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May 2010 The abstract Fatou theorem and the signal transmission on Thomson cables
Hikosaburo KOMATSU
Hokkaido Math. J. 39(2): 157-171 (May 2010). DOI: 10.14492/hokmj/1277385659


The Fatou theorem on the Poisson representation of bounded harmonic functions on a half space is generalized to the bounded solutions $u(t)$ of the second order equation $$ u''(t) = A u(t), 0 < t < \infty, $$ in a dual Banach space $X = X_*{'}$, when $A$ is the dual of a non-negative operator $A_*$ with dense domain in $X_*$. Any bounded weak* solution is represented as $u(t) =$ $\exp(-t\sqrt{A})f$ with the weak* initial value $f$. Its prototype is in A.~V. Balakrishnan's paper in 1960 on fractional powers of non-negative operators. This is applied to prove the uniqueness of solutions in the theory of signal transmission on submarine cables by W. Thomson in 1855.


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Hikosaburo KOMATSU. "The abstract Fatou theorem and the signal transmission on Thomson cables." Hokkaido Math. J. 39 (2) 157 - 171, May 2010.


Published: May 2010
First available in Project Euclid: 24 June 2010

zbMATH: 1198.35053
MathSciNet: MR2665159
Digital Object Identifier: 10.14492/hokmj/1277385659

Primary: 35C15
Secondary: 44A45

Rights: Copyright © 2010 Hokkaido University, Department of Mathematics


Vol.39 • No. 2 • May 2010
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