Fractal Spectral Measures And The Analytic Arcs Of Inner Functions

This thesis consists of two parts.In the first part,we consider the fractal spectral measures.In the second part,we study the analytic arcs of inner functions.Let ? be a compactly supported Borel probability measure on Rn.If the Hilbert space L2(?)admits an exponential orthonormal basis E(A)={e2?i<?,x>:??A},we call ? a spectral measure,and A a spectrum of ?.For Lebesgue measure,the existence of basis has been well studied.For fractal measure,this problem becomes very complicated.Spectral measure is originated from spectral set conjecture which was raised by B.Fuglede in 1974 and then be widely concerned.In 1998,Jorgensen and Pederson discovered the first singular non atomic fractal spectral measure defined on Cantor set.Nowadays,the spectral measure problem has become one of the central problems of Fourier analysis on fractals.In the first Chapter,we introduce the research background,research status and our main results;in the second Chapter,we give the basic knowledge and necessary tools.The core content of this paper is from Chapter 3 to Chapter 5,which are briefly introduced as follows.In Chapter 3,we study the eigenvalue problem of the spectrum of self similar spectral measure ?pb,D with consecutive digits.Fix the typical spectrum of ?pb,D as A(pb,pD).We consider when the integers ? makes TA(pb,pD)also a spectrum.Combining with number theory methods,we obtain some sufficient or necessary conditions.In particular,we give an example to disprove the conjecture that all large primes are spectral eigenvalues.In Chapter 4,we study the spectrality of Moran measure which is more general than self similar measure.We study the spectrality of certain Moran measure generated by expanding integer diagonal matrices Rk and some consecutive-type digits Dk ? Zn.We show that the sequence of Hadamard triples {(Rk,Dk,Sk)} admits a spectrum of the associated Moran measure under specific situations.In Chapter 5,we study the analytic arcs of inner functions.By studying the argument at the singularity of infinite Blaschke products on the circle,we obtain some equivalent conditions for the argument to be finite.As an application,we classify the analytic arcs of inner functions into four types.