The Cardinality Of Orthogonal Exponentials With Some Special Digit Sets

In this paper, the problem of the maximum number of mutually orthogonal exponential functions for planar self-affine measure and spectrality and non-spectrality in R3with some special digit sets will be investigated.The main results are as follows:(1) The cardinality of orthogonal exponential with a class of special digit sets will be discussed. And the conclusion in the literature [15] will be even more accurate. By use of residue class of4and the periodicity of zero-point sets, the maximum number of mutually orthogonal exponential in the will be classified to discussed. Where the matrix is and the digit set is Then the digit set is further improved, so the following conclusion is obtained. When then there are at most four orthogonal exponential functions, and four is the best estimation.(2) By use of the feature of zero-point sets, it is proved that for the matrix and the digit set (I) If P1,(?)2Z,P2,P3(?)Z, then there are infinite orthogonal exponential functions which is decided by (M, D) in L2(μM,D).(II) If P1(?) Z,P2,P3(?)3Z, then there are infinite orthogonal exponential functions which is decided by (M, D) in L2(μM,D).(III) If P1(?)2Z, P2,P3(?)3Z, then there are at most six orthogonal exponential functions in L2(μMD), and six is best.(IV) If P1(?)2Z, p2(?)3Z,P3(?)3Z or P1(?)2Z, P3(?)3Z, P2(?)3Z, then there are at most six orthogonal exponential functions in L2(μM,D), and six is best.In this paper, the spectrality and non-spectrality for self-affine measure in the two-dimension space and three-dimension space is studied and improved on the basis of our predecessors. This result is the effective supplement of related guesses.