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The Klein-Gordon equation, the Hilbert t...
Hedenmalm, Haakan...

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The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps by Hedenmalm, Haakan ( Author )
Lemonjar Open Access
N.A
13-03-2015
A pair (Γ,Λ), where Γ⊂R2 is a locally rectifiable curve and Λ⊂R2 is a {\em Heisenberg uniqueness pair} if an absolutely continuous (with respect to arc length) finite complex-valued Borel measure supported on Γ whose Fourier transform vanishes on Λ necessarily is the zero measure. Recently, it was shown by Hedenmalm and Montes that if Γ is the hyperbola x1x2=M2/(4π2), where M>0 is the mass, and Λ is the lattice-cross (αZ×{0})∪({0}×βZ), where α,β are positive reals, then (Γ,Λ) is a Heisenberg uniqueness pair if and only if αβM2≤4π2. 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∞(R+) if and only if 0<αβ<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,n≥0 spans a weak-star dense subspace of H∞+(R) if and only if 0<αβ≤1. To obtain this result, we need to develop new harmonic analysis tools for the dynamics of Gauss-type maps, related to the Hilbert transform. Some details are deferred to a separate publication.
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Article
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36.88 KB
English
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MYR 0.01
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https://arxiv.org/abs/1503.04038
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