Lacunary polynomials in L1: Geometry of the unit sphere

Konstantin M Dyakonov

doi:10.1016/j.aim.2021.107607

Abstract

Let Λ be a finite set of nonnegative integers, and let P(Λ) be the linear hull of the monomials zk with k∈Λ, viewed as a subspace of L1 on the unit circle. We characterize the extreme and exposed points of the unit ball in P(Λ).

Highlights

Let PN stand for the set of polynomials of degree at most N
Fewnomial, in PN one loosely means a polynomial therein that has “gaps,” in the sense that it is spanned by some selected monomials from the family {zk : k = 0, . . . , N} rather than by all of them
If p is a unit-norm polynomial in P(Λ) satisfying M < m, p is a non-extreme point of ball(P(Λ))

Summary

Introduction

Let PN stand for the set of polynomials (in one complex variable) of degree at most N. It was proved by de Leeuw and Rudin in [3] (see [13, Chapter IV]) that the extreme points of ball(H1) are precisely the outer functions f ∈ H1 with f 1 = 1 By contrast, it is far from clear which unit-norm (and outer) functions in H1 arise as exposed points therein. As far as exposed points of ball(K1(φ)) are concerned, no nice description is available; see, [5, Section 2] for some partial results in the model subspace setting. In both cases, the statements are followed by a brief discussion, and some examples are provided. Only the classical—i.e., nonlacunary—case (where K is an interval) has been studied in this connection; see [9, Section 5]

Extreme points: criterion and examples

Exposed points: criterion and examples

Preliminaries