The Singularity Of Self-affine Measures On Special Digit Set And Its Characterizations

In this paper,the lower bound for Fourier transform of self-affine measures related digit set with four element in R2,and as well as the digit set with eight element in R3are estimated for the first time, the singularity of the related self-affine measure are discussed.By classification discussion of the weight,and Riemann-Lebesgue lemma, we obtained some results of the singularity of self-affine measure mainly decided by its weight in line.Using the property of invariant sub lattice and trigonometric polynomial,we gave a simplified sufficient condition of a certain self-affine measure being singular in R".By use of the property of harmony pair and Parseval identity,the relation between orthogonal pair and spectrum pair was discussed for the first time.Taking use of the property of Unitary matrix,we get a characterization of the digit set,whenever the atomic measure has spectrum.The characterization of the digit set with the prime determinant and related contents are considered.The results of this thesis extend and improve the related conclusions obtained by P.E.T.Jorgensen,K.A.Kornelson,K.L.Shuman, D.E. Dutkay, J.-L.Li, L.-S.Zhang and X.-C.Shen.It is important for the further investigation of the singularity of self-affine measures and the characterizations of the special digit set.