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}$)$,thereexistsasmoothfunction$\Phi$ 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 $\mathrm{\Phi }$ 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 $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n$.