The Spectrality of a Class of Fractal Measures on Rⁿ

Let M =rho I-1 is an element of M-n(R) be an expanding matrix with 0 < |rho| < 1 and D subset of Z(n) be a finite digit set with 0 is an element of D and Z(m(D))- Z(m(D)) subset of Z(m(D)) boolean OR {0}subset of m(-1) Z(n) for a prime m, where Z(mD) := {x : Sigma (d is an element of D) c(2 pi i(lambda,x)) = 0}. Let mu(M,D) be the associated self-similar measure defined by mu(M,D)(center dot) = 1/|D| Sigma(d is an element of D mu M,D)(M(center dot) - d). In this paper, the necessary and sufficient conditions for L-2 ((mu M,D)) to admit infinite orthogonal exponential functions are given. Moreover, by using the order theory of polynomial, we estimate the number of orthogonal exponential functions for all cases that L-2((mu M,D)) does not admit infinite orthogonal exponential functions.