Showing papers on "Biorthogonal system published in 1993"
TL;DR: One more model of aq-harmonic oscillator based on the q-orthogonal polynomials of Al-Salam and Carlitz is discussed in this paper.
Abstract: One more model of aq-harmonic oscillator based on theq-orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form ofq-creation andq-annihilation operators,q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.
72 citations
TL;DR: In this article, a model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz (1965) is discussed.
Abstract: A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz (1965) is discussed. A simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analogue of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed.
72 citations
TL;DR: In this article, the notion of orthonormal wavelet packets introduced by Coifman and Meyer is generalized to the nonorthogonal setting in order to include compactly supported and symmetric basis functions.
Abstract: The notion of orthonormal wavelet packets introduced by Coifman and Meyer is generalized to the nonorthogonal setting in order to include compactly supported and symmetric basis functions. In particular, dual (or biorthogonal) wavelet packets are investigated and a stability result is established. Algorithms for implementations are also developed.
67 citations
TL;DR: In this paper, a compactly supported biorthogonal wavelet basis adapted to some simple differential operators was constructed and the condition numbers of the corresponding stiffness matrices were estimated.
Abstract: In this paper we construct a compactly supported biorthogonal wavelet basis adapted to some simple differential operators. Moreover, we estimate the condition numbers of the corresponding stiffness matrices.
63 citations
TL;DR: A set of signature sequences is presented which optimizes performance in a direct sequence CDMA system with (a) synchronous transmission, (b) no multipath time dispersion, and (c) orthogonal or biorthogonal Walsh-Hadamard coding as a means of spreading the information signal.
Abstract: In code-division multiple-access (CDMA) systems, recent attention has focused on the use of orthogonal coding to provide spreading. Each signal is coded with the same orthogonal or biorthogonal code, followed by a modulo-2 addition of a unique signature sequence. The set of signature sequences used determines how much signals interfere with each other at a receiver, thus determining the performance of the system. An analysis is presented to determine the properties of an optimal set of signature sequences for such a system. Using a Kerdock code, a set of signature sequences is presented which optimizes performance in a direct sequence CDMA system with (a) synchronous transmission, (b) no multipath time dispersion, and (c) orthogonal or biorthogonal Walsh-Hadamard coding as a means of spreading the information signal. For a length-N-binary code (where N is an even power of two), the set contains N/2 signature sequences. Approaches are discussed for the cases when N is an odd power of two and when more sequences are needed. >
42 citations
TL;DR: New general recurrence relations that connect successive pairs in any given subsequence of all existing formally biorthogonal polynomials are presented, which immediately lead to the proposed look-ahead Levinson algorithm for solving Toeplitz systems.
38 citations
01 Jan 1993
TL;DR: In this article, a general set of biorthogonal rational functions, considered previously by Rahman and Wilson, is shown to satisfy a second-order linear difference equation of a nonuniform lattice.
Abstract: A general set of biorthogonal rational functions, considered previously by Rahman and Wilson, is shown to satisfy a second-order linear difference equation of a nonuniform lattice. In the spirit of Hahn’s approach for orthogonal polynomials, raising and lowering operators as well as a Rodriguez-type formula are obtained for these functions which contain the classical orthogonal polynomials as limiting cases. Their biorthogonality in the discrete case is established by means of a Sturm-Liouville type argument. An outline of Wilson’s technique for representing them as Gram determinants is also given.
26 citations
TL;DR: A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz is discussed in this paper, where simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are found.
Abstract: A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed.
19 citations
22 Oct 1993
TL;DR: It is observed that for a comparable peak SNR the contours are better reconstructed with biorthogonal solutions, and the increased regularity has for effect to increase the coding gain, to reach a maximum coding gain for maximally regular Daubechies filters.
Abstract: This paper addresses the problem of joint optimization of wavelet transform, quantization, and data rate allocation according to mathematical criteria for high compression efficiency of image coding algorithms. The relevancy of some filter bank properties for compression purposes is evaluated. Using lattice structures, a large number of orthogonal and biorthogonal wavelet filter banks, with different properties of regularity, coding gain, phase linearity, and cross-correlation between adjacent bands are designed. Scalar and lattice vector quantization is then optimized adaptively to filter bank characteristics and to signal statistics. An appropriate choice of transition bandwidth, decreasing the energy around the Nyquist frequency without constraints of `zeros' in (omega) equals (pi) , provides by the maximum selectivity criterion filter banks close in performance to filters that we found optimum, and designed to satisfy either the maximum coding gain or minimum cross-correlation criterion. For a lower transition bandwidth, the increased regularity has for effect to increase the coding gain, to reach a maximum coding gain for maximally regular Daubechies filters. When comparing results of coding with the optimal orthogonal wavelet filter bank with those provided by a maximally frequency selective biorthogonal solution with same regularity it is observed that for a comparable peak SNR the contours are better reconstructed with biorthogonal solutions.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
15 citations
TL;DR: An analytical criterion evaluating the performances of any perfect reconstruction linear transform in the frame of scene adaptive coding is derived, thereafter, used in order to optimize linear multiresolution transforms.
Abstract: Scene adaptive coders are constituted by the cascade of a linear transform, scalar quantization, entropy coding and a buffer controlled by a feedback loop for bit rate regulation. The main contribution of this paper is to derive an analytical criterion evaluating the performances of any perfect reconstruction linear transform in the frame of scene adaptive coding. This criterion is, thereafter, used in order to optimize linear multiresolution transforms. The optimization adapts the filters parameters to the codec features and to the statistics of the 2-D sources; so, the authors call these transforms adapted multiresolution transforms (AMTs). The transforms under study are implemented by a cascade of separable perfect-reconstruction (PR) FIR two-band filter banks that can change at each resolution level. Two types of filter banks are envisaged: the PR orthogonal quadrature mirror filter (QMF) bank, which allows to implement the orthogonal AMT and the PR linear-phase filter (LPF) bank, which implements the biorthogonal AMT. They perform the optimization of the filters in their factorized lattice form, taking the finite length of the multipliers into account. Their criterion also allows them to show the performances achieved by these two linear multiresolution transforms compared to other linear (multiresolution) transforms. >
01 Jan 1993
TL;DR: A look-ahead variant of the Lanczos process is described that remedies the problems of the original algorithm for computing inverse triangular factorizations of Hankel and Toeplitz matrices.
Abstract: The nonsymmetric Lanczos process can be used to compute approximate eigenvalues of large non-Hermitian matrices, or to obtain approximate solutions of large non-Hermitian linear systems. However, the Lanczos algorithm in its original form is susceptible to possible breakdowns and potential instabilities. We describe a look-ahead variant of the Lanczos process that remedies the problems of the original algorithm. We also discuss related algorithms for the construction of formally orthogonal and biorthogonal polynomials, and for computing inverse triangular factorizations of Hankel and Toeplitz matrices.
TL;DR: In this paper, the use of biorthogonal valence bond reference functions in evaluating the correlation energy is investigated, and the procedure adopted here is a matrix element driven configuration interaction scheme.
Abstract: The use of biorthogonal valence bond reference functions in evaluating the correlation energy is investigated. Since the method is not variationally bound some care must be taken in defining the reference state to ensure that the variational bound is not violated, some discussion is given to this matter. The procedure adopted here is a matrix element driven configuration interaction scheme. To reduce the computational labour involved, a configuration selection criterion is introduced. The method is tested through its application to the symmetric stretching of HF, H2O, (2
B
1) NH2 and the singlet-triplet gap in CH2. Comparison is made with other methods, including full CI. The results show that the current method is quite promising.
01 Jan 1993
TL;DR: Note: M. Farges et al, Eds.
Abstract: Note: M. Farges et al, Eds. Reference LCAV-CHAPTER-2005-011 Record created on 2005-06-27, modified on 2017-05-12
Journal Article•
TL;DR: It is shown that several of these fundamental formulas can be obtained directly from Bessel's equality for biorthogonal systems, aided only by the shift theorem for inner products of Gabor basis functions.
TL;DR: In this article, the authors show how Lynden-Bell's generalization of Mestel's method of flattened spheroids can be adapted and simplified to find biorthogonal potential density sets for non-axisymmetric flat discs.
Abstract: We show how Lynden-Bell's generalization of Mestel's method of flattened spheroids can be adapted and simplified to find biorthogonal potential-density sets for non-axisymmetric flat discs. The simplified formulae turn out to be equivalent to formulae derived by Kalnajs using a quite different method. We then use the simplified formulae to obtain two new biorthogonal potential-density sets. One is related to Gaussian discs and the other to general Toomre models, and both should be useful for stability analyses of these discs. We also give, in an appendix, a large new family of potential-density pairs described by Meijer's G-function. An interesting member of this family has density components that are infinite at the centre, yet decay exponentially at large distance
TL;DR: In this paper, it was shown that for any multi-resolution analysis of L2(R) with multiplicity d, dilation factor A (where A is an integer = 2) and with compactly supported scaling functions, one can associate wavelets.
Abstract: We show that to any multi-resolution analysis of L2(R) with multiplicity d, dilation factor A (where A is an integer = 2) and with compactly supported scaling functions we may associate compactly supported wavelets. Conversely, if (?e,j,k = Aj/2 ?e (Ajx - k)), 1 = e = E and j, k I Z, is a Hilbertian basis of L2(R) with continuous compactly supported mother functions ?e, then it is provided by a multi-resolution analysis with dilation factor A, multiplicity d = E / (A - 1) and with compactly supported scaling functions (which have the same regularity as the wavelets ?e). Those results can be extended to the cases of exponentially localized functions and of biorthogonal wavelets.
01 May 1993
TL;DR: In this paper, a wavelet method for the multiresolution representation of parametric curves and surfaces was developed, and a new family of compactly supported symmetric biorthogonal wavelets with interpolating scaling functions was constructed.
Abstract: We develop wavelet methods for the multiresolution representation of parametric curves and surfaces. To support the representation, we construct a new family of compactly supported symmetric biorthogonal wavelets with interpolating scaling functions. The wavelets in these biorthogonal pairs have properties better suited for curves and surfaces than many commonly used filters. We also give examples of the applications of the wavelet approach: these include the derivation of compact hierarchical curve and surface representations using modified wavelet compression, identifying smooth sections of surfaces and a subdivision-like intersection algorithm for discrete plane curves.
TL;DR: In this paper, the principal parts of the inverse of a holomorphic Fredholm valued operator function can be written as sums of tensor products of biorthogonal canoncial systems of eigenvectors and associated vectors of the operator function and its adjoint.
Abstract: In Banach spaces the principal parts of the inverse of a holomorphic Fredholm valued operator function can be written as sums of tensor products of biorthogonal canoncial systems of eigenvectors and associated vectors of the operator function and its adjoint. If we impose a symmetry condition on the operator function, then for real eigenvalues the eigenvectors and associated vectors of the operator function and its adjoint coincide at this point. We shall show that we can always choose a suitable canonical system of eigenvectors and associated vectors which is orthogonal, i.e., biorthogonal to itself. An application to elliptic differential operators is given.
01 Jan 1993
TL;DR: An analytical criterion evaluating the performances of any perfect reconstruction linear transform in the frame of scene adaptive coding is derived, thereafter, used in order to optimize linear multiresolution transforms.
Abstract: Scene adaptive coders are constituted by the cascade of a linear transform, scalar quantization, entropy coding and a huffer con- trolled by a feedback loop for hit rate regulation. The main contrihu- tion of this paper is to derive an analytical criterion evaluating the performances of any perfect reconstruction linear transform in the frame of scene adaptive coding. This criterion is, thereafter, used in order to optimize linear multiresolution transforms. The optimization adapts the filters parameters to the codec features and to the statistics of the 2-D sources; so, we call these tranforms adapted multiresolution transforms (AMT's). The transforms under study are implemented by a cascade of separable perfect-reconstruction (PR) FIR two-band filter hanks that can change at each resolution level. Two types of filter banks are envisaged: the PR orthogonal quadrature mirror filter (QMF) bank, which allows to implement the orthogonal AMT and the PR linear-phase filter (LPF) bank, which implements the biorthogonal AMT. We per- form the optimization of the filters in their factorized lattice form, tak- ing the finite length of the multipliers into account. Our criterion also allows to show the performances achieved by these two linear multi- resolution transforms compared to other linear (multiresolution) transforms.
TL;DR: A stationary band-limited process is used to construct a wavelet basis and this basis is modified to obtain a biorthogonal sequence which in turn is usedto obtain a series representation of the process with uncorrelated coefficients.
Abstract: A stationary band-limited process is used to construct a wavelet basis. This basis is modified to obtain a biorthogonal sequence which in turn is used to obtain a series representation of the process with uncorrelated coefficients.
01 Nov 1993
TL;DR: In this paper, the Gabor coefficients can be computed with a biorthogonal function or the Zak transform, and relations between a window function and its biorhogonal functions are discussed.
Abstract: In this paper, the theory and computations for the Gabor transform are discussed. The Gabor coefficients can be computed with a biorthogonal function or the Zak transform. Relations between a window function and its biorthogonal function are discussed. The formulas derived for the continuous variable Gabor transform with Zak transforms can be applied to the discrete Gabor transform by replacing the Zak transforms with the discrete Fourier transforms. The generalized Gabor transform are also discussed. Relations between a window function and its biorthogonal functions are presented. In the case of the generalized Gabor transform, the biorthogonal functions are not unique. The optimal biorthogonal functions are discussed. A relation between a window function and its optimal biorthogonal function is presented based on the Zak transform when T/T' is rational. The finite discrete generalized Gabor transform is also derived. The relations between a window function and its optimal biorthogonal function derived for the continuous variable generalized Gabor transform can be extended to the finite discrete case.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
01 Jan 1993
TL;DR: In this paper, the authors consider the case where the closed unit ball of a weak-star Corson compact is an infinite dimensional closed subspace of the Banach space X such that the density character of Y is greater or equal than that of X* for the weak star topology.
Abstract: Let Y be an infinite dimensional closed subspace of the Banach space X such that the density character of Y is greater or equal than the density character of X* for the weak-star topology. If the closed unit ball of Y* is a weak-star Corson compact then there is a total biorthogonal system (xi, μ i)i ∈ I for X such that [xi: i ∈ I] = Y.
TL;DR: In this article, the definitions of basic splines are extended to negative integers and the translates of these negative splines were shown to be biorthogonal to the translations of the usual positive splines.
TL;DR: In this article, a biorthogonal series method is developed to solve Oseen type flow problems, which leads to a new set of eigenfunctions for a specific class of linear non-selfadjoint operators containing the biharmonic one.
01 Jan 1993
TL;DR: The missing label problem complicates considerably the operations of the Wigner-Racah calculus in the irreducible and tensor spaces of the Lie groups of the second and higher ranks, especially when the non-multiplicity-free decompositions of their direct product states appear as discussed by the authors.
Abstract: The missing label problem complicates considerably the operations of the Wigner-Racah calculus in the irreducible and tensor spaces of the Lie groups of the second and higher ranks, especially, when the non-multiplicity-free decompositions of their direct product states appear as well as for non-canonical chains of subgroups used in the nuclear theory and other fields of the contemporary theoretical physics. Beginning from Racah, the impossibility of the analytical constructions of the orthogonal bases and coupling coefficients was supposed1 in the case of the non-trivial multiplicities of the repeating irreducible representations (irreps). Really, the eigenvalue problem for the labeling operators and the diagonalization of the overlap matrices or the κ2 operator matrices of the extended coherent state theory (Rowe, Quesne, Hecht) in general have only the numerical solutions.
TL;DR: One more model of a q-harmonic oscillator based on the q-orthogonal polynomials of Al-Salam and Carlitz is discussed in this article.
Abstract: One more model of a q-harmonic oscillator based on the q-orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.