Moment problems and polynomial approximation

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistic...

Some Polynomial Problems Arising from Padé Approximation

In the convergence theory of Padé approximation, one needs to estimate the size of a set on which a suitably normalized polynomial q is small. For example, one needs to estimate the size of the set of r 2 [0; 1] for which max jtj=1 jq (t)j =min jtj=r jq (t)j is not “too large”. We discuss some old and new problems of this type, and the methods used to solve them. 1. Introduction Let f be a func...

A uniform approximation theorem and its application to moment problems

order that a (real) sequence #, may have the representation /~,=S f dx(t), 0 where X(t) is a function of bounded variation in [0, 1]. HARDY [3] outlines the proof of HAUSDORFF; several alternate proofs of the result of HAUSDORFF are known (see for instance WIDOER [14] and LORENTZ [8]). One of these alternate proofs makes use of the uniform approximation of functions continuous in [0, 1 ] by the...

Optimization and Approximation Problems Related to Polynomial System Solving

Moment Preferences and Polynomial Utility

can be established by a Taylor series argument if U( .) is assumed to be differentiable of a sufficiently high order, we have been puzzled by the absence (to our knowledge) of a direct non-calculus based proof of this latter result in the literature. The purpose of this letter is to provide such a proof, which we hope will help reveal the simple algebraic nature of this result. The next section...