Spectrality Of Self-affine Measures Associated With Three Kinds Of Digit Sets

In this thesis,we mainly discuss the spectrality and non-spectrality of the self-affine measures which generated by a integer expanding matrix and the three kinds of digit sets.First,by using a Criterion of Strichartz,the spectrality of self-affine measures is investigated,when they are spectral measures,and find out some of their spectra.Second,by virtue of the characteristic of the zero-point sets for the Fourier transformation on the self-affine measures,we discuss the non-spectrality,and point out the number of mutually orthogonal exponentials.Specific content is organized:Chapter 2 discusses the spectrality of the self-affine measures generated by a collinear digit set.Based on the characteristic of the zero-point sets for the Fourier transformation on the self-affine measures,we research the non-spectrality of the self-affine measures which is generated by a integer expanding matrix and a collinear digit set.Firstly,the non-spectrality of three-element collinear digit set is discussed.By solving the vanishing sums of three roots of unity,we obtain the zero-point for the Fourier transformation of the self-affine measures.Since the spectrality is invariant under the similarity transformation,the non-spectrality of the self-affine measures is derived.Secondly,we talk about the spectrality of the self-affine measures generated by a triangle expanding matrix and the three-element collinear digit set,when it is a spectral measure,and find out some of its spectra.Finally,the case of q-element collinear digit set is investigated.Using the geometric series summation formula,the zero-point set of the Fourier transformation of the self-affine measures is derived,then the spectrality is obtained.In chapter 3,we study the spectrality of the self-affine measures generated by the digit set which has the direct sum decomposition.In general,spectral self-affine measures are obtained in the light of compatible pair,and determined by Criterion of Strichartz.However,some authors have given some examples of spectral measures which cannot be derived by compatible pair,here more examples are proposed.Base on the structure of vanishing sums of roots of unity,if the number of the digit set is larger than 4,it is generally hard to determine the roots of unity.However,when the digit set has a direct sum decomposition,some spectrality of the self-affine measures are obtained.In chapter 4,the non-spectrality is discussed for the self-affine measures which generated by the digit set that consists of by a standard orthogonal basis on R3 and a zero.Specificly,we discuss the non-spectrality of the self-affine measures which gen-erated by a upper-triangle expanding matrix.On the one hand,the non-spectrality of self-affine measures is proved on generalized 3-dimensional Sierpinski gasket.On the other hand,we get the spectrality of the self-affine measures generated by a diagonal matrix with two elements of which are equal and odd.Finally,the main work of this thesis is summarized,and the future work is given.