On the Truncated Multidimensional Moment Problems in Cn

We consider the problem of finding a (non-negative) measure μ on B(Cn) such that ∫Cnzkdμ(z)=sk, ∀k∈K. Here, K is an arbitrary finite subset of Z+n, which contains (0,…,0), and sk are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. Firstly, one may consider this problem as an extension of the truncated multidimensional moment problem on Rn, where the support of the measure μ is allowed to lie in Cn. Secondly, the moment problem is a particular case of the truncated moment problem in Cn, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials.

Numerical Algorithms for Estimating Probability Density Function Based on the Maximum Entropy Principle and Fup Basis Functions

Estimation of the probability density function from the statistical power moments presents a challenging nonlinear numerical problem posed by unbalanced nonlinearities, numerical instability and a lack of convergence, especially for larger numbers of moments. Despite many numerical improvements over the past two decades, the classical moment problem of maximum entropy (MaxEnt) is still a very demanding numerical and statistical task. Among others, it was presented how Fup basis functions with compact support can significantly improve the convergence properties of the mentioned nonlinear algorithm, but still, there is a lot of obstacles to an efficient pdf solution in different applied examples. Therefore, besides the mentioned classical nonlinear Algorithm 1, in this paper, we present a linear approximation of the MaxEnt moment problem as Algorithm 2 using exponential Fup basis functions. Algorithm 2 solves the linear problem, satisfying only the proposed moments, using an optimal exponential tension parameter that maximizes Shannon entropy. Algorithm 2 is very efficient for larger numbers of moments and especially for skewed pdfs. Since both Algorithms have pros and cons, a hybrid strategy is proposed to combine their best approximation properties.

On the Moment Problem and Related Problems

Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (yj)j∈ℕn of real numbers and a closed subset F⊆ℝn, n∈{1,2,…}, find a positive regular Borel measure μ on F such that ∫Ftjdμ=yj, j∈ℕn. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution μ are the focus of attention. The numbers yj, j∈ℕn are called the moments of the measure μ. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments yj are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.

On Markov Moment Problem and Related Results

We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result. The domain is the Banach lattice of continuous real-valued functions on a compact subset or an space, where is a positive moment determinate measure on a closed unbounded set. The existence and uniqueness of the operator solution are proved. Our solutions satisfy the interpolation moment conditions and are between two given linear operators on the positive cone of the domain space. The norm controlling of the solution is emphasized. The most part of the results are stated and proved in terms of quadratic forms. This type of result represents the first aim of the paper. Secondly, we construct a polynomial solution for a truncated multidimensional moment problem.

THE HAUSDORFF MOMENT PROBLEM IN THE LIGHT OF ILL-POSEDNESS OF TYPE I

The Hausdorf moment problem (HMP) over the unit interval in an L 2 -setting is a classical example of an ill-posed inverse problem. Since various applications can be rewritten in terms of the HMP, it has gathered significant attention in the literature. From the point of view of regularization it is of special interest because of the occurrence of a non-compact forward operator with non-closed range. Consequently, HMP constitutes one of few examples of a linear ill-posed problem of type I in the sense of Nashed. In this paper we highlight this property and its consequences, for example, the existence of an infinite-dimensional subspace of stability. On the other hand, we show conditional stability estimates for the HMP in Sobolev spaces that indicate severe ill-posedness for the full recovery of a function from its moments, because H ̈older-type stability can be excluded. However, the associated recovery of the function value at the rightmost point of the unit interval is stable of H ̈older-type in an H1 -setting. We moreover discuss stability estimates for the truncated HMP, where the forward operator becomes compact. Some numerical case studies illustrate the theoretical results and complete the paper.