  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 $f_{1},...,f_{n}$.

Functional Analysis