A smooth, complex generalization of the Hobby-Rice theorem

TL;DR

This paper extends the Hobby-Rice theorem by constructing smooth, complex-valued multipliers that achieve simultaneous zero integrals and orthogonality for given functions, improving regularity and applicability.

Contribution

It introduces a method to find infinitely differentiable, complex multipliers and functions that generalize the Hobby-Rice theorem with enhanced smoothness properties.

Findings

Existence of smooth multipliers with zero integrals for given functions.

Construction of smooth functions making given functions orthogonal.

Extension of the Hobby-Rice theorem to smooth, complex-valued functions.

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 $f_jh$ 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.