We aim at presenting a new estimate on the cost of observability in small times of the
one-dimensional heat equation, which also provides a new proof of observability for
the one-dimensional heat equation. Our proof combines several tools. First, it uses a
Carleman-type estimate borrowed from our previous work (SIAM J. Control Optim. 56:3
(2018), 1692–1715), in which the weight function is derived from the heat kernel and which
is therefore particularly easy. We also use explicit computations in the Fourier domain to
compute the high-frequency part of the solution in terms of the observations. Finally, we
use the Phragmén–Lindelöf principle to estimate the low-frequency part of the solution.
This last step is done carefully with precise estimations coming from conformal mappings.
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