Function theory, spectral decomposition of operators, probability,
approximation, electrical and mechanical inverse problems, prediction of
stochastic processes, the design of algorithms for signal-processing VLSI
chips—these are among a host of important theoretical and applied topics
illuminated by the classical moment problem. To survey some of these
ramifications and the research which derives from them, the AMS sponsored the
Short Course Moments in Mathematics at the Joint Mathematics Meetings,
held in San Antonio, Texas, in January 1987. This volume contains the six
lectures presented during that course. The papers are likely to find a wide
audience, for they are expository, but nevertheless lead the reader to topics
of current research.
In his paper, Henry J. Landau sketches the main ideas of past work
related to the moment problem by such mathematicians as Caratheodory,
Herglotz, Schur, Riesz, and Krein and describes the way the moment
problem has interconnected so many diverse areas of research.
J. H. B. Kemperman examines the moment problem from a geometric
viewpoint which involves a certain natural duality method and leads to
interesting applications in linear programming, measure theory, and
dilations. Donald Sarason first provides a brief review of the theory
of unbounded self-adjoint operators then goes on to sketch the
operator-theoretic treatment of the Hamburger problem and to discuss
Hankel operators, the Adamjan-Arov-Krein approach, and the theory of
unitary dilations. Exploring the interplay of trigonometric moment
problems and signal processing, Thomas Kailath describes the role of
Szegő polynomials in linear predictive coding methods, parallel
implementation, one-dimensional inverse scattering problems, and the
Toeplitz moment matrices. Christian Berg contrasts the
multi-dimensional moment problem with the one-dimensional theory and
shows how the theory of the moment problem may be viewed as part of
harmonic analysis on semigroups.
Starting from a historical survey of the use of moments in
probability and statistics, Persi Diaconis illustrates the continuing vitality
of these methods in a variety of recent novel problems drawn from such areas as
Wiener-Ito integrals, random graphs and matrices, Gibbs ensembles, cumulants
and self-similar processes, projections of high-dimensional data, and empirical
estimation.