Sets of Uniqueness for Harmonic and Analytic Functions and Inverse Problems for Wave Equations

Several examples of one-dimensional analytic sets of uniqueness for harmonic functions on the sphere in $\mathbb{R}^3$ are given and some examples of analytic sets on the sphere in $\mathbb{R}^n$ which cannot contain sets of uniqueness are presented. Analytic curves which are sets of uniqueness for real-analytic functions in $\mathbb{R}^n$, $n \ge 3$, are constructed. The obtained results are used to justify the inhomogeneity sounding schemes when the inverse problem of acoustic scattering is solved under the conditions that the source and detector coordinates coincide.