Spectral Eigen-subspace and Tree Structure for a Cantor Measure

Abstract

In this work we investigate the question of constructions of the possible Fourier bases $E(Λ)=\{e^{2πi λx}:λ\inΛ\}$ for the Hilbert space $L^2(μ_4)$, where $μ_4$ is the standard middle-fourth Cantor measure and $Λ$ is a countable discrete set. We show that the set $$\mathop \bigcap_{p\in 2\Z+1}\left\{Λ\subset \R: \text{$E(Λ)$ and $E(pΛ)$ are Fourier bases for $L^2(μ_4)$}\right\}$$ has the cardinality of the continuum. We also give other characterizations on the orthonormal set of exponential functions being a basis for the space $L^2(μ_4)$ from the viewpoint of measure and dimension. Moreover, we provide a method of constructing explicit discrete set $Λ$ such that $E(Λ)$ and its all odd scaling sets $E(Λ),p\in2\Z+1,$ are still Fourier bases for $L^2(μ_4)$.