Linear completeness of trajectories in Sobolev spaces and the symmetrised polydisk

TL;DR

This paper develops a framework using infinite analytic block Toeplitz operators to analyze the linear completeness of non-linear trajectories in Sobolev spaces, with applications to classical functions and eigenfunctions of the Gross-Pitaevskii equation.

Contribution

It introduces a novel approach linking non-linear analysis and linear approximation theory to establish linear completeness in Sobolev spaces.

Findings

Linear completeness of dilated Weierstrass functions in Sobolev spaces.

Linear completeness of eigenfunctions of the Gross-Pitaevskii equation.

New insights into methods for analyzing non-linear trajectories.

Abstract

We establish a framework to determine the linear completeness of families of non-linear trajectories in Hilbert spaces, which relies on an infinite analytic block Toeplitz operator formulation. By means of this approach, we show the linear completeness in Sobolev spaces of two families of classical functions. One is the moving family of dilated Weierstrass functions. The other is the family of eigenfunctions of the Gross-Pitaevskii equation with trapping potential in an infinite square well. Our results provide a new insight on the formulation of general methods to examine this intriguing concept, bridging classical non-linear analysis and linear approximation theory.