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 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.

An orthonormal wavelet system in R, d ∈ N, is a countable collection of functions {ψ j,k}, j ∈ Z, k ∈ Z, ` = 1, . . . , L, of the form ψ j,k(x) = | deta|−j/2ψ`(a−jx− k) ≡ (Daj Tk ψ)(x) that is an orthonormal basis for L2(Rd), where a ∈ GLd(R) is an expanding matrix. The first such system to be discovered (almost one hundred years ago) is the Haar system for which L = d = 1, ψ1(x) = ψ(x) = χ[0,1...

There is an apparent renewed interest in application of the Haar wavelet transform in switching theory and logic design.1–6 In particular, the discrete Haar transform appears very interesting for its wavelets-like properties and fast calculation algorithms. The applications have been found in circuit synthesis, verification and testing.1–9 For applications of the Haar transform in logic design,...

I fhI is the respective Haar coefficient, and σ(I) = ±1. This operator, which we denote by Tσ, is a dyadic martingale transform. The martingale transform is bounded as an operator on L(R,C). We want to find a condition on matrix weights, U and V , that implies that all martingale transforms are uniformly bounded as operators from L(R,C, V ) to L(R,C, U) where L(R,C, V ) is the space of function...

Grr ochenig and Madych showed that a Haar-type wavelet basis of L 2 (R n) can be constructed from the characteristic function of a compact set if and only if is an integral self-aane tile of Lebesgue measure one. In this paper we generalize their result to the multiwavelet settings. We give a complete characterization of Haar-type scaling function vectors (x) := 1 (x); : : : ; r (x)] T , where ...

This paper studies the connections between discrete two-dimensional schemes for shift-invariant Haar wavelet shrinkage on one hand, and nonlinear diffusion on the other. We show that using a single iteration on a single scale, the two methods can be made equivalent by the choice of the nonlinearity which controls each method: the shrinkage function, or the diffusivity function, respectively. In...

A method for the design of Fast Haar wavelet for signal processing & image processing has been proposed. In the proposed work, the analysis bank and synthesis bank of Haar wavelet is modified by using polyphase structure. Finally, the Fast Haar wavelet was designed and it satisfies alias free and perfect reconstruction condition. Computational time and computational complexity is reduced in Fas...