A smooth, complex generalization of the Hobby-Rice theorem

Abstract

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 $f_{j}h$ are all simultaneously zero. This multiplier takes values~$\pm 1$ 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 $f_{j}h$ are all zero. We also show the existence of $n$ infinitely differentiable, real functions $g_j$ such that the $n$ functions $f_j{e}^{i{g}_j}$ are pairwise orthogonal.

8 pages, latex. Version V3 has an additional corollary 1.3 plus improved abstract, corrected typos and added references

References in corpus (1)

• Current Densities in Density Functional Theory

Cited by in corpus (5)

• Statistical Mechanics of the Uniform Electron Gas
• Current Densities in Density Functional Theory
• Universal Functionals in Density Functional Theory
• Building Kohn-Sham potentials for ground and excited states
• Pure-state $N$-representability in current-spin-density-functional theory