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The paper has two objectives. On the one hand, we study left Haar null sets, a measure-theoretic notion of smallness on Polish, not necessarily locally compact, groups. On the other hand, we introduce and investigate two classes of Polish groups which are closely related to this notion and to amenability. We show that left Haar null sets form a $\sigma$-ideal and have the Steinhaus property on Polish groups which are ‘amenable at the identity’, and that they lose these two properties in the presence of appropriately embedded free subgroups. As an application we prove an automatic continuity result for universally measurable homomorphisms from inverse limits of sequences of amenable, locally compact, second countable groups to second countable groups. [ABSTRACT FROM AUTHOR]

Despite the commercial success of data mining, a major drawback has been acknowledged across academic, industry, and government sectors, namely, the issue of violating the privacy of individuals. We propose a data transformation method based on wavelets to disguise private data while preserving the original classification patterns. Wavelet transformations have been used extensively in signal processing for data reduction, multiresolution analysis, and removing noise from data. In our implementation, two commonly used wavelet transforms, the Haar and the Daub-4 transforms, are tested for pattern and privacy preservation in classification mining tasks. Empirical results confirm that the Haar and the Daub-4 transforms preserve the classification patterns and preserve the privacy for real valued data. [ABSTRACT FROM AUTHOR]

Time series clustering has attracted increasing interest in the last decade, particularly for long time series such as those arising in the bioinformatics and financial domains. The widely known curse of dimensionality problem indicates that high dimensionality not only slows the clustering process, but also degrades it. Many feature extraction techniques have been proposed to attack this problem and have shown that the performance and speed of the mining algorithm can be improved at several feature dimensions. However, how to choose the appropriate dimension is a challenging task especially for clustering problem in the absence of data labels that has not been well studied in the literature. In this paper we propose an unsupervised feature extraction algorithm using orthogonal wavelet transform for automatically choosing the dimensionality of features. The feature extraction algorithm selects the feature dimensionality by leveraging two conflicting requirements, i.e., lower dimensionality and lower sum of squared errors between the features and the original time series. The proposed feature extraction algorithm is efficient with time complexity O(mn) when using Haar wavelet. Encouraging experimental results are obtained on several synthetic and real-world time series datasets. [ABSTRACT FROM AUTHOR]

In this paper, we study a special perturbation of the function system obtained from the Fejér kernel. It is shown how this relates to the stability of bases and complete systems as well to the stability of the trigonometric system. An approximation algorithm in systems resulting from a perturbation of the original system is given. [ABSTRACT FROM AUTHOR]

We study the universality of a Dirichlet series with periodic coefficients. This property is proved in the case of multiplicative coefficients, and in the general case we establish universality in a certain set of analytic functions related to a probability distribution. [ABSTRACT FROM AUTHOR]

In this paper, we study whether the set A(ϕ) is closed under multiplication f · g, where f and g belong to the class A(ϕ). We also study the problem of the existence of a solution of the equation Bx = C (where B,C ∈ A(ϕ) and B ≠ 0) on the set A(ϕ). [ABSTRACT FROM AUTHOR]

This study addresses the identification of linear time varying systems. The identification is based on the expansion of all time functions in the state equations by Haar wavelets. The unknown time function can thus be identified in terms of Haar wavelets. A Haar wavelet is a set of complete, orthogonal basis and is easy to use computations. Several good properties of Haar wavelets are utilized in the algorithm. Both numerical and experimental results verify the analysis. [ABSTRACT FROM AUTHOR]

A technique is offered to find the wavelet coefficients for mechanical problems. It is based on the Haar wavelet family. A condition for the detail coefficients in discrete wavelet analysis is given to establish orbit stability of mechanical systems [ABSTRACT FROM AUTHOR]

Abstract: A wavelet application to the vibration-based damage evaluation technique is introduced. The proposed method requires only a few of the lower mode shapes before and after a small damage event in order to detect, locate, and size damage on plate-like structures. The proposed method takes account for uncertainty in mass density, surrounding forces, foundation stiffness, etc. Based on a small damage assumption of a thin plate, the two-dimensional damage index (DI) equation is revealed within the context of elasticity. The singularity problem, which occurs in the resulting DI equation, is solved in the two-dimensional multi-resolution wavelet domain with the aid of the singular value decomposition technique. Finally, the desired damage measures are reconstructed from the one in the wavelet space by inverse wavelet transformation. The performance of the proposed method is compared with an existing damage detection method. Finally, an uncertainty analysis of the proposed method is provided. [Copyright &y& Elsevier]

The article considers the m-term L¹-greedy algorithm in the Haar system normalized in L¹ (0,1). It uses the binary numbering of the functions of the Haar system. It describes some notation and proves several lemmas, which will be used in the proof of the theorem. It proves the existence of a function f ε L¹ (0,1) for which there is no L¹-greedy algorithm in the Haar system converging to f.

We present a computational method using Haar wavelets to determine the piecewise constant feedback controls for a finite-time linear optimal control problem of a time-varying state-delayed system. The method is simple and computationally advantageous. The approximate optimal trajectory and optimal control are calculated using the Haar wavelet integral, product, and delay operational matrices. An illustrative example is included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

Due to the rapid development of computer and sensing technology, many measurements of process variables are readily available in manufacturing processes. These measurements carry a large amount of information about process conditions. It is highly desirable to develop a process monitoring and diagnosis methodology that can utilize this information. In this paper, a statistical process control monitoring system is developed for a class of commonly available process measurements—cycle-based waveform signals. This system integrates the statistical process control technology and the Haar wavelet transform. With it, one can not only detect a process change, but also identify the location and estimate the magnitude of the process mean shift within the signal. A case study involving a stamping process demonstrates the effectiveness of the proposed methodology on the monitoring of the profile-type data. Note to Practitioners—Cycle-based signal refers to an analog or digital signal that is obtained through automatic sensing during each operation cycle of a manufacturing process. The cycle-based signal is very common in various manufacturing processes (e.g., forming force in stamping processes, the holding force, and the current signals in spot welding processes, the insertion force in the engine assembly process). In general, cycle-based signals contain rich process information. In this paper, cycle-based signal monitoring will be accomplished by monitoring the wavelet transformation of the signal, instead of monitoring the raw observations themselves. Further, a decision-making technique is developed using the SPC monitoring system to locate where the mean shift occurred and to estimate magnitudes of mean shifts. Thus, this paper presents a generic framework for the enhanced statistical process control technique of cycle-based signals. [ABSTRACT FROM AUTHOR]

Let U m be an m× m Haar unitary matrix and U[ m,n] be its n× n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U[ m,n] as m/ n→ λ and n→∞. The rate function and the limit distribution are given explicitly. U[ m,n] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq. [ABSTRACT FROM AUTHOR]

Abstract: Recently, we have proposed a new multi-rate/multi-media system called wavelet based scale-code division multiple access (W/S-CDMA) that depends on the code, time and scale orthogonality introduced by pseudo-noise (PN) sequences, and wavelets. Wavelets are used as an orthogonal set of symbols for signaling, and their orthogonality is exploited over scale and time. In this system, the channel is divided into different scales, and each scale into time slots. In addition, the PN sequences are used in each scale to accommodate multiple users. In other words, each user encodes its successive information symbols with time-shifted replicas of the same basic wavelet in a specific scale and spreads its scaled and translated wavelets with its PN sequence. The finer the scale used, more symbols are transmitted. In this paper, we analyze the performance of Haar wavelet based S-CDMA (HW/S-CDMA) over a synchronous additive white Gaussian noise (AWGN) channel using a decorrelating multi-user detector. Results reveal that HW/S-CDMA holds promise since the users of HW/S-CDMA can achieve variable and higher data rates than those of direct sequence (DS)-CDMA for a similar bit error rate performance when real-valued PN sequences are used. However, HW/S-CDMA achieves better performance than DS-CDMA when complex-valued PN sequences are used. In addition, for all rates the same performance is achieved in HW/S-CDMA and the multi-user detector has no processing delay for different rates and has the same features as the standard decorrelating detector. Because of the reuse capability introduced by the scales, HW/S-CDMA is also capable of employing many available PN sequences from the optimal PN sequence families with limited number of sequences such as Kasami, Bent, etc. [Copyright &y& Elsevier]

A simple and effective algorithm based on Haar wavelets is proposed to the solution of stiff problems in this article. It can integrate the stiff equation with very accurate results for any length of time. The simulation result shows that the single-term Haar wavelet method is better than the improved Runge–Kutta–Fehlberg method, while the terms of both expansions are the same. [ABSTRACT FROM AUTHOR]

Haar spaces are certain finite-dimensional subspaces of $\cc(K)$, where $K$ is a compact set and $\cc(K)$ is the Banach space of continuous functions defined on $K$ having values in $\C$. We characterize those Haar spaces which are generated by shifts applied to a single, analytic function for $K\subset\C$. This means that an arbitrary finite number of shifts generates Haar spaces by forming linear hulls. We have to distinguish two cases: (a) $K\not=\overline{K^\circ}$; (b) $K=\overline{K^\circ}$. It turns out that, in case (a), an analytic Haar space generator for dimensions one and two is already a universal Haar space generator for all dimensions. The geometrically simplest case that, in case (b), $K$ is convex with smooth boundary turns out to be the most difficult case. There is one numerical example in which the entire function $f:=1/\Gamma$ is interpolated in a shift generated Haar space of dimension four. [ABSTRACT FROM AUTHOR]

Abstract: is defined to be and is called a cyclically ordered 3-subset. A compatibly ordered orthogonal group divisible design (briefly, compatibly ordered-OGDD or COGDD) is a set X and a partition of X into classes (usually called groups), and two sets and of cyclically ordered 3-subsets of X (usually called blocks), so that is an orthogonal group divisible design, and if appears in a block of then appears in a block of , where and . A compatibly ordered orthogonal Steiner triple system of order n (briefly, compatibly ordered- or ) can be viewed as a COGDD of type . In this article, we will show that for , [(i)] there exists a ; [(ii)] there exists a COGDD of type ; and [(iii)] there exists a COGDD of type for or 2 and . [Copyright &y& Elsevier]

We establish the notion of a multiresolution analysis (MRA) on the space of p M -periodic sequences, where p > 1 is a prime number and M > 1 is a positive integer. We give sufficient conditions to ensure the existence of sampling sequences on MRA subspaces. Because each sampling expansion is an MRA projection, we estimate its truncation error. We also present several examples of sampling formulas for computational applications. In particular, we demonstrate an algorithm for edge detection that is based on Haar's sampling formula. [ABSTRACT FROM AUTHOR]

In this paper we present an implementation of the Haar wavelet to the optimal control of linear singularly perturbed systems. The approximated composite control and the slow and fast trajectories with respect to a quadratic cost function are calculated by solving only the linear algebraic equations. The results are illustrated with a simple example. [ABSTRACT FROM AUTHOR]

In spectral interpretation, decision diagrams (DDs) are defined in terms of some spectral transforms. For a given DD, the related transform is determined by an analysis of expansion rules used in the nodes and the related labels of edges. The converse task, design of a DD in terms of a given spectral transform is relatively easy to solve for Kronecker product representable spectral transforms. However, in other cases, it appears a problem of decomposition of basic functions in spectral transform to determine the corresponding expansion rules and labels at the edges. We point out that this problem relates to the assignment of nodes in Pseudo-Kronecker DDs (PKDDs). Then, we show that in the case of Haar transform, the expansion rules and assignment of nodes in Haar Spectral transform DDs (HSTDDs) can be determined from a study and comparison of FFT and DD methods for calculation of the Haar spectrum. This consideration permits to generalize the definition of HSTDDs to multiple-valued (MV) functions considered as functions in finite fields or as subsets of complex-valued functions. Conversely, from such defined HSTDDs, we derive various Haar transforms for MV functions related to Fourier series and polynomial expressions for MV functions. [ABSTRACT FROM AUTHOR]

The Haar system constitutes an unconditional basis for a separable rearrangement invariant (symmetric) spaceEif and only if the multiplier determined by the sequenceforn= 0 andk= 0, 1, . . . , 2n forn>1, is bounded inE. If the Lorentz space ?(?) differs fromL1 andL8 then there is a multiplier with respect to the Haar system which is bounded in ? (?) and unbounded inL8 andL1. [ABSTRACT FROM AUTHOR]

In this paper we discuss the designs of 2-channel orthogonal near symmetric wavelets on the one hand, and the symmetric near orthogonal wavelets on the other. In both types of wavelets we use the Gröbner bases design approach and present examples of both cases. [ABSTRACT FROM AUTHOR]

The main result of this paper is theorem asserting that for a series in the generalized Haar system generated by bounded functions, all of its subseries (and even a wider class of series) are Fourier series in the sense of the A-integral of its sum (although they are not necessarily standard Fourier series). [ABSTRACT FROM AUTHOR]

This article introduces a new method for the estimation of the intensity of an inhomogeneous one-dimensional Poisson process. The Haar-Fisz transformation transforms a vector of binned Poisson counts to approximate normality with variance one. Hence we can use any suitable Gaussian wavelet shrinkage method to estimate the Poisson intensity. Since the Haar-Fisz operator does not commute with the shift operator we can dramatically improve accuracy by always cycle spinning before the Haar-Fisz transform as well as optionally after. Extensive simulations show that our approach usually significantly outperformed state-of-the-art competitors but was occasionally comparable. Our method is fast, simple, automatic, and easy to code. Our technique is applied to the estimation of the intensity of earthquakes in northern California. We show that our technique gives visually similar results to the current state-of-the-art. [ABSTRACT FROM AUTHOR]

Let Hn be an n-dimensional Haar subspace of X=CR[a,b] and let Hn-1 be a Haar subspace of Hn of dimension n-1. In this note we show (Theorem 6) that if the norm of a minimal projection from Hn onto Hn-1 is greater than 1, then this projection is an interpolating projection. This is a surprising result in comparison with Cheney and Morris (J. Reine Angew. Math. 270 (1974) 61 (see also (Lecture Notes in Mathematics, Vol. 1449, Springer, Berlin, Heilderberg, New York, 1990, Corollary III.2.12, p. 104) which shows that there is no interpolating minimal projection from C[a,b] onto the space of polynomials of degree ⩽n, (n⩾2). Moreover, this minimal projection is unique (Theorem 9). In particular, Theorem 6 holds for polynomial spaces, generalizing a result of Prophet [(J. Approx. Theory 85 (1996) 27), Theorem 2.1]. [Copyright &y& Elsevier]

In this note, it is shown that the conclusions of the classical Szegö's theorem on asymptotic distribution of eigen values of finite sections of multiplication operators, do not remain valid when the trigonometric basis is replaced by the Haar basis. [ABSTRACT FROM AUTHOR]

The compression property of wavelets in the analysis of an evolution problem (with unsmooth initial conditions) is investigated. The effectiveness of wavelets both in the reduction of complexity (number of coefficients) and in better approximation is shown. Haar wavelets, having the simplest interpretation of the wavelet coefficients, are used for defining the wavelet solution of an evolution (parabolic-hyperbolic) problem. The approximate solution, at a given fixed scale (resolution), results from the superimposition of (a small set of) fundamental wavelets, thus giving (also) a physical interpretation to wavelets. Since Haar wavelets are not smooth enough, a numerical derivative algorithm, which allows the scale approximation of partial differential evolution operators, is also defined. As application, the heat propagation (of an initial square wave) is explicitly given in terms of wavelets. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new structure for a binary data transmission using the neural network and Haar transform to reduce the noise and distortion in the channel. To solve the problems such as signal distortion and nonlinearity during the transmission and reception of the signal, we use the neural network to design an equalizer to adapt the change of the channel characteristics. Before transmitting digital information, two adjacent binary signals are Haar transformed and are sent to the channel which is assumed to corrupt the signal by an additive white Gaussian noise (AWGN), and then we receive the noise-reduced signal through the inverse Haar transform. We analyze the probability of error and simulate the model to show the validity. [ABSTRACT FROM AUTHOR]

In this paper we study a subgrid model based on extrapolation of a modeling residual, in the case of a linear convection-diffusion-reaction problem Lu=f in two dimensions. The solution u to the exact problem satisfies an equation L[sub h]u=[f][sup h]+F[sub h](u), where L[sub h] is the operator used in the computation on the finest computational scale h, [f][sup h] is the approximation of f on the scale h, and F[sub h](u) is a modeling residual, which needs to be modeled. The subgrid modeling problem is to compute approximations of F[sub h](u) without using finer scales than h. In this study we model F[sub h](u) by extrapolation from coarser scales than h, where F[sub h](u) is directly computed with the finest scale h as reference. We show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a mesh of size less than h/4. [ABSTRACT FROM AUTHOR]

This paper proposes a new watermarking method which applies SVD (singular value decomposition) to the watermark image and the Haar transformation to the host image, respectively. In this method, relatively important information which corresponds to larger singular values is embedded as near to the center of the low-frequency component image of the Haar transformed host image as possible. This method is useful in decreasing degradation of the quality of watermarked host images and thus in making the embedded watermark robust to various attacks. © 2003 Wiley Periodicals, Inc. Syst Comp Jpn, 34(9): 38–47, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.10371 [ABSTRACT FROM AUTHOR]

Describes the proposed hybrid finite-difference time-domain/Haar multiresolution time-domain technique for the time domain analysis. Characterization of microwave structures; Implementation of dynamic mesh refinement; Calculation of the arbitrary number of wavelet levels.

Presents analytical solutions for the problems of granular flow through hoppers and the stress distributions at the base of stock piles. Basic equations of the proper continuum mechanical theory of granular materials; Application of the Haar-von Karman hypothesis to the problem; Numerical solutions for the three-dimensional stock pile.

It is shown that the span of $\{a_ih_i\oplus b_i e_i \}^n_{i=1}$, where $\{h_i\}$ is the Haar system in $L_p$ and $\{e_i\}$ the canonical basis of $\ell_p$, is well isomorphic to a well complemented subspace of $L_p, 2

Presents the survey on basic definitions of Haar wavelet transform. Illustration on the sign versions of transform; Discussion on the efficient symbolic calculation of Haar system; Frequency of ordering trigonometric functions.

Examines the concept of paired Haar transform (PHT) for representation and efficient optimization of systems for incomplete Boolean functions. Calculation on PHT for switching functions through binary decision programs; Advantages of computational and memory requirements of PHT; Emphasis on engineering practices for peculiar properties.

Presents the non-linear transform from ternary vectors into ternary spectral domain. Calculation on the signs of Haar transform; Details on the essential properties of logic functions and variables in spectral domain of quantized transform; Emphasis on the theory of error correcting codes for digital filtering.

The theory of diffusion of reacting solutes in porous media described by a multidimensional differential system is discussed in a multidimensional Haar wavelet basis. [ABSTRACT FROM AUTHOR]

We study the recovery of continuous functions from Fourier coefficients with respect to certain given orthonormal systems, blurred by noise. For deterministic noise this is a classical ill-posed problem. Emphasis is laid on a priori smoothness assumptions on the solution, which allows to apply regularization to reach the best possible accuracy. Results are obtained for systems obeying norm growth conditions. In the white noise setting mild additional assumptions have to be made to have accurate bounds. We finish our study with the recovery of functions from noisy coefficients with respect to the Haar system. [Copyright &y& Elsevier]

We prove that if a locally integrable f has a pointwise bounded dyadic square function, where the square function is defined with respect to a so-called “accretive” weight b, then f is locally exponentially square integrable. We generalize the result to d dimensions by means of “canonical” Haar functions for L2(Rd). [Copyright &y& Elsevier]

Discusses a parallel between Haar multiresolution time-domain and the classic Yee finite-difference time-domain algorithm. Implementation of the incident fields as a plane wave and extend the formulation to inhomogeneous dielectric materials; Results of simulations on some canonical targets; Conclusion.

For a given group Γ with a generating set A, a dipole with |A| parallel arcs (directed edges) labeled by elements of A gives rise to a voltage graph whose covering graph, denoted by H(Γ,A) is a bipartite, regular graph, called a bi-Cayley graph. In the case when Γ is abelian we refer to H(Γ,A) as the Haar graph of Γ with respect to the symbol A. In particular for Γ cyclic the above graph is referred to as a cyclic Haar graph. A basic theory of cyclic Haar graphs is presented. [Copyright &y& Elsevier]

The theory of periodic wavelet transforms presented here was originally developed to deal with the problem of epileptic seizure prediction. A central theorem in the theory is the characterization of wavelets having time and scale periodic wavelet transforms. In fact, we prove that such wavelets are precisely generalized Haar wavelets plus a logarithmic term. It became apparent that the aforementioned theorem could not only be quantified to analyze seizure prediction but could also provide a technique to address a large class of periodicity detection problems. An essential step in this quantification is the geometric and linear algebra construction of a generalized Haar wavelet associated with a given periodicity. This gives rise to an algorithm for periodicity detection based on the periodicity of wavelet transforms defined by generalized Haar wavelets and implemented by wavelet averaging methods. The algorithm detects periodicities embedded in significant noise. The algorithm depends on a discretized version W[SUPp,SUBψ]f(n,m) of the continuous wavelet transform. The version defined provides a fast algorithm with which to compute W[SUPp,SUBψ]f(n,m) from W[SUPp,SUBψ]f(n-1,m) or W[SUPp,SUBψ]f(n,m-1). This has led to the theory of nondyadic wavelet frames in l[SUP2](Z) which was developed by the second author and which will appear elsewhere. [ABSTRACT FROM AUTHOR]

We introduce a definition of a weighted Haar basis which generalizes the notion of the traditional Haar basis. This is used first to derive the characterization of weighted Lp(w) in terms of weighted homogeneous Triebel-Lizorkin sequence space, and second to characterize certain weighted Besov spaces with the approximation error in weighted Lp(w) spaces. [ABSTRACT FROM AUTHOR]

In this paper, a method is presented to model the electromagnetic structure that is smaller than an MRTD cell. As an example, the power-plane resonance problem in a multilayered PCB is analyzed by using a Haar-wavelet-based MRTD. This paper demonstrates the accuracy and efficiency of the MRTD. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 30: 6–9, 2001. [ABSTRACT FROM AUTHOR]

The paper deals with the local Cauchy problem for nonlinear functional differential systems. We investigate a general class of difference methods for this problem. We construct interpolating operators on the Haar pyramid and we give an error estimate for approximate solutions. We adopt nonlinear estimates of the Perron type for given functions with respect to the functional variable. The proof of the stability is based on functional difference inequalities. [ABSTRACT FROM AUTHOR]

Deals with a study which examined the state analysis and parameter estimation of bilinear systems in electronics via Haar wavelets. Properties of Haar wavelets; State analysis; Parameter estimation; Numerical examples; Conclusion.

An algorithm to compute a solution for the classical problem of aperture synthesis is presented. The algorithm uses wavelets as basis functions in order to reach a good approximation of the aperture field constructively from the antenna radiation pattern. Haar wavelet basis is the selected wavelet family for validations. Results prove the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Consider a Markov system of functions whose linear span is dense with respect to the uniform norm in the space of the continuous functions on a finite interval. Gaussian rules are those which correctly integrate as many successive basis functions as possible with the lesser number of nodes. In this paper we provide a simple proof of the fact that such rules converge for all bounded Riemann-Stieltjes integrable functions. The proposed proof is also valid for any sequence of quadrature rules with positive coefficients which converge for the basis functions. Taking the nodes of the Gaussian rules as nodes for Lagrange interpolation, we give a sufficient condition for the convergence in L[sub 2]-norm of such processes for bounded Riemann-Stieltjes integrable functions. [ABSTRACT FROM AUTHOR]

.This paper presents a system for automatically selecting images in an image database to be used as illustrations of an image method or analysis process. This problem is related to the teleteaching of image processing which uses already implemented libraries of algorithms. This is in the context of a teleteaching European project. We first give a Bayesian approach to this problem, by using an image basis. Then we show how to use the Haar transform for this purpose. Finally we give examples and discuss our approach. [ABSTRACT FROM AUTHOR]

Discusses parameter expansion and defines a parameter-expanded data augmentation (PX-DA) algorithm. Data transformation and convergence properties; Group structure and Haar prior for the expansion parameter; Numerical example.