The Spectrality Of Moran Measures

In this thesis, we consider when a Borel probability measure μ in Rn with com-pact support admits an exponential orthonormal basis E(Λ)={e2πi<λ,x>:λ∈Λ} for the Hilbert space L2(μ). In positive case,μ is called a spectral measure and A is called a spectrum of μ. Since the spectral set conjecture was raised by B. Fuglede in 1974, the spectral problem of a Borel measure has been one of the basic ones of Fourier analysis on measures. It plays a key role in Fourier analysis, wavelet, har-monic and non-harmonic analysis. In this thesis, we focus on the spectral property of Moran measures. We will give some new ideas and technics to construct spectrum by means of the theory from number theory and combinatorial mathematics. Those new methods are useful in the study of the spectrality of a Borel measure, the spectrum construction of a spectral measure and the convergence of the spectral expansion, which have been used in our recent work. The main work of this thesis is spread in the following four chapters:In Chapter three, we consider the Moran measure μ{bκ},{Dκ} with consecutive digits. That is, the digit Dκ={0,1, ..., qκ - 1} for some qκ ＞ 1. We show that compatible pair exists if and only if qκ|bκ, and in this case μ{bκ},{Dκ} is a spectral measure. The result has been published on J. Funct. Anal.In Chapter four, we focus on a general Moran measure which generated by a ratio b-1 and a sequence of digits{Dκ}κ=1∞. We assume that there exists an integral digit C such that (b-1Dκ,C) forms a compatible pair for each κ. Then prove it being a spectral measure by studying the b—adic expansion of elements in Z(μb,{Dκ}) and the zero set ofμb,{Dκ}. Several examples are given to explain that the assumptions are essential. We have received the recommendation from Adv. Math, who accept it after a little revision.In Chapter five, we discuss an important measure in fractal and probability-infinite Bernoulli convolution μρ-1,{Dκ}, where each digit set Dκ has two elements. We give a sufficient and necessary condition for that the infinite Bernoulli convolution has an infinite orthogonal set of exponentials. Then we classify the construction of all maximal orthogonal sets and give a sufficient condition for a maximal orthogonal set being a spectrum or non-spectrum. By several examples, we explain that the sufficient condition is essential, and the condition "p-1 € 2N" cannot guarantee its spectral property, which has big difference with Bernoulli measureμp-1,{0,1}·The result is going to be published on J. Funct. Anal. (it is been on the internet.)In Chapter six, we study the spectral construction of a Sierpinski measure in the plane, which is a classic self-similar measure. Firstly, we completely classify the construction of all maximal orthogonal sets. Then we settle the problem of spectral eigen-matrices for the Sierpinski measure. It worth to point out that the problem of spectral eigenvalues for any spectral self-similar measure in one dimension is still open, even for the simplest one.