The Spectral Property Of Moran Measure On R~1

Let ? be a compactly supported Borel probability measure on Rd. One funda-mental problem in Fourier analysis is to find a sequence A C JRd such that the family of complex exponential functions E(?):={e2?<?,x>}??? forms an orthogonal basis(Fourier basis) for L2(?), the space of all square-integrable functions with respect to the measure ?. In this case, the measure ? is called a spectral measure and ?is called a spectrum for ?. We also say that (?,A) is a spectral pair. Research on the spectral measures began with the conjecture of Fuglede in 1970s, and developed rapidly in the turn of the century. And now the spectral theory has been a hot topic in Fourier analysis.The spectral measures are special in measures, however little is known about it,especially the singular spectral measures. The main goals of our thesis are to find or construct more spectral measures and determine the spectral eigenvalues of known spectral measures. For this aim, we focus on the Moran measures. We propose some creative ideas and new techniques when we choose a suitable orthogonal set to be a spectral,and get some new recognition of the spectral construction. Surprisedly,we find that there exists inherent relationship between Weyl's criterion and spectral measures. These ideas and techniques constructed our paper. The thesis is divided into six chapters. In the first chapter, we introduce the background and the actuality of the spectral theory. And the chapter 2 presents some concepts and theorems in spectral theory. The next following four chapters are the main content of the thesis.In the chapter 3, we study the spectrality of the Cantor-Moran measures ?{pn},{dn}with 0 < dn < Pn. We show that if adds the condition of 2|pn/gcd(dn,pn)for all n?1,then ?{pn),{dn} is a spectral measure. It should be mentioned that there is no as-sumption sup{dn} < oo in our study. And we give some examples to explain that those hypotheses are essential. The final result has been published on J. Funct.Anal.In the chapter 4, we characterize the spectrality of the Cantor-Moran measures?pn,{an,bn}with max{an,bn} < pn for n large enough. Then we give the sufficient condition for spectral and Non-spectral measures if we get rid of the condition of max{an, bn} < pn. This special discussion is still lacking to the present, and it more difficult for find out the Non-spectral measure.In the chapter 5, we discuss the spectral eigenvalue problem of a class of random convolution on R. A real number p is called a spectral eigenvalue of ? if there exists a discrete set A such that both A and pA are spectra for ?. Now, we obtained a necessary and sufficient condition for a real number p to be a spectral eigenvalue of some classic spectral measure ? . This article has been submitted.In the chapter 6, we analyze the relationship between spectral measures and Weyl's criterion of ergodic theory. We prove that all the discrete spectral measures,Bernoulli convolution ?2k and the Lebesgue measure restrict on a union of two intervals satisfy the generalized Weyl's criterion. Based on the knowledge of the property of spectral measure, we propose a conjecture: Borel probability measure space (?, T) satisfies the generalized Weyl's criterion if and only if ? is a spectral measure.