Almost-everywhere convergence of expansions in eigenfunctions of the Laplace operator in the sphere
type: Item
instance of: scholarly article
title: Almost-everywhere convergence of expansions in eigenfunctions of the Laplace operator in the sphere
zbMATH Open document ID: 0588.35025
DOI: 10.1007/BF01157464
published in: Mathematical Notes
publication date: 1983-01-01T00:00:00Z
review text: Let \(\Delta\) be the Laplace operator on the unit sphere \(S^ N\), \(N=\dim S^ N\geq 2\). Let \(\{E_{\lambda}\}\) be the spectral resolution for - \(\Delta\), \[ E_{\lambda}f(x)=\sum_{\lambda_ j<\lambda}(f,u_ j)u_ j(x),\quad f\in L^ 2(S^ N), \] where \(u_ j\) is the eigenfunction of (-\(\Delta)\) on \(S^ N\) which corresponds to the eigenvalue \(\lambda_ j\), and (, ) is the inner product in \(L^ 2(S^ N)\). The following statement is proved. Theorem: Let \(f\in L^ 2(S^ N)\) and let \(f\equiv 0\) on some domain \(\Omega \subset S^ N\); then \(\lim_{\lambda \to \infty} E_{\lambda}f(x)=0\) a.e. in \(\Omega\). The proof is based on some auxiliary result which is closely related to a Tauberian type theorem due to L. Hörmander. Also the explicit formula for the eigenvalues \(\lambda_ j\) is used in the proof of the above stated result.
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35P10
zbMATH DE Number: 3944374
zbMATH Keywords: Laplace operator
zbMATH Keywords: spectral resolution
zbMATH Keywords: Tauberian type theorem
author: A. Y. Bastis
MaRDI profile type: Publication
cites work: Q3897569
full work available at URL: https://doi.org/10.1007/bf01157464
OpenAlex ID: W2033318608
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description: scientific article; zbMATH DE number 3944374