A note on the Moment Problem for codimension greater than 1

TL;DR

This paper establishes new convergence conditions for alternating projections in convex feasibility problems involving linear subspaces of finite codimension and lattice cones in Hilbert spaces, with specific results for $ ext{ell}^2( ext{N})$.

Contribution

It introduces novel criteria ensuring norm convergence of alternating projections in complex convex feasibility scenarios, especially for codimension greater than one.

Findings

Convergence of alternating projections is guaranteed under new basis conditions in Hilbert lattices.

Specific convergence results are provided for $ ext{ell}^2( ext{N})$ with particular basis vector properties.

The paper extends the understanding of the moment problem for higher codimension cases.

Abstract

We provide new conditions under which the alternating projection sequence converges in norm for the convex feasibility problem where a linear subspace with finite codimension $N\geq 2$ and a lattice cone in a Hilbert space are considered. The first result holds for any Hilbert lattice, assuming that the orthogonal of the linear subspace admits a basis made by disjoint vectors with respect to the lattice structure. The second result is specific for $\ell^2(\mathbb{N})$ and is proved when only one vector of the basis is not in the cone but the sign of its components is definitively constant and its support has finite intersection with the supports of the remaining vectors.