Haar Basis Research Articles

Renormings and symmetry properties of 1-greedy bases

We continue the study of 1 -greedy bases initiated by Albiac and Wojtaszczyk (2006) [1]. We answer several open problems that they raised concerning symmetry properties of 1 -greedy bases and the improving of the greedy constant by renorming. We show that 1 -greedy bases need not be symmetric or subsymmetric. We also prove that one cannot in general make a greedy basis 1 -greedy as demonstrated for the Haar basis of the dyadic Hardy space H 1 ( R ) and for the unit vector basis of Tsirelson space. On the other hand, we give a renorming of L p ( 1 < p < ∞ ) that makes the Haar basis 1 -unconditional and 1 -democratic. Other results in this paper clarify the relationship between various basis constants that arise in the context of greedy bases.

Dyadic Fefferman–Stein inequalities and the equivalence of Haar bases on weighted Lebesgue spaces

We give sufficient conditions on two dyadic systems to obtain the equivalence of corresponding Haar systems on dyadic weighted Lebesgue spaces on spaces of homogeneous type. In order to obtain these results, we prove a Fefferman–Stein weighted inequality for vector-valued dyadic Hardy–Littlewood maximal operators with dyadic weights in this general setting.

On lower bounds for the [formula omitted]-discrepancy

The L 2 -discrepancy measures the irregularity of the distribution of a finite point set. In this note, we prove lower bounds for the L 2 -discrepancy of arbitrary N -point sets. Our main focus is on the two-dimensional case. Asymptotic upper and lower estimates of the L 2 -discrepancy in dimension 2 are well known, and are of the sharp order log N . Nevertheless, the gap in the constants between the best-known lower and upper bounds is unsatisfactorily large for a two-dimensional problem. Our lower bound improves upon this situation considerably. The main method is an adaption of Roth’s method, using the Fourier coefficients of the discrepancy function with respect to the Haar basis. We obtain the same improvement in the quotient of lower and upper bounds in the general d -dimensional case. Our lower bounds are also valid for the weighted discrepancy.

Fully Automated Reduction of Ocular Artifacts in High-Dimensional Neural Data

The reduction of artifacts in neural data is a key element in improving analysis of brain recordings and the development of effective brain-computer interfaces. This complex problem becomes even more difficult as the number of channels in the neural recording is increased. Here, new techniques based on wavelet thresholding and independent component analysis (ICA) are developed for use in high-dimensional neural data. The wavelet technique uses a discrete wavelet transform with a Haar basis function to localize artifacts in both time and frequency before removing them with thresholding. Wavelet decomposition level is automatically selected based on the smoothness of artifactual wavelet approximation coefficients. The ICA method separates the signal into independent components, detects artifactual components by measuring the offset between the mean and median of each component, and then removing the correct number of components based on the aforementioned offset and the power of the reconstructed signal. A quantitative method for evaluating these techniques is also presented. Through this evaluation, the novel adaptation of wavelet thresholding is shown to produce superior reduction of ocular artifacts when compared to regression, principal component analysis, and ICA.

Inner functions and local shape of orthonormal wavelets

Conditions characterizing all orthonormal wavelets of L 2 ( R ) are given in terms of suitable orthonormal bases (ONBs) related with the translation and dilation operators. A particular choice of the ONBs, the so-called Haar bases, leads to new methods for constructing orthonormal wavelets from certain families of Hardy functions. Inner functions and the corresponding backward shift invariant subspaces articulate the structure of these families. The new algorithms focus on the local shape of the wavelet.

Equivalence of Haar Bases Associated with Different Dyadic Systems

In this note we give sufficient conditions on two dyadic systems on a space of homogeneous type in order to obtain the equivalence of corresponding Haar systems on Lebesgue spaces. The main tool is the vector-valued Fefferman–Stein inequality for the Hardy–Littlewood maximal operator.

The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets

An efficient numerical method based on uniform Haar wavelets is proposed for the numerical solution of second-order boundary-value problems (BVPs) arising in the mathematical modeling of deformation of beams and plate deflection theory, deflection of a cantilever beam under a concentrated load, obstacle problems and many other engineering applications. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. The performance of the Haar wavelets is compared with the Walsh wavelets, semi-orthogonal B-spline wavelets, spline functions, Adomian decomposition method (ADM), finite difference method, and Runge–Kutta method coupled with nonlinear shooting method. A more accurate solution can be obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents the solution of a given problem. Through this analysis the solution is found on the coarse grid points, and then refined towards higher accuracy by increasing the level of the Haar wavelets. Neumann’s boundary conditions which are problematic for most of the numerical methods are automatically coped with. The main advantage of the Haar wavelet based method is its efficiency and simple applicability for a variety of boundary conditions. The convergence analysis of the proposed method alongside numerical procedure for multi-point boundary-value problems are given to test wider applicability and accuracy of the method.

On the convergence of greedy algorithms for initial segments of the Haar basis

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 &lt; p &lt; ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

An Adaptive Wavelet Method for the Chemical Master Equation

An adaptive wavelet method for the chemical master equation is constructed. The method is based on the representation of the solution in a sparse Haar wavelet basis, the time integration by Rothe's method, and an iterative procedure which in each time-step selects those degrees of freedom which are essential for propagating the solution. The accuracy and efficiency of the approach is discussed, and the performance of the adaptive wavelet method is demonstrated by numerical examples.

An example of an almost greedy basis in 𝐿¹(0,1)

We give an explicit construction of an almost greedy basis of L 1 ( 0 , 1 ) L^1(0,1) , complementing the results on existence of such a basis. The basis is described in terms of the Haar basis. We construct a quasi-greedy basis in a Banach space which is isomorphic to L 1 ( 0 , 1 ) L ^1(0,1) , and then we calculate an isomorphic image of a quasi-greedy basis.

Extrapolation of vector-valued rearrangement operators

We study the extrapolation properties of vector valued rearrangement operators acting on the normalized Haar basis in $L^p_X .$

A methodology for the analysis of differential coexpression across the human lifespan

BackgroundDifferential coexpression is a change in coexpression between genes that may reflect 'rewiring' of transcriptional networks. It has previously been hypothesized that such changes might be occurring over time in the lifespan of an organism. While both coexpression and differential expression of genes have been previously studied in life stage change or aging, differential coexpression has not. Generalizing differential coexpression analysis to many time points presents a methodological challenge. Here we introduce a method for analyzing changes in coexpression across multiple ordered groups (e.g., over time) and extensively test its validity and usefulness.ResultsOur method is based on the use of the Haar basis set to efficiently represent changes in coexpression at multiple time scales, and thus represents a principled and generalizable extension of the idea of differential coexpression to life stage data. We used published microarray studies categorized by age to test the methodology. We validated the methodology by testing our ability to reconstruct Gene Ontology (GO) categories using our measure of differential coexpression and compared this result to using coexpression alone. Our method allows significant improvement in characterizing these groups of genes. Further, we examine the statistical properties of our measure of differential coexpression and establish that the results are significant both statistically and by an improvement in semantic similarity. In addition, we found that our method finds more significant changes in gene relationships compared to several other methods of expressing temporal relationships between genes, such as coexpression over time.ConclusionDifferential coexpression over age generates significant and biologically relevant information about the genes producing it. Our Haar basis methodology for determining age-related differential coexpression performs better than other tested methods. The Haar basis set also lends itself to ready interpretation in terms of both evolutionary and physiological mechanisms of aging and can be seen as a natural generalization of two-category differential coexpression.Contact: paul@bioinformatics.ubc.ca

On the unconditional subsequence property

We show that a construction of Johnson, Maurey and Schechtman leads to the existence of a weakly null sequence ( f i ) in ( ∑ L p i ) ℓ 2 , where p i ↓ 1 , so that for all ε > 0 and 1 < q ⩽ 2 , every subsequence of ( f i ) admits a block basis ( 1 + ε ) -equivalent to the Haar basis for L q . We give an example of a reflexive Banach space having the unconditional subsequence property but not uniformly so.

An implicit wavelet sparse approximate inverse preconditioner using block finger pattern

AbstractRecently, an implicit wavelet sparse approximate inverse (IW‐SPAI hereafter) preconditioner has been proposed by Hawkins and Chen for nonsymmetric linear systems. The preconditioning matrix of the IW‐SPAI is characterized by a special sparse structure, the so‐called finger pattern, which makes it possible to construct a good sparse approximate inverse. Since the IW‐SPAI requires a number of QR factorizations with substantial costs to construct the preconditioner, there is a strong need to construct the IW‐SPAI more efficiently. The target of this paper is to reduce the costs of the IW‐SPAI using Haar basis. Under the strategy of classifying the QR factorizations into several groups and reducing the number of QR factorizations, in this paper, we introduce a block finger pattern for Haar basis in the place of the finger pattern as the nonzero pattern of the preconditioning matrix. From the block finger pattern, we propose an implicit wavelet sparse approximate inverse preconditioner using block finger pattern (BIW‐SPAI hereafter). Numerical experiments indicate that the BIW‐SPAI is often more efficient than the IW‐SPAI. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Upscaling as ecological information transfer: a simple framework with application to Arctic ecosystem carbon exchange

Transferring ecological information across scale often involves spatial aggregation, which alters information content and may bias estimates if the scaling process is nonlinear. Here, a potential solution, the preservation of the information content of fine-scale measurements, is highlighted using modeled net ecosystem exchange (NEE) of an Arctic tundra landscape as an example. The variance of aggregated normalized difference vegetation index (NDVI), measured from an airborne platform, decreased linearly with log(scale), resulting in a linear relationship between log(scale) and the scale-wise modeled NEE estimate. Preserving three units of information, the mean, variance and skewness of fine-scale NDVI observations, resulted in upscaled NEE estimates that deviated less than 4% from the fine-scale estimate. Preserving only the mean and variance resulted in nearly 23% NEE bias, and preserving only the mean resulted in larger error and a change in sign from CO2 sink to source. Compressing NDVI maps by 70–75% using wavelet thresholding with the Haar and Coiflet basis functions resulted in 13% NEE bias across the study domain. Applying unique scale-dependent transfer functions between NDVI and leaf area index (LAI) decreased, but did not remove, bias in modeled flux in a smaller expanse using handheld NDVI observations. Quantifying the parameters of statistical distributions to preserve ecological information reduces bias when upscaling and makes possible spatial data assimilation to further reduce errors in estimates of ecological processes across scale.

Discrete Fourier analysis and wavelets: applications to signal and image processing

Preface. Acknowledgments. 1. Vector Spaces, Signals, and Images. 1.1 Overview. 1.2 Some common image processing problems. 1.3 Signals and images. 1.4 Vector space models for signals and images. 1.5 Basic wave forms the analog case. 1.6 Sampling and aliasing. 1.7 Basic wave forms the discrete case. 1.8 Inner product spaces and orthogonality. 1.9 Signal and image digitization. 1.10 Infinitedimensional inner product spaces. 1.11 Matlab project. Exercises. 2. The Discrete Fourier Transform. 2.1 Overview. 2.2 The time domain and frequency domain. 2.3 A motivational example. 2.4 The onedimensional DFT. 2.5 Properties of the DFT. 2.6 The fast Fourier transform. 2.7 The twodimensional DFT. 2.8 Matlab project. Exercises. 3. The discrete cosine transform. 3.1 Motivation for the DCT: compression. 3.2 Initial examples thresholding. 3.3 The discrete cosine transform. 3.4 Properties of the DCT. 3.5 The twodimensional DCT. 3.6 Block transforms. 3.7 JPEG compression. 3.8 Matlab project. Exercises. 4. Convolution and filtering. 4.1 Overview. 4.2 Onedimensional convolution. 4.3 Convolution theorem and filtering. 4.4 2D convolution filtering images. 4.5 Infinite and biinfinite signal models. 4.6 Matlab project. Exercises. 5. Windowing and Localization. 5.1 Overview: Nonlocality of the DFT. 5.2 Localization via windowing. 5.3 Matlab project. Exercises. 6. Filter banks. 6.1 Overview. 6.2 The Haar filter bank. 6.3 The general onstage twochannel filter bank. 6.4 Multistage filter banks. 6.5 Filter banks for finite length signals. 6.6 The 2D discrete wavelet transform and JPEG 2000. 6.7 Filter design. 6.8 Matlab project. 6.9 Alternate Matlab project. Exercises. 7. Wavelets. 7.1 Overview. 7.2 The Haar Basis. 7.3 Haar wavelets versus the Haar filter bank. 7.4 Orthogonal wavelets. 7.5 Biorthogonal wavelets. 7.6 Matlab Project. Exercises. References. Solutions. Index.

A wavelet method for solving singular integral equation of MHD

This paper presents Haar wavelet approximation to solve a singular integral equation which has singularities on a diagonal of the domain R = [ - 1 , 1 ] × [ - 1 , 1 ] . The singularities arise basically due to modified Bessel function K 0 which appears as a part of the kernel. Thus the integral equation is weakly (logarithmically) singular only. The problem is solved considering all the singularities of the kernel and results are examined for approximations of different orders. Our interest to solve the problem using Haar wavelet basis is due to its simplicity and efficiency in numerical approximation. The results show convergence trend as mesh is refined. Comparison is made with some available results obtained earlier by partial consideration of the singularities.

MULTIRESOLUTION ANALYSIS AND HAAR WAVELETS ON THE PRODUCT OF HEISENBERG GROUP

In this paper, the properties of multiresolution analysis on the product of Heisenberg group are investigated. Moreover, orthonormal Haar wavelet bases for L2(ℋn) are constructed.

BCR algorithm and the T(b) Theorem

We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on Lp, 1 < p < ∞, and of a perfect dyadic singular integral operator. This allows to deduce a local T(b) theorem for singular integral operators from the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Tao, Thiele and the first author.

Multiresolution Analysis and Haar Wavelets on the Laguerre Hypergroup

Let ℍn be the Heisenberg group. The fundamental manifold of the radial function space for ℍn can be denoted by [0, +∞) × ℝ, which is just the Laguerre hypergroup. In this paper the multiresolution analysis on the Laguerre hypergroup 𝕂 = [0, +∞) × ℝ is defined. Moreover the properties of Haar wavelet bases for are investigated.