More properties of optimal polynomial approximants in Hardy spaces

This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, $H^p$ ($1<p<\mathrm{\infty }$). For fixed $f\in H^p$ and $n\in \mathbb{N}$, the OPA of degree $n$ associated to $f$ is the polynomial which minimizes the quantity $\mathrm{\parallel }qf-1{\mathrm{\parallel }}_p$ over all complex polynomials $q$ of degree less than or equal to $n$. We begin with some examples which illustrate, when $p\mathrm{\ne }2$, how the Banach space geometry makes these problems interesting. We then weave through various results concerning limits and roots of these polynomials, including results which show that OPAs can be witnessed as solutions of certain fixed point problems. Finally, using duality arguments, we provide several bounds concerning the error incurred in the OPA approximation.