On Generalized Gaussian Quadratures for Exponentials and Their Applications
01 May 2002-Applied and Computational Harmonic Analysis (Academic Press)-Vol. 12, Iss: 3, pp 332-373
TL;DR: A generalization of a representation theorem due to Caratheodory is used to derive new families of Gaussian-type quadratures for weighted integrals of exponential functions and their applications to integration and interpolation of bandlimited functions.
About: This article is published in Applied and Computational Harmonic Analysis.The article was published on 2002-05-01 and is currently open access. It has received 89 citations till now. The article focuses on the topics: Toeplitz matrix & Bessel function.
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TL;DR: This paper further develops the separated representation by discussing the variety of mechanisms that allow it to be surprisingly efficient; addressing the issue of conditioning; and presenting algorithms for solving linear systems within this framework.
Abstract: Nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension. The separated representation, introduced previously, allows many operations to be performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by
(i) discussing the variety of mechanisms that allow it to be surprisingly efficient;
(ii) addressing the issue of conditioning;
(iii) presenting algorithms for solving linear systems within this framework; and
(iv) demonstrating methods for dealing with antisymmetric functions, as arise in the multiparticle Schrodinger equation in quantum mechanics.
Numerical examples are given.
393 citations
Cites methods from "On Generalized Gaussian Quadratures..."
...15) is addressed in [5] by extending methods in [4]....
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TL;DR: A new approach is introduced for the efficient approximation of functions and sequences by short linear combinations of exponential functions with complex-valued exponents and coefficients with significantly fewer terms than Fourier representations.
319 citations
Cites background or methods from "On Generalized Gaussian Quadratures..."
...Even though most of the results in [11] are based on the particular properties of band functions, and as such, cannot be directly obtained by the general method of this paper, som in [11] are immediate consequences of the general approach presented here....
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...The approach in this paper has grown from that in [11] where we used properties of band functions and of Hermitian Toeplitz matrices to construct solutions of (7)....
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...In our paper [11] weights are computed via a fast algorithm based on implementation of the re (19) for nodes on the unit circle....
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...As in our paper [11], we reformulate the continuous problem (6) as a discrete problem....
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...We also showed in [11] that even though Vande systems can be arbitrarily ill-conditioned, the approximation problem on the unit circle is well pose to the particular location of the nodes and the specific right-hand sides....
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TL;DR: A multiresolution solver for the all-electron local density approximation Kohn-Sham equations for general polyatomic molecules to a user-specified precision and the computational cost of applying all operators scales linearly with the number of parameters.
Abstract: We describe a multiresolution solver for the all-electron local density approximation Kohn-Sham equations for general polyatomic molecules. The resulting solutions are obtained to a user-specified precision and the computational cost of applying all operators scales linearly with the number of parameters. The construction and use of separated forms for operators (here, the Green’s functions for the Poisson and bound-state Helmholtz equations) enable practical computation in three and higher dimensions. Initial applications include the alkali-earth atoms down to strontium and the water and benzene molecules.
255 citations
TL;DR: The efficient approximation of functions by sums of exponentials or Gaussians in Beylkin and Monzon (2005) is revisited to discuss several new results and applications, and the Poisson summation is used to discretize integral representations of e.g., power functions r − β, β > 0.
167 citations
Cites background from "On Generalized Gaussian Quadratures..."
...We note that the special case of purely imaginary exponents ηm in (67) is treated in [15]....
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TL;DR: In this article, a wavelet basis consisting of sinc functions is replaced by one based on prolate spheroidal wave functions (PSWFs) which have much better time localization than the sinc function.
Abstract: The article is concerned with a particular multiresolution analysis (MRA) composed
of Paley–Wiener spaces. Their usual wavelet basis consisting of sinc functions is replaced by
one based on prolate spheroidal wave functions (PSWFs) which have much better time localization
than the sinc function. The new wavelets preserve the high energy concentration in both
the time and frequency domain inherited from PSWFs. Since the size of the energy concentration
interval of PSWFs is one of the most important parameters in some applications, we modify the
wavelets at different scales to retain a constant energy concentration interval. This requires a slight
modification of the dilation relations, but leads to locally positive kernels. Convergence and other
related properties, such as Gibbs phenomenon, of the associated approximations are discussed. A
computationally friendly sampling technique is exploited to calculate the expansion coefficients.
Several numerical examples are provided to illustrate the theory.
80 citations
Cites methods from "On Generalized Gaussian Quadratures..."
...However, modern computational facilities and the recently developed numerical algorithms of Beylkin and Monzon [ 1 ] and of Xiao, Rokhlin and Yarvin [26] promise to provide elegant numerical results with satisfactory complexity....
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References
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Book•
12 Jul 2010TL;DR: The field of values as discussed by the authors is a generalization of the field of value of matrices and functions, and it includes singular value inequalities, matrix equations and Kronecker products, and Hadamard products.
Abstract: 1. The field of values 2. Stable matrices and inertia 3. Singular value inequalities 4. Matrix equations and Kronecker products 5. Hadamard products 6. Matrices and functions.
7,013 citations
Posted Content•
18 Dec 2005
TL;DR: In this paper, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed and orthogonality on the unit circle is not discussed.
Abstract: In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed
5,648 citations
TL;DR: In this paper, the authors apply the theory developed in the preceding paper to a number of questions about timelimited and bandlimited signals, and find the signals which do the best job of simultaneous time and frequency concentration.
Abstract: The theory developed in the preceding paper1 is applied to a number of questions about timelimited and bandlimited signals. In particular, if a finite-energy signal is given, the possible proportions of its energy in a finite time interval and a finite frequency band are found, as well as the signals which do the best job of simultaneous time and frequency concentration.
2,498 citations
TL;DR: In this article, Toeplitz forms are used for the trigonometric moment problem and other problems in probability theory, analysis, and statistics, including analytic functions and integral equations.
Abstract: Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.
2,279 citations