arXiv subjects

Yan-Song Fu

Publications and source records attributed to Yan-Song Fu.

2 recordsLinked to original sources

arXiv2026

Characterization on spectra of self-similar measures and the dual spectral set conjecture

A discrete set $\Lambda$ is called a {\it spectrum} of a Borel probability measure $\mu$ if the exponential functions $\{e^{2\pi i \langle\lambda, x\rangle}:\lambda\in\Lambda\}$ form an orthonormal basis for $L^2(\mu)$. In this work, we first give necessary and sufficient conditions for a self-replicating translation set to be a spectrum of the self-similar measure associated with a product-form Hadamard triple, which were discovered by the first-named author and Lai [Adv. Math. 431 (2023) Paper No.109257]. These results extend the studies by {\L}aba and Wang [J. Funct. Anal. 193 (2002) 409-420], Dutkay and Lai [J. Math. Pures Appl. 107 (2017) 183-204] in $\mathbb R$. As an application, we can explicitly construct such a self-replicating spectrum. Finally, we prove that the dual spectral set conjecture holds for a self-replicating translation set.

math.CA↗

arXiv2024

Spectral Eigen-subspace and Tree Structure for a Cantor Measure

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 $L^2(\mu_4)$.

math.FA↗