Showing papers on "Orthonormal basis published in 1990"

A Karhunen-Loeve-like expansion for 1/f processes via wavelets

Abstract: While so-called 1/f or scaling processes emerge regularly in modeling a wide range of natural phenomena, as yet no entirely satisfactory framework has been described for the analysis of such processes. Orthonormal wavelet bases are used to provide a new construction for nearly 1/f processes from a set of uncorrelated random variables. >

Receptive field families

Abstract: It is generally agreed upon that size invariance and the absence of spurious resolution are two requirements that characterize well behaved spatial samping in visual systems We show that these properties taken together constrain the structure of receptive fields to a very large degree Only those field structures that arise as solutions of a certain linear partial differential equation of the second order prove to satisfy these general constraints The equation admits of complete, orthonormal families of solutions These families have to be regarded as the possible receptive field families subject to certain symmetry conditions They can be transformed into each other via unitary transformations Thus a single representation suffices to construct all the others This theory permits us to classify the possible linear receptive field structures exhaustively, and to define their internal and external interrelations This induces a principled taxonomy of linear receptive fields

The power radiated by a vibrating body in an acoustic fluid and its determination from boundary measurements

Abstract: The power radiated into a specified angular sector by a vibrating object immersed in a fluid is expressed as a quadratic functional of the boundary normal velocity field. The diagonalization of the functional, obtained through the singular value decomposition of its discretized version, identifies a set of orthonormal boundary velocity patterns, each corresponding to a far‐field pattern belonging to a set of functions orthonormal in the angular sector of interest. Any boundary normal velocity field can be represented as a linear superposition of the orthonormal patterns. The velocity patterns having high radiation efficiencies form a subset, whose dimension depends upon the object boundary shape and size in wavelengths. The other velocity patterns do not radiate efficiently and contribute mainly to the evanescent field in the neighborhood of the object. Assuming that some noise is present, only the radiation patterns associated with the efficiently radiating velocity patterns are observable in the far field. Therefore, the dimension of their set defines the number of degrees of freedom of the far field. The efficiently radiating velocity patterns constitute a set of spatial filtering functions, separating the radiating from the essentially nonradiating components of an arbitrary boundary normal velocity field.

Tight frames of compactly supported affine wavelets

Abstract: This paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by Daubechies [Commun. Pure Appl. Math. 41, 909 (1988)]. For each integer N≥1, a family of wavelet functions ψ having support [0,2N−1] is constructed such that {ψjk(x)=2j/2ψ(2jx−k) kj,k∈Z} is a tight frame of L2(R), i.e., for every f∈L2(R), f=c∑jk 〈ψjk‖f〉ψjk for some c>0. This family is parametrized by an algebraic subset VN of R4N. Furthermore, for N≥2, a proper algebraic subset WN of VN is specified such that all points in VN outside of WN yield orthonormal bases. The relationship between these tight frames and the theory of group representations and coherent states is discussed.

Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation

Abstract: The Burgers equation with a small viscosity term, initial and periodic boundary conditions is resolved using a spatial approximation constructed from an orthonormal basis of wavelets. The algorithm is directly derived from the notions of multiresolution analysis and tree algorithms. Before the numerical algorithm is described these notions are first recalled. The method uses extensively the localization properties of the wavelets in the physical and Fourier spaces. Moreover, the authors take advantage of the fact that the involved linear operators have constant coefficients. Finally, the algorithm can be considered as a time marching version of the tree algorithm. The most important point is that an adaptive version of the algorithm exists: it allows one to reduce in a significant way the number of degrees of freedom required for a good computation of the solution. Numerical results and description of the different elements of the algorithm are provided in combination with different mathematical comments on the method and some comparison with more classical numerical algorithms.

Stability For a Multidimensional Inverse Spectral Theorem

Abstract: We consider the eigenvalue problem in Ω Where Ω is a bounded domain in Rd with smooth boundary,a nd q is a bounded, measurable function on Ω The eigenvalue problem has discrete spectrum; we denote by and a nondecreasing sequence of eigenvalue and corresponding (orthonormal) eigenfunctions. It is known ([N–S–U]) that knowledge of the eigenvalues and the boundary values of the normal derivatives of the corresponding eigenfunctions is sufficient to uniquely determine a coefficient, q.

Approximate system identification using system based orthonormal functions

Abstract: The problem of approximate system identification is addressed. The use of prefilters that are based on a special class of system based orthonormal functions is proposed. It is shown that every linear finite dimensional time invariant discrete time system gives rise to two sets of orthonormal functions and that both can be used as a basis of the space l/sub 2/. The derivation of these functions, to be considered as a generalization of the Laguerre polynomials, is based on the properties of discrete time all-pass functions. Transformation of the input/output signals of a linear system in terms of these functions leads to new system descriptions, and new possibilities arise for the construction of approximate identification methods, with favorable properties allowing the use of simple estimation techniques and a systematic choice of prefilters. >

A new implementation of the Lanczos method in linear problems

Abstract: The Lanczos algorithm has proved to be a powerful solution method not only for finding the eigenvalues but for solving linear systems of equations. In this work a new implementation of the algorithm is presented for solving linear systems of equations with a sequence of right-hand sides. The versions of the method proposed in the past treat the right-hand side vectors successively by keeping the tridiagonal matrix and the orthonormal basis in fast or secondary storage. The new technique handles all approximations to the solution vectors simultaneously without the necessity for keeping the tridiagonal matrix or the orthonormal basis in fast or secondary storage. Thus, when the first solution vector has converged to a required accuracy good approximations to the remaining solution vectors have simultaneously been obtained. It then takes fewer iterations to reach the final accuracy by working separately on each of the remaining vectors.

A general solution of the weighted orthonormal procrustes problem

Abstract: A general solution for weighted orthonormal Procrustes problem is offered in terms of the least squares criterion. For the two-demensional case. this solution always gives the global minimum; for the general case, an algorithm is proposed that must converge, although not necessarily to the global minimum. In general, the algorithm yields a solution for the problem of how to fit one matrix to another under the condition that the dimensions of the latter matrix first are allowed to be transformed orthonormally and then weighted differentially, which is the task encountered in fitting analogues of the IDIOSCAL and INDSCAL models to a set of configurations.

Robust eigensystem assignment for second-order dynamic systems

Abstract: A novel approach for the robust eigensystem assignment of flexible structures using full state or output feedback is developed. Using the second-order dynamic equations, the approach can assign the eigenvalues of the system via velocity and displacement feedbacks, or acceleration and velocity feedbacks. The eigenvalues and eigenvectors of the system are assigned, via the second-order eigenvalue problem for the structural system, in two steps. First, an orthonormal basis spanning the attainable closed-loop eigenvector space corresponding to each desired closed-loop eigenvalue is generated using the Singular Value or QR decompositions. Second, a sequential procedure is used to choose a set of closed-loop eigenvectors that are as close as possible to the column space of a well-conditioned target matrix. Among the possible choices of the target matrix, the closest unitary matrix to the open-loop eigenvector matrix appears to be a suitable choice. A numerical example is given to illustrate the proposed algorithm.

Relative asymptotics for polynomials orthogonal on the real axis

Abstract: Given a positive Borel measure on the real line and a function on , the main purpose of the paper is to prove (under certain assumptions on ) relative asymptotic formulas of the type where , is Szego's function corresponding to and the function , and are polynomials orthonormal relative to the measures and respectively.Bibliography: 15 titles.

Perfect reconstruction binomial QMF-wavelet transform

Abstract: This paper describes a class of orthogonal binomial filters which provide a set of basis functions for a bank of perfect reconstruction Finite Impulse Response Quadrature Mirror Filters (FIR-QMF). These Binomial QMFs are shown to be the same filters as those derived from a discrete orthonormal wavelet approach by Daubechies [13]. The proposed filters can be implemented very efficiently with output scaling but otherwise no multiply operations. The cornpaction performance of the proposed signal decomposition technique is computed and shown to be better than that of the DCT for the AR(1) signal models and also for standard test images.© (1990) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A Fourier-Series Method for Solving Soliton Problems

Abstract: A Fourier–Galerkin method with an earlier proposed complete orthonormal system of functions in $L^2 ( - \infty ,\infty )$ as the set of trial functions is developed and displayed for the problem of calculating the shape of the one-soliton solution of the Korteweg–de Vries equation. The convergence of the method is investigated through comparison with the analytic solution, which appears to be very good. The truncation and discretization errors are assessed pointwise. The technique developed is also applied to the soliton problem for the so-called Kuramoto–Sivashinsky equation and the obtained soliton shape is compared to the existing difference solution. The quantitative agreement between the Fourier-series-method result and the numerical one is good. In the present paper, however, the soliton solution is obtained for a significantly wider range of phase velocities, which suggests that the spectrum might be continuous. The new technique can also be applied to a variety of other problems involving identification of homoclinic solutions.

Hilbert lattices: New results and unsolved problems

Abstract: The class of Hilbert lattices that derive from orthomodular spaces containing infinite orthonormal sets (normal Hilbert lattices) is investigated. Relevant open problems are listed. Comments on form-topological orthomodular spaces and results on arbitrary orthomodular spaces are appended.

Explicit orthonormal Clebsch-Gordan coefficients of SU(3)

Abstract: The construction of the explicit algebraic‐polynomial expressions for the nonmultiplicity‐free orthonormal Clebsch–Gordan (Wigner) coefficients of SU(3)⊇U(2) is completed in the case of the paracanonical coupling scheme related with the explicit minimal biorthogonal systems by means of the Hecht or Gram–Schmidt process. The direct and inverse orthogonalization coefficients (the first of them being equivalent to the boundary orthonormal isofactors) are expressed, up to explicitly given multiplicative factors, in terms of the numerator and denominator polynomials related with the auxiliary Aλ function of Louck, Biedenharn, and Lohe that appears as a fragment of the denominator G‐functions of canonical SU(3) tensor operators.

Stochastic integration, trace and the skeleton of Wiener functionals

Abstract: It is shown that the notion of trace induced by a given complete orthonormal system relates the Skorohod integral with a corresponding Ogawa‐type integral evaluated with respect to the same orthonormal systems. Similarly the multiple Wiener‐Ito integral is shown to be related to a multiple Ogawa‐type integral induced by a complete orthonormal system via the Hu‐Meyer formula with suitably defined multiple traces. The notion of skeleton of a Wiener functional relative to a given orthonormal system is defined and yields what seems to be a “natural” extension of Wiener functionals to the Cameron Martin space and the Wiener processes with a different scale.

Diffraction by multiple slits

Abstract: Diffraction of a plane wave by multiple slits is investigated by a method that uses a set of orthonormal functions and Fourier transformations. A solution satisfying the wave equation and the boundary conditions is obtained. Expressions for the transmission coefficient and the far field are derived from the solution, and some numerical results are given.

Classical solutions of the many-time functional equations of motion of the nambu string

Abstract: In this paper the many-time approach for the open Nambu string is developed This enables discussion of the model in a general gauge The corresponding functional equations of motion are solved; the restriction to the orthonormal gauge gives the standard results These solutions are a preliminary and necessary step in the search for a complete set of observables, which will be the argument of a subsequent paper

Convergencia en Lp con pesos de la Serie de Fourier respecto de algunos sistemas ortogonales

Abstract: Convergence of the fourier series with respect to several orthogonal systems abstract Given an orthogonal complete system with respect to a measure µ on an interval, we approach the convergence in weighted Lp spaces of the corresponding Fourier series The orthogonal systems we analyze are generalized Jacobi, generalized Hermite, several systems of orthogonal polynomials with respect to weights with Dirac's deltas and Bessel and Dini's systems In this study we use good estimates of the orthonormal functions and results on Ap-theory in order to solve the boundedness of the Hilbert transform with one or two weights

Feature identification of ECG waveforms via orthonormal functions

Abstract: A new method, the orthonormal function model (OFM) is presented to identify the underlying features of electrocardiogram (ECG) waveforms. This method is based on the approximation properties of orthonormal functions. Among the set of all orthonormal functions, the Chebyshev polynomial has been selected, because it can uniformly approximate a broad class of functions and it gives the strongest convergence among all ultraspherical polynomials. Simulation results from normal and abnormal (ST depression) ECG waveforms indicate that the OFM has the following properties: (1) it has a residual error that decays to zero faster than for the linear predictor model (LPM): (2) it only needs a small model order for feature identification: (3) the model order for high resolution is smaller than for the LPM; and (4) most importantly the OFM can discriminate features from the ECG waveforms. The OFM can be successfully used for feature identification and as an aid in the classification and diagnosis of normal and abnormal patterns in the electrical activity of the heart. >

Bimodules over Cartan subalgebras

Abstract: Given a Cartan subalgebra A of a von Neumann algebra M, the techniques of Feldman and Moore are used to analyze the partial isometries v in M such that v∗Av is contained in A. Orthonormal bases for M consisting of such partial isometries are discussed, and convergence of the resulting generalized fourier series is shown to take place in the Bures A-topology. The Bures A-topology is shown to be equivalent to the strong topology on the unit ball of M. These ideas are applied to A-bimodules in M to prove the existence of orthonormal bases for bimodules and to give a simplified and intuitive proof of the Spectral Theorem for Bimodules first proven by Muhly, Saito, and Solel.

Receptive field taxonomy

Abstract: Given very general constraints, only a few receptive field types are admissible. These can be characterized completely through a certain second order partial differential equation. Families of receptive fields satisfying certain symmetry conditions can be constructed that are both orthonormal and complete.

Gabor and Wavelet Expansions

Abstract: This paper is an examination of techniques for obtaining Fourier series-like expansions of finite-energy signals using so-called Gabor and wavelet expansions. These expansions decompose a given signal into time a frequency localized components. The theory of frames in Hilbert spaces is used as a criteria for determining when such expansions are good representations of the signals. Some results on the existence of Gabor and wavelet frames in the Hilbert space of all finite-energy signals are presented.

Orthogonal discriminant analysis for interactive pattern analysis

Abstract: In general, a two-dimensional display is defined by two orthogonal unit vectors. In developing the display, discriminant analysis has the shortcoming that, in general, the extracted axes are not orthogonal. In order to overcome this shortcoming, the authors propose a discriminant analysis which provides an orthonormal system in the transformed space. The transformation preserves the discriminatory ability in terms of the Fisher ratio. The authors present a necessary and sufficient condition under which discriminant analysis in the original space provides an orthonormal system. Relationships between orthogonal discriminant analysis and the Karhunen-Loeve expansion in the original space are investigated. >

Theory of projected probabilities on non-orthogonal states: Application to electronic populations in molecules

Abstract: Often it is important to consider the expansion of a quantum state ⋎ ψ) in terms of physically meaningful basis states. For example, molecular orbitals can be expressed as linear combinations of atomic orbitals, or vibrational states can be expressed as super positions of local or normal mode eigenstates. In such expansions, it then becomes desirable to determine how much “character” a quantum state has in one of these basis states. One way of accimplishing this task is to calculate the projected probability of |ψ) on basis state |j). In this paper, we consider this general quantum mechanical problem. If the basis states are orthonormal, then the projected probability of|ψ) on |j) is of course | |2. However, if the basis states are not orthogonal, then this result is no longer valid and one must develop a more general theory to calculate these projected probabilities. An earlier paper used one-dimensional projection operators to initiate this theory and gave closed form results for the case of two non-orthogonal basis states [1]. One- and many-dimensional projection operators, together with linear algebraic techniques, are used to extend this theory to the n non-orthogonal basis state case. Explicit closed form results are given for the two- and three-state cases, and a general algorithm is developed for the case of four or more basis states. Application of the theory is made to atomic populations in three- to six-atom molecules, and comparisons are made to the related work of Mulliken.