An inequality for bi-orthogonal pairs

TL;DR

This paper establishes a lower bound for partial sums of bi-orthogonal series in Hilbert spaces, with applications to $L^{1}$ norms in orthogonal expansions and linear algebra, using Salem's method.

Contribution

It introduces a new lower bound for bi-orthogonal series partial sums and applies Salem's method to derive results in orthogonal expansions and linear algebra.

Findings

Lower bounds for partial sums of bi-orthogonal series

Lower bounds on $L^{1}$ norms for orthogonal expansions

Applications to linear algebra

Abstract

We use Salem's method to prove that there is a lower bound for partial sums of series of bi-orthogonal vectors in a Hilbert space, or the dual vectors. This is applied to some lower bounds on $L^{1}$ norms for orthogonal expansions. There is also an application concerning linear algebra.