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

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 \(f_{1},...,f_{n}\).