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 $E(\hat{I}›)=\{e^{2\ddot{I}€i\hat{I}»x}:\hat{I}»\in \hat{I}›\}$ for the Hilbert space $L^2(\hat{I}{¼}_4)$, where $\hat{I}{¼}_4$ is the standard middle-fourth Cantor measure and $\hat{I}›$ 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 $L^2(\hat{I}{¼}_4)$ from the viewpoint of measure and dimension. Moreover, we provide a method of constructing explicit discrete set $\hat{I}›$ such that $E(\hat{I}›)$ and its all odd scaling sets $E(\hat{I}›),p\in 2\mathbb{Z}+1,$ are still Fourier bases for $L^2(\hat{I}{¼}_4)$.

31 pages, 3 figures