Some properties of spectral measures
Spectral measures, first introduced by \textit{P. E. T. Jørgensen} and \textit{S. Pedersen} [J.\ Anal.\ Math.\ 75, 185--228 (1998; Zbl 0959.28008)], are a natural extension of spectral sets. A finite Borel measure \(\mu\) in \(R^d\) is called a \textit{spectral measure} if there exists a set \(\Lambda \subset R^d\) such that the set of exponentials \(\{\exp(2 \pi i \lambda \cdot x); \lambda \in \Lambda\}\) forms an orthogonal basis for \(L^2(\mu)\). Let \(\Omega\) be a measurable set in \(R^d\), and let \(\mu\) denote the restriction of the \(d\)--dimensional Lebesgue measure to \(\Omega\). We say that \(\Omega\) is a \textit{spectral set} if \(\mu\) is a spectral measure. The main interest for studying spectral sets comes from their misterious connection to tiling, first formulated in a conjecture by \textit{B. Fuglede} [J. Funct. Anal. 16, 101--121 (1974; Zbl 0279.47014)], and known today as the Fudledge Conjecture or the Spectral Set Conjecture: A measurable set \(\Omega\) in \(R^d\) is a spectral set if and only if it tiles \(R^d\) by translation. The conjecture had baffled the mathematicians that study spectral sets for years until very recently, when \textit{T. Tao} [Math. Res. Lett. 11, No. 2--3, 251--258 (2004; Zbl 1092.42014)] exhibited a spectral set in dimensions \(d \geq 5\) that is not a tile, and \textit{M. N. Kolountzakis} and \textit{M. Matolcsi} [``Tiles with no spectra'', preprint, arXiv: math.CA/0406127] exhibited tiles that are not spectral sets in dimensions \(d \geq 5\). Despite the counterexamples, the connection between spectral sets and tiling is strongly evident, especially in low dimensions, as indicated by earlier works, including the original article by Fuglede. The study of spectral measures so far has focused on self--similar measures. These measures have the advantage that their Fourier transforms can be expressed as infinite products, which permits the computation of their zeros. In the article under review the authors study measures that are not self--similar. Without knowing the zeros of their Fourier transforms, the characterization of spectral measures becomes difficult and the results obtained herein, although fundamental and quite general, are modest. They shoud, however, offer valuable guidance to future studies in the area.
scientific article; zbMATH DE number 5013516
scientific article