In this paper, a minimization of Haar wavelet series for simplification of circuits and Haar based decision diagrams representing discrete multiple-valued functions is proposed. The minimization is performed by permutation of indices of generalized Haar functions. Experimental results show that this method provides reasonable reduction in the number of non-zero coefficients. The Haar series red...

We give a new generalization for Haar functions. The generalization starts from the Walsh-like functions and based on the connection between the original Walsh and Haar systems. We generalize the Haar– Fourier Transform too.

In this article,we present a wavelet method for solving stochastic Volterra integral equations based on Haar wavelets. First, we approximate all functions involved in the problem by Haar Wavelets then, by substituting the obtained approximations in the problem, using the It^{o} integral formula and collocation points then, the main problem changes into a system of linear or nonlinear equation w...

The new class of weights called A p weights is introduced. We prove that a characterization and an unconditional basis of the weighted Lp space Lp(Rn,w(x)dx) with w ∈ A p (1 < p < ∞) are given by the Haar wavelets and the Haar scaling function. As an application of these results, we establish a greedy basis by using the Haar wavelets and the Haar scaling function again.

Unnormalized Haar spectra and Ordered Binary Decision Diagrams (OBDDs) are two standard representations of Boolean functions used in logic design. In this article, mutual relationships between those two representations have been derived. The method of calculating the Haar spectrum from OBDD has been presented. The decomposition of the Haar spectrum, in terms of the cofactors of Boolean function...

We prove that in Polish, abelian, non-locally-compact groups the family of Haar null sets of Christensen does not fulfil the countable chain condition, that is, there exists an uncountable family of pairwise disjoint universally measurable sets which are not Haar null. (Dougherty, answering an old question of Christensen, showed earlier that this was the case for some Polish, abelian, non-local...

An orthogonal Haar scattering transform is a deep network computed with a hierarchy of additions, subtractions and absolute values over pairs of coefficients. Unsupervised learning optimizes Haar pairs to obtain sparse representations of training data with an algorithm of polynomial complexity. For signals defined on a graph, a Haar scattering is computed by cascading orthogonal Haar wavelet tr...

Abstract The concept of Rademacher type p (1 ≤ p ≤ 2) plays an important role in the local theory of Banach spaces. In [3] Mascioni considers a weakening of this concept and shows that for a Banach space X weak Rademacher type p implies Rademacher type r for all r < p. As with Rademacher type p and weak Rademacher type p, we introduce the concept of Haar type p and weak Haar type p by replacing...

The efficient and discriminating feature extraction is a significant problem in pattern recognition and computer vision. This paper presents a novel Discriminating Haar (D-Haar) features for eye detection. The DHaar feature extraction starts with a Principal Component Analysis (PCA) followed by a whitening transformation on the Haar feature space. A discriminant analysis is then performed on th...

The classical Haar wavelet system of L2(R) is commonly considered to be very local in space. We introduce and study in this paper piecewise-constant framelets (PCF) that include the Haar system as a special case. We show that any bi-framelet pair consisting of PCFs provides the same Besov space characterizations as the Haar system. In particular, it has Jackson-type performance sJ = 1 and Berns...