A necessary and sufficient condition for the finite µ M,D -orthogonality

Abstract

The self-affine measure µM,D associated with an expanding matrix M ∈ Mn(ℤ) and a finite digit set D ⊂ ℤn is uniquely determined by the self-affine identity with equal weight. The set of orthogonal exponential functions E(Λ) := {e2πi〈λ, x〉 : λ ∈ Λ} in the Hilbert space L2(µM,D) is simply called µM,D-orthogonal exponentials. We consider in this paper the finiteness of µM,D-orthogonality. A necessary and sufficient condition is obtained for the set E(Λ) to be a finite µM,D-orthogonal exponentials. The research here is closely connected with the non-spectrality of self-affine measures.