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On Gabor frames generated by B-splines, totally positive functions, and hermite functions

APPLIED NUMERICAL MATHEMATICS 2025年 207卷 1-23页

作者： Ghosh, Riya Selvan, A. Antony Indian Inst Technol Dhanbad Dhanbad 826004 India

The frame set of a window phi is an element of L-2(R) is the subset of all lattice parameters (alpha,beta)is an element of R-+(2) such that G(phi,alpha,beta) ={e(2 pi i beta m)& sdot;phi(& sdot;-alpha k):k,m is an element of Z} forms a frame for L-2(R). In this paper, we investigate the frame set of B-splines, totally positive functions, and hermite functions. We derive a sufficient condition for Gabor frames using the connection between sampling theory in shift-invariant spaces and Gabor analysis. As a consequence, we obtain a new frame region belonging to the frame set of B-splines and hermite functions. For a class of functions that includes certain totally positive functions, we prove that for any choice of lattice parameters alpha, beta > 0 with alpha beta < 1, there exists a gamma>0 depending on alpha beta such that G(phi(gamma & sdot;),alpha,beta) forms a frame for L-2(R)

关键词： B-splines Gabor frames hermite functions Shift-invariant spaces Totally positive functions Two-sided exponential function

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Fourier transforms of time correlation functions using hermite functions

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COMPUTER PHYSICS COMMUNICATIONS 2025年 308卷

作者： Li, Hui Wang, Shaojie Zhang, Youjun Dove, Martin T. Sichuan Univ Inst Atom & Mol Phys Chengdu 610065 Sichuan Peoples R China Queen Mary Univ London Sch Phys & Chem Sci Mile End Rd London E1 4NS England

We present an alternative to standard Fourier transform methods in order to obtain the power spectrum from a time correlation function. Our approach involves fitting the correlation function with a sum of hermite functions, and recombining these to obtain the power spectrum directly. Although Fourier transform methods have been used for many decades, our approach avoids some ambiguities and uncertainties that face the user, and also allow for a more flexible form of the power spectrum to be obtained. We present a few examples to show the quality of the method.

关键词： Fourier transforms Time correlation functions hermite functions

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hermite functions and weighted spaces of generalized functions

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MANUSCRIPTA MATHEMATICA 2006年第3期119卷 269-285页

作者： Langenbruch, M Univ Oldenburg Dept Math D-26111 Oldenburg Germany

We prove that the hermite functions are an absolute Schauder basis for many weighted spaces of (ultra)differentiable functions and ultradistributions including the space of Fourier hyperfunctions. The coefficient spac... 详细信息

关键词： Generalized functions hermite functions Fourier

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Numerical approximations for population growth model by rational Chebyshev and hermite functions collocation approach: A comparison

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2010年第17期33卷 2076-2086页

作者： Parand, K. Rezaei, A. R. Taghavi, A. Shahid Beheshti Univ Dept Comp Sci GC Tehran Iran

This paper aims to compare rational Chebyshev (RC) and hermite functions (HF) collocation approach to solve Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions, which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro-differential equations. Copyright (C) 2010 John Wiley & Sons, Ltd.

关键词： collocation method spectral method Volterra's population model nonlinear integro-differential equation hermite functions rational Chebyshev numerical approximations population growth model

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A spectral viscosity method based on hermite functions for nonlinear conservation laws

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2008年第2期46卷 1060-1078页

作者： Aguirre, Julian Rivas, Judith Univ Basque Country Fac Ciencia & Tecnol Dept Matemat E-48080 Bilbao Spain

We consider the approximation by a spectral method of the solution of the Cauchy problem for a scalar conservation law in one dimension posed in the whole real line. We analyze a spectral viscosity method in which the orthogonal basis considered is the one of hermite functions. We prove the convergence of the approximate solution to the unique entropy solution of the problem by using compensated compactness arguments.

关键词： hyperbolic conservation laws hermite functions spectral methods spectral viscosity compensated compactness

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An approximate solution of the MHD Falkner-Skan flow by hermite functions pseudospectral method

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2011年第1期16卷 274-283页

作者： Parand, K. Rezaei, A. R. Ghaderi, S. M. Shahid Beheshti Univ Med Sci Dept Comp Sci GC Tehran Iran

Based on a new approximation method, namely pseudospectral method, a solution for the three order nonlinear ordinary differential laminar boundary layer Falkner-Skan equation has been obtained on the semi-infinite domain. The proposed approach is equipped by the orthogonal hermite functions that have perfect properties to achieve this goal. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, this method reduces solution of the problem to solution of a system of algebraic equations. We also present the comparison of this work with numerical results and show that the present method is applicable. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Falkner-Skan Pseudospectral method Semi-infinite Nonlinear ODE hermite functions Boundary layer flow Magnetohydrodynamics (MHD) flow

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Numerical solution of the Vlasov-Poisson system using generalized hermite functions

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COMPUTER PHYSICS COMMUNICATIONS 2006年第8期175卷 528-544页

作者： Le Bourdiec, S. de Vuyst, F. Jacquet, L. CEA DAM Ile France F-91680 Bruyeres Le Chatel France Ecole Cent Paris Lab MAS F-92295 Chatenay Malabry France ENS Cachan CMLA F-94235 Cachan France CEA DAM ECP Lab Rech Convent F-91680 Bruyeres Le Chatel France

Two different spectral approaches for solving the nonlinear Vlasov-Poisson equations are presented and discussed. The first approach is based on a standard spectral Galerkin method (SGM) using hermite functions in the velocity space. The second method which belongs to the family of pseudospectral methods (SCM) uses Gauss-hermite collocation points for the velocity discretization. The high-dimensional feature of these equations and the suspected presence of small scales in the solution suggested us to employ these methods that provide high order accuracy while considering a "small" number of ad hoc basis functions. The scaled hermite functions allow us to treat the case of unbounded domains and to properly recover Gaussian-type distributions. Some numerical results on usual test cases are shown and prove the good agreement with the theory. (C) 2006 Elsevier B.V. All rights reserved.

关键词： Vlasov equation spectral method pseudospectral method hermite functions collocation scaling factor

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Rayleigh-Ritz analysis for localized buckling of a strut on a softening foundation by hermite functions

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INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES 2003年第26期40卷 7463-7474页

作者： Chen, G Baker, G Univ So Queensland Fac Engn & Surveying Toowoomba Qld 4350 Australia

This paper proposes a Rayleigh-Ritz procedure for localized buckling of a strut on a non-linear elastic foundation. Firstly, the deflected shape of a strut is expanded into a series of hermite orthogonal functions, which are proved energy-integrable in an infinite region. Secondly, the errors of the numerical integrations of hermite functions on the infinite region are investigated and the suitable integral limit is proposed. Through the numerical investigation, it is demonstrated that the first thirty hermite functions are usually enough to approximate the localized buckling pattern. The proposed method overcomes the disadvantages of the traditional methods, in which the trial functions in either Rayleigh-Ritz or Galerkin analysis are based on the perturbation analyses of the corresponding non-linear differential equation. (C) 2003 Elsevier Ltd. All rights reserved.

关键词： localized buckling non-linear elastic foundation hermite functions Rayleigh-Ritz analysis

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An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using hermite functions collocation method

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COMPUTER PHYSICS COMMUNICATIONS 2010年第6期181卷 1096-1108页

作者： Parand, K. Dehghan, Mehdi Rezaei, A. R. Ghaderi, S. M. Amir Kabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran 15914 Iran Shaheed Beheshti Univ Med Sci Dept Comp Sci G C Tehran Iran

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain They are categorized as singular initial value problems The proposed approach is based on a Herniae function collocation (HFC) method To illustrate the reliability of the method, some special cases of the equations are solved as test examples The new method reduces the solution of a problem to the solution of a system of algebraic equations hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable COD (C) 2010 Elsevier B V All rights reserved

关键词： Lane-Emden type equations Nonlinear ODE Collocation method hermite functions Isothermal gas spheres Astrophysics

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A lead dependent ischemic episodes detection strategy using hermite functions

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BIOMEDICAL SIGNAL PROCESSING AND CONTROL 2010年第4期5卷 271-281页

作者： Rocha, T. Paredes, S. Carvalho, P. Henriques, J. Harris, M. Morais, J. Antumes, M. Inst Super Engn Coimbra Dept Engn Informat & Sistemas Coimbra Portugal Univ Coimbra Ctr Informat & Sistemas Coimbra Portugal Philips Res Europe Aachen Germany Hosp Santo Andre Serv Cardiol Leiria Portugal Univ Hosp Ctr Cirurg Cardiotoracia Coimbra Portugal

In this work a new strategy for automatic detection of ischemic episodes is proposed. A new measure for ST deviation based on the time-frequency analysis of the ECG and the use of a reduced set of hermite basis functions for T wave and QRS complex morphology characterization, are the key points of the proposed methodology. Usually, ischemia manifests itself in the ECG signal by ST segment deviation or by QRS complex and T wave changes in morphology. These effects might occur simultaneously. Time-frequency methods are especially adequate for the detection of small transient characteristics hidden in the ECG, such as ST segment alterations. A Wigner-Ville transform-based approach is proposed to estimate the ST shift. To characterize the alterations in the T wave and the QRS morphologies, each cardiac beat is described by expansions in hermite functions. These demonstrated to be suitable to capture the most relevant morphologic characteristics of the signal. A lead dependent neural network classifier considers, as inputs, the ST segment deviation and the hermite expansion coefficients. The ability of the proposed method in ischemia episodes detection is evaluated using the European Society of Cardiology ST-T database. A sensitivity of 96.7% and a positive predictivity of 96.2% reveal the capacity of the proposed strategy to perform ischemic episodes identification. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： ECG analysis Ischemic episodes detection hermite functions Wigner-Ville transform Neural networks classifier

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