Spectral Eigen-subspace and Tree Structure for a Cantor Measure

TL;DR

This paper explores the construction of Fourier bases for the Hilbert space associated with the middle-fourth Cantor measure, revealing a rich set of such bases and methods for explicit construction.

Contribution

It characterizes the set of all Fourier bases for the Cantor measure and provides explicit methods to construct such bases with specific scaling properties.

Findings

The intersection of bases under odd scalings has continuum cardinality.

Characterizations of Fourier bases via measure and dimension are provided.

Explicit constructions of Fourier bases with odd scaling invariance are given.

Abstract

In this work we investigate the question of constructions of the possible Fourier bases $E(\Lambda)=\{e^{2\pi i \lambda x}:\lambda\in\Lambda\}$ for the Hilbert space $L^2(\mu_4)$, where $\mu_4$ is the standard middle-fourth Cantor measure and $\Lambda$ is a countable discrete set. We show that the set $$\mathop \bigcap_{p\in 2\Z+1}\left\{\Lambda\subset \R: \text{$E(\Lambda)$ and $E(p\Lambda)$ are Fourier bases for $L^2(\mu_4)$}\right\}$$ has the cardinality of the continuum. We also give other characterizations on the orthonormal set of exponential functions being a basis for the space $L^2(\mu_4)$ from the viewpoint of measure and dimension. Moreover, we provide a method of constructing explicit discrete set $\Lambda$ such that $E(\Lambda)$ and its all odd scaling sets $E(\Lambda),p\in2\Z+1,$ are still Fourier bases for $L^2(\mu_4)$.