On Some Related Problems Of Fractals And Frames

The related problems of fractals and frames are both important and interesting in fractal geometry. In this thesis, we study the fractal series, obtain some characterizations of Fourier frames and tilings, and construct a class of generalized continuous frames by using iterated function systems.This thesis contains three parts.In the first part of this thesis, we describe a natural way to associate fractals to a certain class of absolutely convergent series in Rd, and portray the structures of the sets. In the case for d= 1, we give two sufficient conditions for the set associated to series being an interval or a homogeneous Moran set, respectively. Finally, for d≥1, we give a sufficient condition to calculate the Hausdorff dimension of some fractals with overlapping, and present some examples.In the second part of this thesis, we give the definition of a generalized continuous frame for a separable Hilbert space. For such a generalization, after obtaining some basic results about these frames, we construct a class of generalized continuous frames by using iterated function systems, probability functions and window functions.In the last part of this thesis, we introduce the spectral set conjecture. In their investi-gation and study of the conjecture, Jorgensen and Pedersen give a conjecture. Motivated by the conjecture, Lagarias, Reeds and Wang established a characterization of spectra and tilings that can be used to prove the conjecture of Jorgensen and Pedersen by Keller's criterion. Different techniques to prove these facts have also been developed by Iosevich, Pedersen and Kolountzakis. In this part, we present an elementary approach to give some characterizations of Fourier frames and tilings, and obtain a more general form of spectra and tilings.