Let \(I\) be a spherical line segment and \(\kappa\) a parametrization of \(I\). A spherical polynomial \(f\) restricted on a spherical line segment \(\gamma\) is a exponential polynomial of the type

\[ f\circ \kappa(t)=\sum_{k=1}^{n} \beta_k \e^{i\lambda_k t} \]

for some \(n\in \mathbb{N}\), \(\beta_1,\ldots ,\beta_n\in \mathbb{C}\) and \(\lambda_1,\ldots ,\lambda_n\in \mathbb{R}\). [1, Lemma 2.2]

References Link to heading

1. A. Dicke and I. Veselic, Spherical Logvinenko-Sereda-Kovrijkine type inequality and null-controllability of the heat equation on the sphere, 2024. doi:10.48550/arXiv.2207.01369