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A smooth, complex generalization of the Hobby-Rice theorem

The Hobby-Rice Theorem states that, given $n$ functions $f_j$ on ${\mathbb{R}}^N$, there exists a multiplier $h$ such that the integrals of $f_jh$ are all simultaneously zero. This multiplier takes values~$\pm1$ and is discontinuous. We show how to find a multiplier $h=e^{ig}$ that is infinitely differentiable, takes values on the unit circle, and is such that the integrals of $f_jh$ are all zero. We also show the existence of $n$ infinitely differentiable, real functions $g_j$ such that the $n$ functions $f_j e^{ig_j}$ are pairwise orthogonal.