Quasi-analyticity and determinacy of the full moment problem from finite to infinite dimensions

This paper is aimed to show the essential role played by the theory of quasi-analytic functions in the study of the determinacy of the moment problem on finite and infinite-dimensional spaces. In particular, the quasi-analytic criterion of self-adjointness of operators and their commutativity are crucial to establish whether or not a measure is uniquely determined by its moments. Our main goal is to point out that this is a common feature of the determinacy question in both the finite and the infinite-dimensional moment problem, by reviewing some of the most known determinacy results from this perspective. We also collect some properties of independent interest concerning the characterization of quasi-analytic classes associated to log-convex sequences.Comment: 28 pages, Stochastic and Infinite Dimensional Analysis, Chapter 9, Trends in Mathematics, Birkh\"auser Basel, 201

Reconstruction of the early Universe as a convex optimization problem

We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales -- a few Mpc -- where multi-streaming becomes significant. Above 6 Mpc/h we propose and implement an effective Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge (1781). After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than cubic time complexity in N and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60% of exactly reconstructed points. We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge-Ampere equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. Self-contained discussion of relevant notions and techniques is provided.Comment: 26 pages, 14 figures; accepted to MNRAS. Version 2: numerous minour clarifications in the text, additional material on the history of the Monge-Ampere equation, improved description of the auction algorithm, updated bibliography. Version 3: several misprints correcte