Best Approximation Pair of Two Linear Varieties via an (In)Equality by (Fan-Todd) Beesack

TL;DR

This paper introduces a method to find the closest points between two linear varieties using an Ostrowski-type inequality, providing explicit formulas and characterizations for the best approximation pair.

Contribution

It presents a novel approach leveraging an (in)equality to explicitly compute the best approximation points between linear varieties.

Findings

Explicit formulas for closest points are derived.

Characterization of the best approximation pair is provided.

The method applies to linear varieties using an Ostrowski-type inequality.

Abstract

The closest point of a linear variety to an external point is found by using the equality case of an Ostrowski's type inequality. This point is given in closed form as the quotient of a (formal) and a (scalar) Gram determinant. Then, the best approximation pair of points onto two linear varieties is given, as well as characterization of this pair of best approximation points.