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The paper is mainly concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the unit disk in    $${\mathbb {R}}^2$$  R 2 and the unit ball in    $${\mathbb {R}}^3$$  R 3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. The basis is the theory of semigroups of linear operators mapping a Banach space  X  into itself. The classical one-parameter theory for semigroups applies in the present particular applications, actually for    $$X= L^2_{2\pi }$$  X = L 2 π 2 in case of the unit disk, and    $$X=L^2(S)$$  X = L 2 ( S ) in the three dimensional setting,  S  being the unit sphere in    $${\mathbb {R}}^3$$  R 3 . Another tool is a Drazin-like inverse operator  B  for the infinitesimal generator  A  of a semigroup that arises naturally in ergodic theory. This operator  B  is a closed, not necessarily bounded, operator. It was introduced in a paper with Butzer and Westphal (Indiana Univ Math J 20:1163–1174, 1970/1971) and extended to a generalized setting with Butzer and Koliha (J Oper Theory 62:297–326, 2009).

Exterior Dirichlet and Neumann Problems and the Linked Ergodic Inverse Problems in the Entire Space