The Cardinality Of Orthogonal Exponential Functions Of Several Kinds Of Self-affine Measures

Let ? be a Borel probability measure with compact support o Rn,if there exists a discrete set ? C Rn such that E?:={e2?i<?,x>:???} forms an orthogonal basis for L2(?),then we call ? a spectral measure,? a spectrum of? and(?,?)a spectral pair.It is well know that the affine measure ?M,D is determined by the expanding matrix M and the finite digit set D uniquely.The first spectral measure,the 1/4-Cantor measure,was given by Jorgensen and Perdersen[13].Since then,whether the self-affine measure supported on fractal set is spectral measure caused a great deal of discussion among mathematicians.This paper mainly studies the spectral property of two kinds of self-affine measures.The arrangements of this paper are as follows:In chapter 1,we mainly introduce the main achievements of fractal measure's spectral problems in recent years,as well as the required preparatory knowledge and the main results of this paper.In chapter 2,we mainly study the non-spectral problem of the selfaffine measure ?M,D which is generated by a expanding real matrix M=?-1I and the digit set D(?)Rn,where 0<|?|<1,I is an n x n unit matrix,and ZDn(?)Emn,ZDn-ZDn\{0}(?)ZDn(mod Zn).We give the necessary and sufficient condition that L2(?M,D)has infinite orthogonal system.Meanwhile,if L2(?M,D)has only finite orthogonal exponential functions,we give the exact maximal cardinality.In chapter 3,we mainly study the non-spectral problem of the affine-measure ?M,D generated by the expanding matrix M ?M2(Z)and the digit set(?)with ?1?2-?2?1?0.If ?1?2-?2?1(?)2Z,then L2(?M,D)has infinite orthogonal exponentials if and only if det(M)? 2Z.If det(M)(?)2Z,we obtain that there exist at most 4 mutually orthogonal exponential functions in L2(?M,D).If ?1?2-?2?1?2Z and det(M)(?)2Z,we estimate the number of mutually orthogonal exponential functions in L2(?M,D).But we fail to find the exact maximal cardinality.