Analysis of a class of spectral pair conditions

作者：Li JianLin^*

来源：Science China Mathematics, 2011, 54(10): 2099-2110.

DOI：10.1007/s11425-011-4227-4

• iterated function system
• self-affine measure
• spectral pair
• compatible pair
• digit set

For an expanding integer matrix M a M (n) (a"currency sign) and two finite digit sets D, S aS, a"currency sign (n) with 0 a D a (c) S, we shall investigate and study the possible conditions on the spectral pair (A mu (M, D) , I >(M, S)) associated with the iterated function systems {I center dot (d) (x) = M (-1)(x + d)} (daD) and {psi (s) (x) = M*x + s} (saS) in the case when |D| = |S| = |det(M)|. Under the condition that (M (-1) D, S) is a compatible pair, we obtain a series of necessary and sufficient conditions for (A mu (M, D) , I >(M, S)) to be a spectral pair. These conditions include how to characterize the invariant sets I >(M, S) and T(M, D) such that I >(M, S) = a"currency sign (n) and A mu (L) (T(M, D)) = 1 which play an important role in the number system research and in the construction of Haar-type orthogonal wavelet basis respectively.

• 出版日期2011-10
• 单位陕西师范大学