✍ Scribed by Liangpan Li; Lan Tang
No coin nor oath required. For personal study only.
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),
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When X = 𝕊^n^ (the unit sphere in ℝ^n +1^) and μ is the standard area measure of 𝕊^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^ (𝕊^n^ , μ),
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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)
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