The Existence Of Fourier Basis For Some Moran Measures

It is well known that the family of complex exponential functions {e2?i?x}??Z is an orthonormal basis for L2([0,1]).It is natural to ask the following question:given a Borel probability measure ? with compact support,does L2(?)admit a Hilert basis consisting of exponential functions E(A):= {e2?i?x:? ? A}?If it does exists,we say that ? is a spectral measure,A is called a spectrum of ? and(?,?)is called a spectral pair.Researches on the spectral measures started with the conjecture of Fuglede(1970s).After a rapid development during about a half century,the spectral theory has been a hot topic in Fourier analysis.There are two goals of our thesis.The first goal is to find or construct more spectral measures,especially Moran measures.The second goal of the thesis is to study the uniqueness of the spectrum which is not obtained by translations of each other or construct all spectra for a given spectral measure.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 main content of this thesis is contained in the following three chapters:In the Chapter three,we focus on the Moran measure ?b,{Dk} = ?b-1D1*?b-2D2*…with#Dk = 4.We prove the following result:"Let {Dk}k=1? be a uniformly bounded sequence of 4-digit spectral sets,b = 2l+1 q with q>1 an odd integer.Then the Moran measure ?b,{Dk} is a spectral measure if l sufficiently large(depends on Dk)."The result has been published on J.Math.Anal.Appl.In the chapter four,we consider the more general Moran measure ?{bn},{Dn}=?b1-1D1*?(b1b2)-1D2*…*?(b1b2…bn)-1Dn*…,where Dn = {0,rn,2rn,…,(qn-1)rn}.The result is as follows:Let {bn}n=1? be a sequence of integers bigger than 1,and let{Dn}n=1? be a sequence of digit sets with Dn=?{0,rn,2rn,…,(qn-1)rn} in Z.Then the associated Moran measure?{bn},{Dn}=?b1-1D1*?(b1b2)-1D2*…*?(b1b2…bn)-1Dn*…is a spectral measure if rnqn|bn.The result was also published on J.Math.Anal.Appl.In the chapter five,We study equally-weighted Moran measures ?{bn},{0,1,2,…,q-1}arising from Moran iterated function system of the form {fn,i(x)= bn-1(x + i):i ? {0,1,2,…,q-1}},where q<bn.Firstly,we classify the(bn,q)and study the existence of infinitely many mutually orthogonal exponentials in L2(?{bn},{0,1,2,…,q-1}).When gcd(bk,q)= q,we characterize all the maximal orthogonal sets ? for L2(?{bn},D)via a maximal mapping on the q-adic.Secondly,we give two sufficient conditions for some maximal mapping to generate a spectrum(an orthonormal basis)or not.These results not only provide us an easy and efficient way to construct various of new spec-tra for these Moran measures but also are contributed to the convergence of Fourier series.