The Existence And Spectrality Of Moran Measures

Let ? be a Borel probability measure on Rn with compact support,and let L2(?)be the Hilbert space of all square-integrable functions with respect to ?.? is called a spectral measure if there exists a countable set(?)Rn such that the family of complex exponential functions EA:={e2?i(?,x):? ??}forms an orthonormal basis for L2(?),where(·,·)is the standard inner product of Rn.The study of spectral measure is a generalization of classical Fourier analysis on general measure,which is of great significance to fractal geometry,wavelet analysis and harmonic analysis.Jorgensen and Pederson discovered thefirst singular non-atomic spectral measure(1/4-Cantor measure ?1/4,{0,2}),which has aroused a great interest in the study of spectral properties of fractal mea-sures.Strichartz first studied the spectral properties of Moran measures and introduced the spectral measure problem to the general infinite Bernoulli convo-lution measure,which formed a new mathematical research hotspot.This thesis consists of five chapters,mainly studying the existence and spectral properties of Moran measures.In Chapter 1,firstly,we briefly introduce the development history and present situation of fractal geometry,and then summarize the background of spectral measures and spectral properties of fractal measures,and finally list the main results of this thesis.In Chapter 2,we first introduce some basic knowledge of infinite Bernoulli convolution and weak convergence,and then introduce some fundamental the-orems for judging spectral properties and some theorems about expansion of real numbers.In Chapter 3,using Levy theorem and Portmanteau theorem,we give some equivalent conditions for a general Moran measure to have a compact support and answer the guess of An et al.in J.Funct.Anal.[85].By proving the weak convergence of Fourier transforms of convolutions,we give a sufficient and necessary condition for the existence of Moran measures under the constraint of infk?1inf{p:p?Pk}>0.A sufficient condition for the existence of Moran measure is obtained if we remove the constraint.In addition,we give some examples to illustrate our conclusions.In Chapter 4,we mainly study the sufficient condition for spectrality of Moran measures ?{b?}{D?}{D?} with compression ratio of the reciprocal of an integer and arithmetic integers digit sets.We generalize the conclusions of[69],[70].We first prove a necessary and sufficient condition for compatible pair,then construct the orthogonal set of ?{b?}{D?} by using compatible pairs,and finally prove the completeness of the orthogonal set using the criterion for spectrum.We give some examples to show that our conclusions.In Chapter 5,we mainly study the necessary condition for spectrality of Moran measures ??,{D?} with compression ratio of real number and finite arith-metic integers digit sets.We generalize the conclusions of[75].By analyzing the zero properties of Fourier transform ??,{D?}and reordering the orthogonal set,we obtain that the compression ratio ? must be the reciprocal of an integer.We also give a sufficient condition for ??{Dk} to be a spectral measure.