Metric projections, zeros of optimal polynomial approximants, and some extremal problems in Hardy spaces

TL;DR

This paper explicitly calculates metric projections in Hardy spaces $H^p$, revealing new behaviors of best approximations and zeros of optimal polynomial approximants, extending classical results beyond $H^2$.

Contribution

It provides explicit formulas for metric projections in $H^p$, highlighting differences from $H^2$ and exploring the structure of shift-invariant subspaces.

Findings

In $H^p$, the best approximation of conjugate inner functions can be zero or non-constant outer functions.

The exact distance between the constant function and shift-invariant subspaces is determined.

Zeros of optimal polynomial approximants in $H^p$ exhibit new behaviors.

Abstract

The well-known proof of Beurling's Theorem in the Hardy space $H^2$, which describes all shift-invariant subspaces, rests on calculating the orthogonal projection of the unit constant function onto the subspace in question. Extensions to other Hardy spaces $H^p$ for $0 < p < \infty$ are usually obtained by reduction to the $H^2$ case via inner-outer factorization of $H^p$ functions. In this paper, we instead explicitly calculate the metric projection of the unit constant function onto a shift-invariant subspace of the Hardy space $H^p$ when $1<p<\infty$. This problem is equivalent to finding the best approximation in $H^p$ of the conjugate of an inner function. In $H^2$, this approximation is always a constant, but in $H^p$, when $p\neq 2$, this approximation turns out to be zero or a non-constant outer function. Further, we determine the exact distance between the unit constant and any shift-invariant subspace and propose some open problems. Our results use the notion of Birkhoff-James orthogonality and Pythagorean Inequalities, along with an associated dual extremal problem, which leads to some interesting inequalities. Further consequences shed light on the lattice of shift-invariant subspaces of $H^p$, as well as the behavior of the zeros of optimal polynomial approximants in $H^p$.