Spectra Of Riesz Product Measures And Beurling Dimension

Let ? be a Borel probability measure with compact support in Rd.One of the foundationary problems in those issues is that whether there exist a countable subset A(?)Rd such that the family of complex exponential functions E?:= {e2?i<?,x>:? ??} is a Fourier frame/Riesz basis/orthogonal basis for L2(?).If E? is a frame/Riesz basis/orthogonal basis for L2(?),then ? is called aa frame/Riesz basis/orthogonal spectral measure and A is called the Fourier frame/Riesz basis/orthogonal spectrum of ?.The frame theory has become a pillar of applied harmonic analysis,including Gabor analysis,wavelets,compressive sensing,interpolation and sampling theory,signal processing,and has been developed rapidly in recent years in both theory and applications.It has a long history to study a normolized Lebesgue measure,restricted on a set,to be a frame spectral measure.This could be traced back at least to 1967.The spectral measure problems attract more attention due to the famous Fuglede(spectral set)conjecture.From the beginning of this topic,the Beurling density plays a key role.Let A ? Md(Z)be an expanding matrix with integer entries,D=(d0 =0,d1,...,dq-1}(?)Zd a finite digit set and {Pk={pk,0,Pk,1,...,pk,q-1}}k=1 ? a family of probability vectors.We call the Borel probability measure? A,D,{Pk} = ?A-1 D,P1*? A-2D,P2*? A-3D,P3...,Riesz product measure generated by A,D and {Pk},where the convergence is in the sense of weak convergence.Clearly,the Riesz measure contains all self-affine measure and self-similar measure.In this paper,we mainly consider the spectrality of such measure and the Beurling dimension of the spectrum of a self-similar measure.The thesis consists of five chapters.In the first chapter,we introduce the back-ground and the actuality of this paper.As the existence of the Riesz measure is related to the weak convergence of measures,some basic knowledge about conver-gence of measures are included in the second chapter.In the chapter 3,we present some concepts and theorems which will be used in the next chapters.Our main results are list in the next two chapters.In the chapter 4,we study the relationship of spectrality between the Riesz product measure ? A,D,{Pk} and the self-affine measure ? A,D Under the assumptions inf k?1 Pk>0,the Riesz product measure ?=?A,D,(Pk)is a frame/Riesz spectral measure if and only if ? A,D is a frame/Riesz spectral measure and the sequence {Pk}satisfies the equivalent product condition.In particular,the condition inf Pk->0 could be dropped off in some special case.In the chapter 5,we discuss the Beurling dimension of Bessel spectra(A set A is called Bessel spectrum for measure ? if E? is a Bessel sequence for L2(?))and frame spectra of some self-similar measures on Rd.We obtain their exact upper bound of the dimensions.As for some special frame spectra A,we state a necessary and sufficient condition for which the Hausdorff dimension of the self-similar set is equal to the Beurling dimension of A.