UW AGD Seminar: Nicholas Marshall

We consider a region in the complex plane enclosed by a deltoid curve inscribed in the unit circle, and define a family of polynomials that satisfy the same recurrence relation as the Faber polynomials for this region. We use this family of polynomials to give a constructive proof that z^n is approximately a polynomial of degree ~sqrt(n) within the deltoid region. Moreover, we show that this family of polynomials is bounded in this deltoid region and has a useful rapid growth property. We illustrate our polynomial approximation theory with an application to iterative linear algebra, and discuss generalizations of the presented framework. (This talk is based on joint work with Peter Cowal and Sara Pollock).