Least Squares Problems in Orthornormalization

TL;DR

This paper introduces a unique orthonormal basis in Hilbert spaces that minimizes the sum of squared deviations from a given linearly independent set, and explores its stability and applications.

Contribution

It constructs a unique orthonormal basis minimizing the squared deviations from a given set and analyzes its stability in Hilbert spaces.

Findings

The orthonormal basis is unique and optimal in the least squares sense.

The paper provides stability analysis of the constructed orthonormalization.

Applications and examples demonstrate the practical relevance of the method.

Abstract

For any $n$-tuple $(\alpha_1,...,\alpha_n)$ of linearly independent vectors in Hilbert space $H$, we construct a unique orthonormal basis $(\epsilon_1,...,\epsilon_n)$ of $span\{\alpha_1,...,\alpha_n\}$ satisfying: $$\sum_{i=1}^n\|\epsilon_i-\alpha_i\|^2\le\sum_{i=1}^n\|\beta_i-\alpha_i\|^2$$ for all orthonormal basis $(\beta_1,...,\beta_n)$ of $span\{\alpha_1,...,\alpha_n\}$. We study the stability of the orthornormalization and give some applications and examples.