Almost-everywhere convergence of expansions in eigenfunctions of the Laplace operator in the sphere
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.
scientific article; zbMATH DE number 3944374
scientific article