| Abstract: |
| Any locally compact Abelian (LCA) group acts on a torus $\mathbb{T}^n$ of arbitrary dimension. We show that this action admits a decomposition into uniquely ergodic subsystems. As application, the famous Kronecker theorem on Diophantine approximations (density) is then strengthened into a Weyl-type theorem (equidistribution). This allows us to show that the characters of an LCA group are orthogonal in the sense of Bohr and other consequences follow.
Furthermore, we show that the algebra of quasi-periodic functions with spectra in a $\mathbb{Z}$-module of finite rank n is isometrically isomorphic to the algebra C($\mathbb{T}^n$) of continuous periodic functions. This is the theoretical foundation upon which the projection method in scientific computation is based. As a theoretical application of this algebra isomorphism, we can prove, in a very simple way, the Hausdorff-Young inequalities for almost periodic functions in the sense of Besicovitch. This last part is a joint work with Kai JIANG and Pingwen ZHANG. |
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