Truncated Moment Problems for the Unit Disk and Unit Circle
by
Raúl E. Curto
University of Iowa

In joint work with Lawrence A. Fialkow, we have recently found criteria for the existence of finitely atomic measures interpolating a given collection of complex numbers, and with support inside a prescribed semi-algebraic set in the complex plane. Our study includes the construction of a new ''localizing matrix, '' and it uses previous work on

• positivity for square matrices;
• extendibility of matrices obtained by adding a prescribed number of rows and columns;
• recursiveness; and
• the structure of the real or complex algebraic variety associated to the given moment matrix.

When these four basic ingredients interact in appropriate ways, aided by symbolic manipulation, the result is the construction of algorithms that often describe in detail the space of all possible representing measures. We apply the new criteria to the specific cases of the unit circle and unit disk, and we describe concrete algorithms for the construction of interpolating measures for the quadratic moment problem, possessing one atom inside the disk, and the remaining atoms on the circle.

Date received: May 7, 1999