Orthonormal Systems in Linear Spans

TL;DR

This paper demonstrates that for any N-dimensional subspace of L^2 on the torus, one can construct an orthonormal system minimizing the square variation operator's L^2 norm, with specific improvements for trigonometric systems.

Contribution

It introduces a method to construct orthonormal systems in L^2 that minimize the square variation operator, improving upon the properties of classical trigonometric systems.

Findings

Existence of orthonormal systems with minimal V^2 norm in L^2

Construction of such systems for N-dimensional subspaces

Significant reduction of V^2 operator size for trigonometric polynomials

Abstract

We show that any $N$-dimensional linear subspace of $L^2(\mathbb{T})$ admits an orthonormal system such that the $L^2$ norm of the square variation operator $V^2$ is as small as possible. When applied to the span of the trigonometric system, we obtain an orthonormal system of trigonometric polynomials with a $V^2$ operator that is considerably smaller than the associated operator for the trigonometric system itself.