In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem. Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications. While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples illustrating our results.

We prove that in dimension n≥ 2 , within the collection of unit-measure cuboids in Rn (i.e. domains of the form ∏i=1n(0,an)), any sequence of minimising domains RkD for the Dirichlet eigenvalues λk converges to the unit cube as k→ ∞. Correspondingly we also prove that any sequence of maximising domains RkN for the Neumann eigenvalues μk within the same collection of domains converges to the unit cube as k→ ∞. For n= 2 this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for n= 3 was recently treated by van den Berg and Gittins. In addition we obtain stability results for the optimal eigenvalues as k→ ∞. We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first k eigenvalues.

A long series of previous papers have been devoted to the (one-dimensional) moment problem with nonnegative rational measure. The rationality assumption is a complexity constraint motivated by applications where a parameterization of the solution set in terms of a bounded finite number of parameters is required. In this paper we provide a complete solution of the multidimensional moment problem with a complexity constraint also allowing for solutions that require a singular measure added to the rational, absolutely continuous one. Such solutions occur on the boundary of a certain convex cone of solutions. In this paper we provide complete parameterizations of all such solutions. We also provide errata for a previous paper in this journal coauthored by one of the authors of the present paper.

Let C denote a closed convex cone in R-d with apex at 0. We denote by E'(C) the set of distributions on R-d having compact support contained in C. Then E'(C) is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on (f) over cap (1), ..., (f) over cap (n) for f(n) is an element of E'(C) to generate the ring E'(C). (Here (center dot) over cap denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Lars Hormander. En route we answer an open question posed by Yutaka Yamamoto.