Generalized Riesz products produced from orthonormal transforms
Volume 126 / 2012
Colloquium Mathematicum 126 (2012), 141-154
MSC: Primary 42A55; Secondary 42C10.
DOI: 10.4064/cm126-2-1
Abstract
Let $\mathcal M_p=\{m_k\}_{k=0}^{p-1}$ be a finite set of step functions or real valued trigonometric polynomials on $\mathbb T=[0,1)$ satisfying a certain orthonormality condition. We study multiscale generalized Riesz product measures $\mu$ defined as weak-$^*$ limits of elements $\mu_N \in V_N$ $(N\in \mathbb N)$, where $V_N$ are $p^N$-dimensional subspaces of $L_2(\mathbb T)$ spanned by an orthonormal set which is produced from dilations and multiplications of elements of $\mathcal M_p$ and $\overline{\bigcup_{N \in \mathbb N}V_N}=L_2(\mathbb T)$. The results involve mutual absolute continuity or singularity of such Riesz products extending previous results on multiscale Riesz products.
