On Linear Functional Operations and the Moment Problem for a Finite Interval in One or Several Dimensions

The note by T. H. Hildebrandt2 on the moment problem for a finite interval, which deduces the Hausdorff theorem on moments from the theorem of Riesz on linear functional operations, suggests the question whether conversely the Riesz result is derivable from the Hausdorff theorem in a simple way, thus giving a new proof of the Riesz theorem. In the first part of this paper we show that this is actually the case. Incidentally, there is suggested a very simple and elegant method of proving the Riesz theorem directly. The object of the second part of this paper is to extend to the case of several variables, the methods of the note by I. J. Schoenberg3 on the moment problem, as well as the results indicated above, for the onedimensional case. Detailed consideration of this extension seems also to be justified by the fact that it requires the deduction of properties of functions of bounded variation as related to Stieltjes integrals in several variables. For convenience, the results obtained by this extension are stated and proved for the case of two variables only. The extension to more than two variables presents no essential difficulty. 1. A derivation of Riesz' theorem from the Hausdorff theorem. Let L [f ] be a linear continuous functional operation on continuous functions f(t) on (0, 1). Then there exists an M such that for every f: