Spectral Property Of Infinite Convolutions

Let ?, be a Borel probability measure on the Euclidean space Rn. We call the measure ? a spectral measure, if there exists a discrete set A such that the exponential functions system {e27ri<?,x>:?? A} forms an orthonormal basis for the Hilbert space L2(?). In this case, we call the set A a spectrum of the measure?, and (?, A) a spectral pair.In this thesis, we mainly consider when infinite convolutions ? admits an exponential orthonormal basis in the Hilbert space L2(?), the structure of spectra for the Bernoulli con-volutions and the convergence problem of the Fourier series with respect to the spectra. We will introduce the background on the study of measure spectral theory in Chapter One, and we provide necessary results for our later chapters in Chapter Two. The main part of the thesis is from Chapter three to Chapter six, we arrange it as follows:In Chapter three, we consider the spectral property of a class of random convolutions. Based on the idea of constructing spectrum in the literature [1] and [2], we provide two sufficient conditions to guarantee the random convolutions to be spectral ones, which help us construct more spectra with concrete form. In addition, these conditions are suitable to self-affine measures.In Chapter four, we consider the spectral structure of Bernoulli convolutions, we also call it spectral eigenvalue problem. We show that any odd number p will determine a discrete set A such that the sets A, pA form spectra for Bernoulli convolutions. It is worthy noting that for any odd integer p, the corresponding discrete sets is at least countable. Furthermore, we talk about the convergence problem with respect to the spectra we constructed.In Chapter five, we consider the spectral property of the random convolutions ?p,{Dn} generated by a fixed contraction rate p and the digit sets{Dn}?n=1 with three-elements. If the digit sets are all bounded from upper and lower, we give a complete characterization to the spectral property of the measure ??i{Dn}.In Chapter six, we consider the spectral property of the lacunary infinite convolutions. More precisely, we fix a contraction constant and a digit set, but the contradiction rate is determined by a sequence of lacunary sequence. In this chapter, we provide some sufficient conditions to ensure the measure be spectral one and formulate a conjecture on it.In Chapter seven, we list some topics of further research.