The Fourier Transform And Fourier Coefficient Of Some Self-similar Measures

Fractal geometry is a popular research subject.In recent yaer,there are a lot of results in the crossing field of Fourier analysis and fractal geometry.In 2019,for some 1-d IFS,Slomyak[65]proved that,while all contractive ratios belonging to a set(0,1)m minus a set of zero Housdorff dimension,the decay rate of the Fourier transform of self-similar measure ? depends on |t|-?.In this thesis,we studied that the decay rate of the Fourier transform and the Fourier coefficient of some self-similar measures.It shows that the study of the decay rate of the Fourier transform of some self-similar measures can be transformed into the study of the decay rate of the Fourier coefficient of it.T.Y.Hu[72]generalized the Erdos Salem theorem.Theorem 1.9 can be used to further generalize Erdos Salem theorem and to prove the necessity of generalized Erdos Salem theorem.We also find some singular measures ? which were not Rajchman measures.And the Positive real part analytic function which associated with ? are of class Hp(0<p<1),but are not of class H1.This thesis consists of five chapters,arranged as follows:In Chapter 1,we give a brief summary of the research background related to the Fourier transform and coefficient of some self-similar measures,and state some main results of this thesis.And we introduce some related definitions,such as iterated function system,self-similar measure,the Fourier transform and coefficient of self-similar measures etc.In Chapter 2,we introduced the prepared knowledge of my research.In Chapter 3,we proved,under certain circumstance,the decay rate of the Fourier transform and coefficient of infinitely convolved Bernoulli measures ? are equal.We also proved that ? is not Rajchman measure.Moreover,we obtain that some Positive real part analytic function which associated with ? are of class Hp(0<p<1),but are not of class H1.In Chapter 4,we obtain the decay rate of the Fourier transform and Fourier coefficient of some self-similar measures ? are equal.Meanwhile ? is not Rajchman measure.Moreover,we obtain that some Positive real part analytic function which associated with ? are of class Hp(0<p<1),but are not of class H1.In Chapter 5,we study the problem of the decay rate of the Fourier transform of some self-similar measures which generated by the IFS,that associated with Pisot number.