The initial parameters we use in constructing a unitary filter bank are the number of channels N and the flatness M of the squared magnitude of the frequency response. With N = 2 and M ≥ 1 we get the classical Daubechies wavelet family, including the class of Haar wavelets for M = 1. Any choice of these two parameters leads to a polynomial of degree n = N · M − 1 in the variable z−1 containing ...

We study the problem of the approach to equilibrium in a macroscopic quantum system in an abstract setting. We prove that, for a typical choice of “nonequilibrium subspace”, any initial state (from the energy shell) thermalizes, and in fact does so very quickly, on the order of the Boltzmann time τB := h/(kBT ). This apparently unrealistic, but mathematically rigorous, conclusion has the import...

Methods based on hypothesis tests (HTs) in the Haar domain are widely used to denoise Poisson count data. However, facing large datasets or real-time applications, Haar-based denoisers have to use the decimated transform to meet limited-memory or computation-time constraints. Unfortunately, for regular underlying intensities, decimation yields discontinuous estimates and strong “staircase” arti...

Let G be an abelian Polish group, e.g. a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure μ on G such that μ(B+g) = 0 for every g ∈ G. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if G...

Let X1, X2, . . . be independent identically distributed random elements of a compact group G. We discuss the speed of convergence of the law of the product Xl · · ·X1 to the Haar measure. We give poly-log estimates for certain finite groups and for compact semi-simple Lie groups. We improve earlier results of Solovay, Kitaev, Gamburd, Shahshahani and Dinai. 2010 Mathematics Subject Classificat...

Let M be a unitary matrix with eigenvalues t j , and let f be a function on the unit circle. Deene X f (M) = P f(t j). We derive exact and asymptotic for-mulae for the covariance of X f and X g with respect to the measures j(M)j 2 dM where dM is Haar measure and an irreducible character. The asymptotic results include an analysis of the Fej er kernel which may be of independent interest.

Let M be a unitary matrix with eigenvalues t j , and let f be a function on the unit circle. Deene X f (M) = P f(t j). We derive exact and asymptotic for-mulae for the covariance of X f and X g with respect to the measures j(M)j 2 dM where dM is Haar measure and an irreducible character. The asymptotic results include an analysis of the Fej er kernel which may be of independent interest.