# Completeness of the set $\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}$

## TL;DR

This paper proves that the set of exponential functions with directions on the unit sphere is complete in the space of square-integrable functions on a smooth surface, under certain spectral conditions.

## Contribution

It establishes the completeness of a specific exponential set in L^2 space on a surface, under the condition that k^2 is not a Dirichlet eigenvalue.

## Key findings

- The set $\
- is total in L^2(S).
- Completeness holds for fixed k > 0 when $k^2$ is not a Dirichlet eigenvalue of the Laplacian in D.

## Abstract

It is proved that the set $\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, $k>0$ is a fixed constant, $k^2$ is not a Dirichlet eigenvalue of the Laplacian in $D$, $s\in S$, is total in $L^2(S)$. Here $S$ is a smooth, closed, connected surface in $\mathbb{R}^3$.