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The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps | | Haakan Hedenmalm
; Alfonso Montes-Rodriguez
; | | Date: |
13 Mar 2015 | | Abstract: | A pair $(Gamma,Lambda)$, where $GammasubsetR^2$ is a locally rectifiable
curve and $LambdasubsetR^2$ is a {em Heisenberg uniqueness pair} if an
absolutely continuous (with respect to arc length) finite complex-valued Borel
measure supported on $Gamma$ whose Fourier transform vanishes on $Lambda$
necessarily is the zero measure. Recently, it was shown by Hedenmalm and Montes
that if $Gamma$ is the hyperbola $x_1x_2=M^2/(4pi^2)$, where $M>0$ is the
mass, and $Lambda$ is the lattice-cross $(alpha imes{0}) cup
({0} imeseta)$, where $alpha,eta$ are positive reals, then
$(Gamma,Lambda)$ is a Heisenberg uniqueness pair if and only if $alphaeta
M^2le4pi^2$. The Fourier transform of a measure supported on a hyperbola
solves the one-dimensional Klein-Gordon equation, so the theorem supplies very
thin uniqueness sets for a class of solutions to this equation. The case of the
semi-axis $R_+$ as well as the holomorphic counterpart remained open. In this
work, we completely solve these two problems. As for the semi-axis, we show
that the restriction to $R_+$ of the above exponential system spans a
weak-star dense subspace of $L^infty(R_+)$ if and only if $0<alphaeta<4$,
based on dynamics of Gauss-type maps. This has an interpretation in terms of
dynamical unique continuation. As for the holomorphic counterpart, we show that
the above exponential system with $m,nge0$ spans a weak-star dense subspace of
$H^infty_+(R)$ if and only if $0<alphaetale1$. To obtain this result, we
need to develop new tools for the dynamics of Gauss-type maps, related to the
Hilbert transform. We study series of powers of transfer operators in a
completely new setting, combining ideas from Ergodic Theory with ideas from
Harmonic Analysis. | | Source: | arXiv, 1503.4038 | | Services: | Forum | Review | PDF | Favorites | |
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