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Characterization of Eigenfunctions by Boundedness Conditions

Published online by Cambridge University Press:  20 November 2018

Ralph Howard  and
Margaret Reese
Ralph Howard
Affiliation:
Department of Mathematics University of South Carolina Columbia, South Carolina 29208
Margaret Reese
Affiliation:
Department of Mathematics Sa in t Olaf Co liege Northfield, Minnesota 55057
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Abstract

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Suppose is a sequence of functions on ℝn with Δfk = fk+1 (where Δ is the Laplacian) that satisfies the growth condition: |fk(x)| ≤ Mk{1 + |x|)a where a ≥ 0 and the constants have sublinear growth Then Δf0 = —f0- This characterizes eigenfunctions f of Δ with polynomial growth in terms of the size of the powers Δkf, —∞ < k < ∞. It also generalizes results of Roe (where a = 0, Mk = M, and n = 1 ) and Strichartz (where a = 0, Mk = M for n). The analogue holds for formally self-adjoint constant coefficient linear partial differential operators on ℝn.

Keywords

42B1035B35EigenfunctionsLaplacianconstant coeffcient differential operators
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 35 , Issue 2 , 01 June 1992 , pp. 204 - 213
Copyright
Copyright © Canadian Mathematical Society 1992 

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