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Matrix-Valued Moment Problems Matrix-valued moment problems A Thesis Submitted to the Faculty of Drexel University by David P. Kimsey in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics June 2011 c Copyright 2011 David P. Kimsey. All Rights Reserved. ii Acknowledgments I wish to thank my advisor Professor Hugo J. Woerdeman for being a source of mathematical and professional inspiration. Woerdeman's encouragement and support have been extremely beneficial during my undergraduate and graduate career. Next, I would like to thank Professor Robert P. Boyer for providing a very important men- toring role which guided me toward a career in mathematics. I owe many thanks to L. Richard Duffy for providing encouragement and discussions on fundamental topics. Professor Dmitry Kaliuzhnyi-Verbovetskyi participated in many helpful discussions which helped shape my understanding of several mathematical topics of interest to me. In addition, Professor Anatolii Grinshpan participated in many useful discussions gave lots of encouragement. I wish to thank my Ph.D. defense committee members: Professors Boyer, Lawrence A. Fialkow, Grinshpan, Pawel Hitczenko, and Woerde- man for pointing out any gaps in my understanding as well as typographical mistakes in my thesis. Finally, I wish to acknowledge funding for my research assistantship which was provided by the National Science Foundation, via the grant DMS-0901628. iii Table of Contents Abstract ................................................................................ iv 1. INTRODUCTION ................................................................. 1 1.1 Measure theory .............................................................. 6 1.2 Integration theory ........................................................... 7 2. THE FULL MOMENT PROBLEM ON A LOCALLY COMPACT ABELIAN GROUP................................................................ 10 2.1 Positive semidefinite functions on a locally compact Abelian group ..... 10 2.2 Matrix-valued positive semidefinite functions on a locally compact Abelian group ................................................................ 13 2.3 The full moment problem on a locally compact Abelian group .......... 18 3. TRUNCATED MATRIX-VALUED MOMENT PROBLEMS .................. 20 3.1 The truncated matrix-valued K-moment problem on Rd ................. 20 3.2 The truncated matrix-valued K-moment problem on Cd ................. 53 3.3 The truncated matrix-valued K-moment problem on Td ................. 65 3.4 The cubic complex moment problem ....................................... 81 3.5 Hermitian matrix-valued generalization of Tchakaloff's theorem......... 88 List of References...................................................................... 96 APPENDIX A: FREQUENTLY USED SETS AND NOTATION ................. 103 VITA ...................................................................................104 iv Abstract Matrix-valued moment problems David P. Kimsey Hugo J. Woerdeman, Ph.D. We will study matrix-valued moment problems. First, we will study matrix-valued positive semidefinite function on a locally compact Abelian group. We will show that matrix-valued positive semidefinite functions and matrix-valued moment functions are in a one-to-one correspondence on a locally compact Abelian group. Next, we will explore the truncated matrix-valued K-moment problem on Rd, Cd, and Td. As a consequence, we get a solution criteria for data which is indexed up to an odd-degree to admit a minimal solution. Moreover, the representing measure is easily constructed from the given data. We will use this criterion to partially solve the complex cubic moment problem and show that it is different from the results that were obtained in the quadratic case. Finally, we will extend Tchakaloff's theorem, for the existence of cubature rules, to the matrix-valued case. CHAPTER 1. INTRODUCTION 1 1. Introduction The moment problem is a question in classical analysis which has given rise to rich theory in pure and applied mathematics. It is elegantly connected to orthogo- nal polynomials, spectral representation of operators, matrix factorization problems, probability and statistics, prediction of stochastic processes, polynomial optimization, inverse problems in financial mathematics, and function theory. A related object is the notion of a positive semidefinite function (sometimes called positive definite func- tion), which show up in probability theory. As we will see, the moment problem and ascertaining whether or not a function is positive semidefinite are equivalent in some settings. The moment problem bifurcates in the following way. First, the full moment problem asks for necessary and sufficient condition(s), called a solution, on an infinite ³ sequence of real numbers to be the power moments, which are given by xmdσpxq, m 0; 1;:::, of a positive Borel measure σ. We call σ a representing measure for the given sequence. Another important question is the following: if a sequence has a representing measure exists is it unique? If not, then one would like to understand the set of positive measures whose power moments are the given sequence. More generally, the K-moment problem has the additional requirement that the measure have support that lies in some given set K. Stieltjes [47], Hamburger [21], and Hausdorff [23], [22] studied moment problems when K is the non-negative real numbers, real numbers, and a closed and bounded interval, respectively. Second, the truncated moment problem is the same as the full moment problem save for the fact that one is given a finite sequence of real numbers. For excellent references confer Akhiezer [1], Krein and Nudel0man [30]. Curto and Fialkow [9] have a summary of the truncated K-moment problem. In particular, the truncated Stieltjes, Hamburger, and Hausdorff moment CHAPTER 1. INTRODUCTION 2 problems are considered as well. t u8 If a given sequence sm m1 are the power moments of a positive Borel measure σ. It is not hard to see that ¸n zaz¯bsa b ¥ 0; (1.1) a;b0 t u for all choices of subsets z1; : : : ; zn C. Indeed, » ¸n ¸n a b zaz¯bsa b zaz¯b x dσpxq R a;b 0 a;b §0 § » § §2 §¸n § § a§ p q § zax § dσ x R a0 ¥ 0: Note that (1.1) is equivalent to the Hankel matrix ¤ ¤ ¤ ¤ ¦s0 sn ¦ p qn ¦ : :: : © sa b a;b0 : ¦ : : : 0; ¥ sn ¤ ¤ ¤ s2n p qn P i.e., sa b a;b0 is positive semidefinite for all n N. Hamburger [21] showed that this necessary condition is also sufficient via a 150 page proof. A much shorter proof is based on the theory of self-adjoint extensions of symmetric operators, confer e.g. Ñ Lax [35]. One can define a positive semidefinite function s: N0 C, on N0, via t u8 (1.1). Hamburger's theorem can be formulated as follows. Every sequence sm m1 with a representing measure (also called a moment function on N0) is also a positive semidefinite function and the converse is also true. Bochner's theorem allows one to associate, via an integral representation, a unique finite positive Borel measure to any positive semidefinite function defined on a locally compact Abelian group G. The case when G is the set of d-tuples of integers is CHAPTER 1. INTRODUCTION 3 due to Herglotz [25]. While the case when G is the all d-tuples of real numbers is due to Bochner [5]. Weil [51] proved the result when G is an arbitrary locally compact Abelian group. Falb [18] extended Bochner's theorem to the case when the positive semidefinite function takes bounded linear operators on a complex separable Hilbert space as values. Thus the convex cone of positive semidefinite functions and the convex cone of moment functions are in one-to-one correspondence on a locally compact Abelian group. Beyond the classical work described above, several generalizations to the moment problem have occurred. We first catalog some of the multivariable generalizations. The main difficulty with the multidimensional moment problem is that there exist positive semidefinite functions which are not moment problems, confer Berg [4] for details. Thus an inequality similar to (1.1) is not enough to guarantee the existence of a representing measure. Haviland [24] has a solution to the full multivariable Hamburger moment problem which involves checking positivity of functionals which depend on the given data. We remark that this solution is not very tractable. An easy consequence of this result is a solution to the full multivariable K-moment problem, where K is semialgebraic. Subsequent improvements to this result have been made by Schm¨udgen[43] when K is assumed to be compact. We now discuss literature on multivariable truncated moment problems. Over the course of several papers, Curto and Fialkow [10], [11], and [12] developed their flat extension theory which solves certain multivariable truncated moment problems in the real and complex setting. The given data is assumed to be indexed up to an even total degree. A multivariable Hankel matrix is built from the given data, which must be necessarily positive semidefinite. A solution to the multivariable truncated Hamburger moment problem and multivarible truncated complex moment problem was obtained in terms of the ability to eventually construct rank preserving extension. CHAPTER 1. INTRODUCTION 4 This approach also leads to a solution to the truncated K-moment problem in both the multivariable real and multivariable complex settings, see [14], [16]. When K is semialgebraic, the truncated K-moment problem has applications in polynomial optimization (confer [31]) and also cubature problems. Stroud [49] has a canonical reference on cubature rules.
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