Optimal Polynomial Approximants in $H^p$

This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, $H^p$ ($1<p<\mathrm{\infty }$). In particular, we uncover some estimates concerning the OPAs of degree zero and one. It is also shown that if $f\in H^p$ is an inner function, or if $p>2$ is an even integer, then the roots of the nontrivial OPA for $1\mathrm{/}f$ are bounded from the origin by a distance depending only on $p$. For $p\mathrm{\ne }2$, these results are made possible by the novel use of a family of inequalities which are derived from a Banach space analogue of the Pythagorean theorem.