynomial previously proved by Blekherman and Teitler [BT15]. d R d+1 In [Ric57] Richter proved that for A = f1; x; : : : ; x g on the Carath´eodory number is 2 , i.e., our lower bound is the Carath´eodory number in this case. This study was also extended to unions of intervals of R, see e.g. [KN77]. But for systems with gaps little is known. In Theorem 3.68 we gave a d2 dm R m criterion when A = f1; x ; : : : ; x g on also has Carath´eodory number CA = 2 at our lower bound. In Example 3.69 with A = f1; x2; x3; x5; x6g this condition is fulfilled, i.e., the Carath´eodory number is 3. A nice little byproduct is Corollary 3.70: Every non-negative polynomial a + bx2 + cx3 + dx5 + ex6 has at most 2 real zeros. Example 3.71 shows that when the condition in Theorem 3.68 is not meet, then m CA > 2 is possible. Our method is suited to study non-negative polynomials with gaps further. In Section 6.2 we treat the convex hulls of curves (a special case of the one-dimensional truncated moment problem) and especially their Carath´eodory numbers.
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