Given \(p\in [1,\infty]\), a spherical polynomial \(f\) of degree at most \(\mathcal{N}\in \mathbb{N}\) and a \(\gamma\)-thick set \(S\subseteq \mathbb{S}^{d-1}\). Then

\[ \lVert f\rVert_{L^p(\mathbb{S}^{d-1})}\le \Bigl(\frac{c}{\gamma}\Bigr)^{2\mathcal{N}+1/p} \lVert f\rVert_{L^p(S)}, \]

where \(c>0\) is a constant depending on \(d\). [1, Theorem 1.3]

Proof
The proof is similar to the one for Kovrijkine’s uncertainty principle . It starts with the observation that spherical polynomials restricted on line segments are exponential polynomials . Then we may apply (0x669905bf) and there is no need to calculate the maximum as in Kovrijkine’s proof. However, in this case a dependence on the radius of the sensors of the thick set is lost.
Remarks
  • The inequality does not depend on the radius of the sensors of the thick set. Therefore, it doesn’t matter how the subset \(S\) is distributed. Either it is concentrated in the poles or equally distributed, the inequality is the same.

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