Sign intermixing for Riesz bases and frames measured in the Kantorovich-Rubinstein norm

TL;DR

This paper investigates how the Kantorovich-Rubinstein norm measures sign interlacing in Bessel sequences and frames in L^2 spaces, revealing that convergence rates depend on the measure space's dimension and Bernstein n-widths.

Contribution

It establishes a quantitative relationship between the decay of the Kantorovich-Rubinstein norm and Bernstein n-widths, providing sharp convergence rate results for frames in d-dimensional cubes.

Findings

Convergence rate of KR norms depends on measure space dimension.

Sharp bounds for worst and best convergence rates of frames.

Dependence of sign interlacing phenomena on Bernstein n-widths.

Abstract

We measure a sign interlacing phenomenon for Bessel sequences $ (u_{k})$ in $ L^{2}$ spaces in terms of the Kantorovich--Rubinstein mass moving norm $ \Vert u_{k}\Vert_{KR}$. Our main observation shows that, quantitatively, the rate of the decreasing $ \Vert u_{k}\Vert_{KR}\longrightarrow 0$ havily depends on S. Bernstein $ n$-widths of a compact of Lipschitz functions. In particular, it depends on the dimension of the measure space. We have sharp results on the worst and the best rate of convergence of Kantorovich--Rubinstein norms of frames on $d$-dimensional cube. Those rates are sharp.