Title: On the Fourier Analysis of Measures with Meyer Set Support Show Chinese title

Title: 关于具有Meyer集支撑的测度的傅里叶分析

Abstract: In this paper we show the existence of the generalized Eberlein decomposition for Fourier transformable measures with Meyer set support. We prove that each of the three components is also Fourier transformable and has Meyer set support. We obtain that each of the pure point, absolutely continuous and singular continuous components of the Fourier transform is a strong almost periodic measure, and hence is either trivial or has relatively dense support. We next prove that the Fourier transform of a measure with Meyer set support is norm almost periodic, and hence so is each of the pure point, absolutely continuous and singular continuous components. We complete the paper by discussing some applications to the diffraction of weighted Dirac combs with Meyer set support. Show Chinese abstract

Abstract: 在本文中，我们证明了具有Meyer集支撑的可傅里叶变换测度的广义Eberlein分解的存在性。 我们证明了三个组成部分中的每一个也是可傅里叶变换的，并且具有Meyer集支撑。 我们得出结论，傅里叶变换的纯点、绝对连续和奇异连续组成部分中的每一个都是强几乎周期测度，因此要么是平凡的，要么具有相对稠密的支撑。 接下来，我们证明具有Meyer集支撑的测度的傅里叶变换是范数几乎周期的，因此纯点、绝对连续和奇异连续组成部分中的每一个也是如此。 最后，我们通过讨论一些应用来完成本文，这些应用涉及具有Meyer集支撑的加权Dirac梳的衍射。