Multiscale Riesz Products and their Support Properties
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Multiscale Riesz Products and their Support Properties JOHN J. BENEDETTO1, , ERICA BERNSTEIN2 and IOANNIS KONSTANTINIDIS3,
1 Department of Mathematics, University of Maryland, College Park, MD 20742, USA e-mail: [email protected] 2 University of Hawaii at Hilo, 200 W. Kawili St., Hilo, HI 96720, USA 3 Institute for Digital Informatics and Analysis, University of Houston, Houston, TX 77204-3010, USA. e-mail: [email protected]
To John Horváth, our friend and teacher and colleague. Years ago, he gave the first named author his calligraphically magnificent and mathematically perfect notes of Raphaël Salem’s lectures on Riesz products. (Received: 16 June 2004; accepted: 7 May 2005) 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. Mathematics Subject Classifications (2000): Primary 43A05, 43A45; secondary 43A25, 43A46. Key words: Riesz products, thin sets.
1. Introduction Riesz products are examples of a construction originating in a celebrated paper by Frédéric Riesz. In his 1918 paper, “Über die Fourierkoeffizienten einer stetigen Funktion von beschränkter Schwankung” [33], Riesz proved the existence of a continuous function F of bounded variation on T, whose Fourier–Stieltjes coefficients Supported in part by NSF-DMS Grant 0139759 (2002-2005) and the General Research Board of the University of Maryland. Supported in part by DARPA Grant MDA 972011003.
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do not vanish at infinity. His example essentially was the pointwise limit of the functions x N (1 + cos(2π 4n t)) dt. FN (x) = 0
n=1
Notationally, T is the quotient group R/Z of the real numbers R by the integers Z. Generally our notation is standard, e.g., [36]; in particular, M(T) is the Banach algebra of bounded Radon measures on T, see (1). Riesz’ construction proved to be the source of powerful ideas that can be used to produce concrete examples of measures with a number of desired properties (singularity being the goal of the original construction), and the literature on the subject is extensive. It starts with Antoni Zygmund [38] (1932), [39], who generalized Riesz’ example by introducing a bounded sequence of coefficients αn and a sequence of scales λn , and sh
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