Hausdorff Moment Problem Research Articles

A Multi-Dimensional Hausdorff Moment Problem: Regularization by Finite Moments

We consider the multi-dimensional Hausdorff moment problem over the unit cube: to reconstruct an unknown function from the (inaccurately) given values of the integrals of the unknown function multiplied by all power-products of the independent variables. We describe a regularization scheme using orthogonalization by the tensor product of (shifted) Legendre polynomials and "approximation" of the unknown function by a finite sum, the dimension of the space of approximation playing the role of the regularization parameter. For the case of square integrability of the unknown function we present an estimate of the regularization error that implies convergence if the data error tends to zero.

The Hausdorff Moment Problem with Complex Exponents and a Müntz‐Szasz Theorem for Distributions

AbstractThis paper deals with a Hausdorff moment problem with complex exponents, that is, given a sequence of complex numbers (zn)n and a fixed space X of functions denned on [0, 1], we ask under which conditions on a sequence (an)n the moment problem an = ∫ tZn f(t) dt for n ∈ IN has a solution in the space X. Mainly, we study the solution belonging to the space D[0, 1] The results which we will obtain will lead us to prove a Müntz‐Szasz Theorem for distributions.

A note about minimum relative entropy solutions of finite moment problems

The finite Hausdorff moment problem admits minimum relative entropy solutions if and only if the data satisfy some simple geometrical conditions. A procedure to compute such solutions is rigorously analyzed here.

Error bounds in maximum entropy approximations

A useful technique in underdetermined inverse problems is that of maximum entropy. A simple error bound for averages over a distribution approximated by the maximum entropy method in the case of the undetermined Hausdorff moment problem was devised. Under the conditions specified, the error bound for averages over such an approximate distribution can be very tight. Numerical examples to illustrate are presented.

The Hausdorff moments in statistical mechanics

A new method for solving the Hausdorff moment problem is presented which makes use of Pollaczek polynomials. This problem is severely ill posed; a regularized solution is obtained without any use of prior knowledge. When the problem is treated in the L2 space and the moments are finite in number and affected by noise or round-off errors, the approximation converges asymptotically in the L2 norm. The method is applied to various questions of statistical mechanics and in particular to the determination of the density of states. Concerning this latter problem the method is extended to include distribution valued densities. Computing the Laplace transform of the expansion a new series representation of the partition function Z(β) (β=1/kBT) is obtained which coincides with a Watson resummation of the high-temperature series for Z(β).

On the Solution of an Ill-Posed Non-Linear Fredholm Integral Equation Connected with an Inverse Problem of Thin Film Optics

We carry out a theoretical analysis of the simultaneous identification of geometrical thick-ness and refractive index profile for inhomogeneous single layer systems from indirect mea-surements. The problem leads to a non-linear integral equation of the first kind with smooth kernel. We present a uniqueness theorem for monotone solutions referring to the Hausdorff moment problem.

Solution of the Hausdorff moment problem by the use of Pollaczek polynomials

We present a new method for solving the Hausdorff moment problem which makes use of Pollaczek polynomials. This problem is ill-posed in the sense of Hadamard and therefore it is unstable from a numerical viewpoint. We shall prove that the expansion, which we obtain, is asymptotically convergent in the L 2-norm when the data (i.e., the moments) are perturbed by noise and finite in number. This result is largely used in the numerical analysis of the problem.

Moment problems for compact sets

The solvability of the Hausdorff moment problem for an arbitrary compact subset of Euclidean n n -space is shown to be equivalent to the nonnegativity of a family of quadratic forms derived from the given moment sequence and the given compact set. A variant theorem for the one-dimensional case and an analogous theorem for the trigonometric moment problem are also given. The one-dimensional theorems are similar to theorems of Kreĭn and Nudel’man [11], but the proofs, unlike those in [11], do not depend on the existence of a standard form for polynomials which are nonnegative on a given compact set.

Maximum entropy in the problem of moments

The maximum-entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem. In important special cases, such as the Hausdorff moment problem, we establish necessary and sufficient conditions for the existence of a maximum-entropy solution and examine the convergence of the resulting sequence of approximations. A number of explicit illustrations are presented. In addition to some elementary examples, we analyze the maximum-entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems. We also briefly indicate possible applications to the Lee–Yang theory of Ising models, to the summation of divergent series, and so on. The general conclusion is that maximum entropy provides a valuable approximation scheme, a serious competitor of traditional Padé-like procedures.

An algorithm for the Hausdorff moment problem

Given 0?x?1 and the moments $$\mu _n (f) = \int\limits_0^1 {t^n f(t)dt} $$ of a suitably smooth functionf, a sequence of approximations is presented which converges tof(x). An asymptotic expansion of the error is established. This shows how to construct a useful acceleration of the convergence.

Hausdorff's moment problem and expansions in Legendre polynomials

A new proof is given for Hausdorff's condition on a set of moments which determines when the function generating these moments is in L 2. The proof uses Legendre polynomials and their discrete extensions found by Tchebychef. Then an extension is given to a weighted L 2 space using Jacobi polynomials and their discrete extensions.

Stabilization and error bounds for the inverse laplace transform

We present some theorems on the stabilization of the inverse Laplace transform. We assume the Laplace transform is measured at N points to within some error ∊. We prove error bounds for the inverse Laplace transform in terms of ∊ and N, under suitable a-priori constraints. This is achieved by proving parallel stabilization results for a related Hausdorff moment problem.

The Hausdorff Moment Problem

Suppose throughout thatand that {μn}(n≥ 0) is a sequence of real numbers. The (generalized) Hausdorff moment problem is to determine necessary and sufficient conditions for there to be a function x in some specified class satisfying.

A Hausdorff moment problem for operators in locally convex spaces

A Hausdorff moment problem for operators in locally convex spaces

Quadratic forms and chain sequences

It has been shown that a monotone Hausdorff moment sequence can be characterized by Stieltjes type quadratic forms. This was a result of the theory of chain sequences [4]. In the present paper we shall characterize an extended monotone Hausdorff moment sequence by Jacobi type quadratic forms. This will be done by extending the properties of ordinary chain sequences. The paper is in two sections. In the first section a relation between chain sequences and Jacobi type quadratic forms will be established. The connection between the extended monotone Hausdorff moment problem and chain sequences will be discussed in the second section.

A Remark on Completely Monotonic Sequences, with an Application to Summability

A sequence of non-negative numbers, μ0 μ1 … μn, is called completely monotonic [5, p. 108] if (-1)n Δn μk ≧ 0 for n, k = 0, 1, 2, …. Such sequences occur in many connexions, such as the Hausdorff moment problem and Hausdorff summability [1, Chapter XI, 5, Chapters III and IV].It is natural to inquire as to the circumstances under which the inequality "≧" above can be strengthened to ">". As it happens, this can be done always, except for sequences all of whose terms past the first are identical. The formal statement follows. (In it, as above, Δnμk is the n-th forward difference, i. e. Δ0μk = μk; Δnμk = Δn-1 μk+1 - Δn-1μk.)