Determining Projection Constants of Univariate Polynomial Spaces

TL;DR

This paper presents computational methods to accurately determine projection constants of univariate polynomial spaces, advancing understanding of minimal projections through linear and semidefinite programming techniques.

Contribution

It introduces a computational framework combining linear and semidefinite programming to estimate projection constants with high precision, addressing a longstanding problem.

Findings

Achieved high-accuracy bounds for projection constants of specific polynomial spaces.

Demonstrated the effectiveness of combined linear and semidefinite programming methods.

Provided insights into the uniqueness and shape-preservation of minimal projections.

Abstract

The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a linear program, and the lower bound, produced by a semidefinite program exploiting the method of moments, are often close enough to deduce the projection constant with reasonable accuracy. The implementation of these programs makes it possible to find the projection constant of several three-dimensional spaces with five digits of accuracy, as well as the projection constants of the spaces of cubic, quartic, and quintic polynomials with four digits of accuracy. Beliefs about uniqueness and shape-preservation of minimal projections are contested along the way.