On self-similar spectral measures

作者：An, Lixiang; Wang, Cong^*

来源：Journal of Functional Analysis, 2021, 280(3): 108821.

DOI：10.1016/j.jfa.2020.108821

• Spectral measure
• Self-similar measure
• Orthogonal basis
• Product-form digit set

A probability measure mu on R is called a spectral measure if it has an exponential orthogonal basis for L-2 (mu). In this paper, we study the spectrality of the self-similar measure generated by an iterated function system {tau d(.) = rho( + d)}(d is an element of D )dED associated with a real number 0 < rho < 1 and a finite set D subset of R. It can also be expressed as @@@ mu(rho,D) = delta(rho D) * delta(rho 2D) * delta(rho 3D) * ... @@@ = mu(k) * mu(rho,D )(rho(-k).), @@@ where mu k is the convolutional product of the first k discrete measures. Until now, all known self-similar spectral measures are obtained from rho(-1) is an element of N and spectral measures mu(k). We will show that these two conditions are also necessary under some natural assumptions. It improves significantly many results studied by recent research. As an application, we characterize a self-similar spectral measure associated with an integer tile.