Generalized Riesz products produced from orthonormal transforms

Nikolaos Atreas ¹   ; Antonis Bisbas ²

¹ Department of Mathematics, Physics and Computer Sciences Faculty of Engineering Aristotle University of Thessaloniki 54-124 Thessaloniki, Greece
² School of Technological Applications Department of General Sciences Institute of West Macedonia Kila 50-100 Kozani, Greece

Colloquium Mathematicum, Tome 126 (2012) no. 2, pp. 141-154

Résumé

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.

Détail

DOI : 10.4064/cm126-2-1

Keywords: mathcal p finite set step functions real valued trigonometric polynomials mathbb satisfying certain orthonormality condition study multiscale generalized riesz product measures defined weak * limits elements mathbb where n dimensional subspaces mathbb spanned orthonormal set which produced dilations multiplications elements mathcal overline bigcup mathbb mathbb results involve mutual absolute continuity singularity riesz products extending previous results multiscale riesz products