Multiscale Riesz Products and their Support Properties

Abstract

This work defines and investigates the properties of multiscale Riesz product measures. These are product measures constructed on general locally compact Abelian groups in a process similar to that of the original example of Riesz. The multiscale element of the construction is the use of a general homomorphism of the group in place of the dilation factor. Furthermore, this construction allows for the use of generating functions that are piecewise constant, reminiscent of a wavelet approach, as well as trigonometric polynomials, which is a more classical Fourier approach. Results obtained include the characterization of the mutual absolute continuity or singularity of such Riesz products, in the spirit of Zygmund's original dichotomy result. In addition, the differences regarding the support properties of measures based on each approach are analyzed and examples are constructed. (2000): Primary 43A05, 43A45; secondary 43A25, 43A46. Mathematics Subject Classifications

The singular continuous diffraction measure of the Thue–Morse chain

“…Note that these limits exist due to unique ergodicity, compare [10], with η(0) = 1. The autocorrelation coefficients satisfy η(−m) = η(m) and, due to (1), the recursions (4) η(2m) = η(m) and η(2m…”

“…It is interesting to note that the set of potential plateau locations coincides with the set of potential (but in our case extinct) Bragg peak positions, so the (extinct) Bragg peaks appear to 'repel' the continuous diffraction spectrum. Nevertheless, one has supp( dF ) = supp(ν) = [0, 1] by [4,Prop. 28], which also implies that F is a strictly increasing function.…”

“…This shows that there is no gap around 0 (and none around 1 by symmetry). The general argument uses that ν is a regular Borel measure; see [4] and references therein for details. The methods used above can also be applied to other substitutions of constant length that fail to have pure point spectrum.…”

Multiscale Haar Orthonormal Matrices with the Corresponding Riesz Products and a Characterization of Cantor-Type Languages

“…The common thread among the aforementioned approaches is a multiscale Riesz product defined in Benedetto-Bernstein and Konstantinidis work (see [3]). Their definition consists of a homomorphism T : G → G, G being a locally compact Abelian group with Haar measure m and a real valued function H on G called generating function, such that:…”

Methods from Multiscale Theory and Wavelets Applied to Nonlinear Dynamics

“…Independently of Dutkay-Jorgensen [45], John Benedetto and his co-authors Erica Bernstein and Ioannis Konstantinidis [12] have developed a new Fourier/infinite-product approach to the very same singular measures that arise in the study in [45]. The motivations and applications are different.…”