Determining Projection Constants of Univariate Polynomial Spaces
This work provides a practical computational method for a long-standing problem in approximation theory, enabling accurate determination of projection constants for low-degree polynomial spaces.
The paper presents computational techniques to determine bounds on projection constants of univariate polynomial spaces, achieving five-digit accuracy for three-dimensional spaces and four-digit accuracy for cubic, quartic, and quintic polynomial spaces. It also challenges existing beliefs about uniqueness and shape-preservation of minimal projections.
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.