Hokkaido Mathematical Journal

KOMATSU Hikosaburo,

The abstract Fatou theorem and the signal transmission on Thomson cables.

Hokkaido Mathematical Journal, 39 (2010) pp.157-171

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Abstract

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.

MSC(Primary)
MSC(Secondary)35C15(MSC2000), 44A45(MSC2000)
Uncontrolled KeywordsFatou theorem; Thomson cable; uniqueness;