The Frame Spectral Properties Of Absolutely Continuous Measures

In this article, we introduce the Fourier frame and Beurling density, prove some important properties of Fourier frame and show the complete characterization of absolutely continuous frame spectral measure by two different methods. This paper consists of four chapters. The first chapter is introduction and main content. In this part, we introduce the origin, development and application of Fourier frame as well as the main results of this paper. The second chapter is preliminaries. In this part, we give some fundamental definitions and properties about uniformly separated sets, relatively separated sets, Beurling density, Fourier frame and frame spectral measures. In the third chapter, we introduce some elementary properties about Fourier frame, especially its relationship with Beurling density and Bessel sequence, which is well prepared for the following proof of the main results. In the fourth chapter, we show the complete characterization of absolutely continuous frame spectral measure by two different methods.