Uniqueness theorem for Fourier transformable measures on LCA groups
We show that if points of supports of two discrete ”not very thick” Fourier transformable measures on locally compact abelian (LCA) groups tend to one another at infinity and the same is true for the masses at these points, then these measures coincide. The result is valid for discrete almost periodic measures on LCA groups too. Also, we show that the result is false for some discrete ”thick” measures. To do this, we construct a discrete almost periodic measure on the real axis, whose masses at the points of support tend to zero as these points approach infinity.
Kurasov P., Suhr R., Asymptotically isospectral quantum graphs and generalised trigonometric polynomials, J. Math. Anal. Appl., 488 (2020), 1–15.
Gil de Lamadril J., Argabright L., Almost periodic measures, Mem. Amer. Math. Soc., 428 (1990), 221p.
Levitan B.M., Zhikov V.V., Almost periodic functions and differensial equations, Cambridge University Press, Cambridge, 1982.
Moody R.V., Strungaru N., Almost Periodic Measures and their Fourier Transforms. In: Aperiodic Order, V.2, Crystallography and Almost Periodicity, eds. M. Baake and U. Grimm, Cambridge University Press, Cambridge, 2017, p. 173–270.
Strungaru N., Terauds V. Diffraction theory and almost periodic distributions, J. Stat. Phys., 164, No5, 1183–1216, 2016. arXiv:1603.04796.
Strungaru N., On the Fourier analysis of measures with Meyer set support, arXiv:1807.03815v2 (2019).
Lenz D., Spindler T., Strungaru N., Pure point spectrum for dynamical sistems and mean almost periodicity, arXiv:2006.10825v1, (2020).
Baake M., Strungaru N., Terauds V., Pure point measure with sparse support and sparse Fourier-Bohr support, arXiv:1908.00579v2, (2020).
Copyright (c) 2020 S.Yu. Favorov
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.