The

moment problem Let I ⊆ R be an interval. For a positive measure µ on I the nth moment is defined as ∫ I xndµ(x) – provided the integral exists. If we suppose that (sn)n≥0 is a sequence of real numbers, the moment problem on I consists of solving the following three problems: (I) Does there exist a positive measure on I with moments (sn)n≥0? In the affirmative, (II) is this positive measure uniquely determined by the moments (sn)n≥0? If this is not the case, (III) how can one describe all positive measures on I with moments (sn)n≥0? Without loss of generality we can always assume that s0 = 1. This is just a question of normalizing the involved measures to be probability measures. When µ is a positive measure with moments (sn)n≥0, we say that µ is a solution to the moment problem. If the solution to the moment problem is unique, the moment problem is called determinate. Otherwise the moment problem is said to be indeterminate. On the following pages we give an introduction to the classical moment problem on the real line with special focus on the indeterminate case. For a more detailed discussion the reader is referred to Akhiezer [1], Berg [3] or Shohat and Tamarkin [23]. There are three essentially different types of (closed) intervals. Either two end-points are finite, one end-point is finite, or no end-points are finite. In the last case the interval is simply R and in the first two cases one can think of [0, 1] and [0,∞). For historical reasons the moment problem on [0,∞) is called the Stieltjes moment problem and the moment problem on R is called the Hamburger moment problem. Moreover, the moment problem on [0, 1] is referred to as the Hausdorff moment problem. It is elementary linear algebra to verify that a positive measure with finite support is uniquely determ-ined by its moments. Applying the approximation theorem of Weierstrass and the Riesz representation theorem, one can extend this result to hold for positive measures with compact support. The Hausdorff moment problem is therefore always determinate. As regards existence, Hausdorff [13] proved in 1923 that the moment problem has a solution on [0, 1] if and only if the sequence (sn)n≥0 is completely monotonic. Stieltjes introduced the moment problem on [0,∞) and solved the problems about existence and unique-ness in his famous memoir “Recherches sur les fractions continues ” from 1894-95, see [24]. The memoir is devoted to the study of continued fractions of the form 1 m1z