On a maximum principle in discretization with spherical harmonics

@misc{reusken2026maximum,
  title = {{On a maximum principle in discretization with spherical harmonics}},
  author = {Arnold Reusken},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13540},
  eprint = {2610.13540},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13540},
}

TY  - UNPB
TI  - On a maximum principle in discretization with spherical harmonics
AU  - Arnold Reusken
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13540
UR  - https://arxiv.org/abs/2610.13540
AB  - Spherical harmonics have been studied extensively and are used in many fields of application. We consider the probably earliest application of spherical harmonics, namely for a representation of the solution of the Laplace equation on the d-dimensional unit sphere where the boundary data are represented in a spherical harmonics basis. We consider general d ≥ 2. For the solution u of the Laplace equation the fundamental maximum principle holds. If one approximates the boundary data in the finite dimensional space spanned by all spherical harmonics of degree at most N, the resulting discrete solution, denoted by u_N, does in general not satisfy the maximum principle. In this paper we derive sharp estimates for r_N^max such that the maximum principle holds for u_N(x) for all x ∈ BbbR^d with |x| ≤ r_N^max. We show that for d=3 one has $r_N^{\rm max} \gtrsim 1- 3 N^{-1} \ln\big( 2^\frac13 N \big) $. Similar estimates are derived for d ≠ 3. We also study the sharpness of these estimates.
ER  -