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

• Published in: The Journal of fourier analysis and applications Vol. 28; no. 2
• Main Authors: Butzer, Paul L., Stens, Rudolf L.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2022; Springer; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Banach spaces; Dirichlet problem; Ergodic processes; Fourier Analysis; Harmonic analysis; Inverse problems; Linear operators; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Neumann problem; Operators (mathematics); Partial Differential Equations; Semigroups; Signal,Image and Speech Processing; 47D03; 31B25; Primary 35J25; 47A35; 31B20; 42B37; Secondary 31B05
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-022-09929-3

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 R 2 and the unit ball in 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 π 2 in case of the unit disk, and X = L 2 ( S ) in the three dimensional setting, S being the unit sphere in 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).