On the Lazarev-Lieb Extension of the Hobby-Rice Theorem

TL;DR

This paper provides an alternative proof of Lazarev and Lieb's theorem on the existence of smooth circle-valued functions that annihilate a span of integrable functions, with bounds on the Sobolev norm, and discusses limitations for p>1.

Contribution

It offers a new proof with explicit bounds for the $W^{1,1}$ norm and clarifies the non-existence of such bounds for $W^{1,p}$ when p>1.

Findings

Bound on $W^{1,1}$ norm by $5\pi n+1$

No bounds for $W^{1,p}$ norm when p>1

Alternative proof of Lazarev-Lieb theorem

Abstract

O. Lazarev and E. H. Lieb proved that given $f_{1},...,f_{n}\in L^{1}([0,1];\mathbb{C})$, there exists a smooth function $\Phi$ that takes values on the unit circle and annihilates ${span}\{f_{1},...,f_{n}}$. We give an alternative proof of that fact that also shows the $W^{1,1}$ norm of $\Phi$ can be bounded by $5\pi n+1$. Answering a question raised by Lazarev and Lieb, we show that if $p>1$ then there is no bound for the $W^{1,p}$ norm of any such multiplier in terms of the norms of $f_{1},...,f_{n}$.