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

Abstract

We consider the multi-dimensional Ilausdorif 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.

Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

“…The numerical computation for the Hausdorff moment problem (HMP) has been proposed in [1,20,22]. In this paper, we use a Tikhonov type regularization method to solve (3.12) and the basic idea comes from paper [24].…”

Numerical Computation of a Cauchy Problem for Laplace's Equation

“…For the numerical computation of the Hausdorff moment problem, Talenti in [19] and Viano in [21] gave a practical algorithm in two-dimensional cases. A numerical method for the multi-dimensional Hausdorff moment problem can be found in Ang, Gorenflo, and Trong [2].…”

“…This is one kind of regularization with N being the regularizing parameter (cf. [2,3]). A-priori choice of N for convergence is given in the following corollary.…”

Laguerre polynomials and the inverse Laplace transform using discrete data

“…Another method is developed by Saitoh and his group [4,5,20,21], where the function f is approximated by integrals having the form…”