Self-affine Measure Index Orthogonal Spectra

The concept of spectral measure is first introduced by P.E.T. Jorgensen and S. Pedersen in 1998, and it is a natural generalization of spectral set. Thus the research of spectral measure theory in recent years has also become an issue. Laba, Jorgensen, Hutchinson, Pedersen, Li and others have deeply studied the spectral measure, which provides the basis for establishing Fourier analysis theory under the self-affine measure theory. However, due to involving wide range of knowledge, and covering lots of contents, many questions remain unresolved. The main conclusions are as follows:(1) When self-affine measure is spectral measure, using the Fourier transform of self-affine measure and Parseval identity, there is another way to prove the sufficient condition of self-affine measure for the formation of spectrum in the case of high-dimensional.(2) When self-affine measure is non-spectrality measure, first the matrix Nα∈N3(Z) can be viewed as an operator on R3 according to the residue class modul-3 and the periodicity of Na when it acts on the sets Z0 and Z0. For the matrix and the digit set it is to estimate the number of orthogonal exponentials in L2 (μN,D), and to generalize the conclusion.Secondly by using the properties of the matrix operator N*j which acts on the sets Z0 and Z0, For the matrix and the digit set it is to estimate the number of orthogonal exponentials in L2(μN.D), and to generalize the result.