A note on total degree polynomial optimization by Chebyshev grids

Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization

“…by discrete optimization of a trigonometric polynomial of degree n on approximately 2mn subperiodic Chebyshev-like angles. This is exactly the error bound obtainable for algebraic polynomial optimization on a real interval by approximately mn Chebyshev nodes, in view of the classical Ehlich-Zeller estimates in [10] (see also [4] and [15,20] with the references therein).…”

“…In particular, C 1 = 1 + O(m −1 ) whereas C * = 1 + O(m −2 ). Using the trigonometric Dubiner distance, a similar improvement can be obtained also for the constants of the general Jacobi-like norming meshes in ( [15]).…”

“…The following proposition shows that the Dubiner distance is also intimately related with the notions of norming set and polynomial mesh. It is proved in [15] and the proof is essentially outlined also in [1].…”

“…Moreover, polynomial meshes have also begun to be used in the framework of fully discrete approaches for polynomial optimization. We refer the reader, e.g., to [2,6,12,15] and the references therein.…”

An Elementary Approach to Polynomial Optimization on Polynomial Meshes

“…for example [8,9] and the references therein), whereas nonuniform grids (of Chebyshev type) together with the corresponding polynomial inequalities have been considered only on multidimensional boxes, for example in [10] and more recently in [11].…”