We propose a duality method that reduces the problem to suitable new observability inequalities, which consist of estimating the boundary traces of waves on part of the boundary from boundary measurements done on another subset of the boundary. These inequalities lead to novel questions that do not seem to be treatable by the classical techniques employed in the field, such as Carleman inequalities, non-harmonic Fourier series, microlocal analysis and multipliers.

We propose a genuinely 1-d solution method, based on sidewise energy propagation estimates yielding a complete sharp solution. The obtained observability results can be reinterpreted in terms of Fourier series. This leads to new non-standard questions in the context of non-hamonic Fourier series}, editor = {Zuazua Iriondo, Enrique}, faupublication = {yes}, peerreviewed = {unknown}, publisher = {Documents Mathématiques of the French Mathematical Society (SMF)}, series = {Volume in honor of Yves Meyer, Documents Mathématiques of the French Mathematical Society (SMF)}, title = {{Fourier} series and sidewise control of 1-d waves}, url = {https://arxiv.org/abs/2308.04906}, volume = {22}, year = {2024} }