On a new form of the ergodic theorem for the unit sphere with application to spectral theory

Let 1 ≤ p < ∞ and let T be an ergodic measure‐preserving transformation of the finite measure space ( X , μ ). The classical L p ergodic theorem of von Neumann asserts that for any f ϵ L p ( X , μ ),When X = n (the unit sphere in ℝ n +1 ) and μ is the standard area measure of n , we establish a new form of the ergodic theorem. That is, we can replace the sequence of finite subsets of O ( n + 1) (the orthogonal transformation group of ℝ n +1 ) with a sequence of some fixed finite subsets of O ( n + 1) which are independent of any measure‐preserving transformation, such that for any f ϵ L p ( n , μ ),As an application, we can completely determine the point spectrum of the Laplace operator in L p (ℝ n ) (1 ≤ p < ∞, n ≥ 1) spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)