8 1. MATHEMATICAL SHAPES OF UNCERTAINTY zero, i.e. measures mutually singular with Lebesgue measure (singular measures). Let us start with the following classical result. A sequence an 0 is calm if can am Can for all m [n, 2n] and some fixed c, C 0. Theorem 3 (Ivashov-Musatov, 1952). Let μ be a finite singular measure on [0, 1]. If ˆ(n) l2 then μ 0. However, if an 0 is any calm sequence that is not in l2 then there exists a singular measure μ with |ˆ(n)| an. The calmness condition can be replaced with other regularity requirements. In this regard let us mention the result by orner [90, 1987] where an l2 is even and decreasing for positive n. The fact that a singular measure may have a Fourier transform that tends to zero at infinity was known much earlier. It belongs to an interesting branch of Fourier analysis that we would like to discuss in the remainder of this subsection. We will call a finite complex measure on R an R-measure if ˆ(t) 0 as t ±∞. Similarly, for measures on the unit circle T, an R-measure is the measure whose Fourier coefficients ˆ(n) = zndμ(z) tend to zero as |n| ∞. In this name ‘R’ stands for A. Rajchman who first undertook a systematic study of such measures in 1920’s. The basic facts about R-measures are as follows. Every finite absolutely con- tinuous measure, i.e. a measure of the form f(x)dx, f L1(R), is an R-measure. This was proved by Riemann in 1854 for Riemann-integrable f and extended by Lebesgue in 1903 to Lebesgue integrable densities. The property of being and R-measure is ‘spectral,’ i.e. if μ is an R-measure then for all f L1(|μ|) , is an R-measure. Hence, any measure splits into R- and non-R- parts: μ = μR + μnR where μR is an R-measure and μnR is completely non-R, in the sense that any measure absolutely continuous with respect to μnR is not an R-measure, unless it is trivial. Such a decomposition appears in physical settings: if μ is a spectral measure of a Hamiltonian then μR and μnR correspond to bound and transient states of the quantum system, see for instance [124]. Another important property is that ˆ(t) 0 as |t| if and only if ˆ(t) 0 as t (−∞) and similarly for the circle case. The first UP-style problem in this area, motivated by the Riemann theorem above, is whether measures suppported on zero sets (singular measures) can be R- measures. The answer is positive and the first example of a singular R-measure was given by Menshov in 1916 [111]. On the other hand, we have the following theorem by Wiener, 1924 [151]. We denote by μp the pure-point part of the measure μ. Theorem 4. Let μ be a finite measure on R. Then lim n→∞ 1 2n + 1 n −n |ˆ(k)|2 = ||μp||. In particular, the measure is continuous if and only if the limit is 0. Hence, measures with point masses cannot be R-measures. Wiener’s theorem immediately raises another historical problem: do there exist continuous measures that are not R-measures? To answer this question in the positive F. Riesz developed his products in 1918 [131]. He seemingly overlooked a simple fact that the standard 1 3 -Cantor measure satisfies ˆ(3n) = ˆ(n) = 0 and therefore provides an example of such a measure.
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