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115
THE EXPONENTIALLY CONVERGENT TRAPEZOIDAL RULE
"... Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods ..."
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Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions
, 707
"... This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. T ..."
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This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is
A Derivation of Extrapolation Algorithms Based on Error Estimates
, 1993
"... In this paper, we shall emphasize the role played by error estimates and annihilation difference operators in the construction of extrapolations processes. It is showed that this approach leads to a unified derivation of many extrapolation algorithms and related devices, to general results about the ..."
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Cited by 16 (8 self)
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In this paper, we shall emphasize the role played by error estimates and annihilation difference operators in the construction of extrapolations processes. It is showed that this approach leads to a unified derivation of many extrapolation algorithms and related devices, to general results about their kernels and that it opens the way to many new algorithms. Their convergence and acceleration properties could also be studied within this framework.
Umbral calculus, discretization, and Quantum Mechanics on a lattice
 J. Phys. A: Math. Gen
, 1996
"... ‘Umbral calculus ’ deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie algebras on a lattice. Related ideas appeared in recent publicati ..."
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‘Umbral calculus ’ deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie algebras on a lattice. Related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus. This observation then suggests various generalizations of these examples. A special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail. 1
Computing the Algebraic Relations of Cfinite Sequences and Multisequences
, 2006
"... We present an algorithm for computing generators for the ideal of algebraic relations among sequences which are given by homogeneous linear recurrence equations with constant coefficients. Knowing these generators makes it possible to use Gröbner basis methods for carrying out certain basic operatio ..."
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We present an algorithm for computing generators for the ideal of algebraic relations among sequences which are given by homogeneous linear recurrence equations with constant coefficients. Knowing these generators makes it possible to use Gröbner basis methods for carrying out certain basic operations in the ring of such sequences effectively. In particular, one can answer the question whether a given sequence can be represented in terms of other given sequences. A collection of examples, which were done with an implementation of our algorithm, is included. 1
Entire Majorants via EulerMaclaurin summation
 Trans. Amer. Math. Soc
"... Abstract. It is the aim of this article to give extremal majorants of type 2pi for the class of functions fn(x) = sgn(x)x n where n ∈ N. As applications we obtain positive definite extensions to R of ±(it)−m defined on R\[−1, 1] where m ∈ N, optimal bounds in Hilberttype inequalities for the class ..."
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Abstract. It is the aim of this article to give extremal majorants of type 2pi for the class of functions fn(x) = sgn(x)x n where n ∈ N. As applications we obtain positive definite extensions to R of ±(it)−m defined on R\[−1, 1] where m ∈ N, optimal bounds in Hilberttype inequalities for the class of functions (it)−m, and majorants of type 2pi for functions whose graphs are trapezoids.
qBernoulli and qEuler Polynomials, an Umbral approach II
, 2009
"... We proceed with pseudoqAppell polynomials in the spirit of [12]. It turns out that these qBernoulli numbers are the same as BJHC,ν,q. As in [12] we find qanalogues of many formulas in [38], the umbral calculus works remarkably well also for pseudoqAppell pol., only the q is put up instead of d ..."
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We proceed with pseudoqAppell polynomials in the spirit of [12]. It turns out that these qBernoulli numbers are the same as BJHC,ν,q. As in [12] we find qanalogues of many formulas in [38], the umbral calculus works remarkably well also for pseudoqAppell pol., only the q is put up instead of down corresponding to inversion of basis. We also find new qEulerMaclaurin expansions.
Invariant manifolds and their zeroviscosity limits for NavierStokes equations
 Dynamics of PDE
"... Abstract. First we prove a general spectral theorem for the linear NavierStokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis i ..."
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Abstract. First we prove a general spectral theorem for the linear NavierStokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in H ℓ (ℓ = 0, 1, 2, · · ·). Then we prove the existence of invariant manifolds. We are also interested in a more challenging problem, i.e. studying the zeroviscosity limits (ν → 0 +) of the invariant manifolds. Under an assumption, we can show that the sizes of the unstable manifold and the centerstable manifold of a steady state are O ( √ ν), while the sizes of the stable manifold, the center manifold, and the centerunstable manifold are O(ν), as ν → 0 +. Finally, we study three examples. The first example is defined on a rectangular periodic domain, and has only one unstable eigenvalue which is real. A complete estimate on this eigenvalue is obtained. Existence of an 1D unstable manifold and a codim 1 stable manifold is proved without any assumption. For the other
Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations
, 2007
"... The Painlevé property is closely connected to differential equations that are integrable via related isomonodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finiteorder meromorphic solutions ..."
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Cited by 8 (1 self)
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The Painlevé property is closely connected to differential equations that are integrable via related isomonodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finiteorder meromorphic solutions appears to be a good analogue of the Painlevé property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and qdifference operators. New results are presented concerning equations of the form w(z + 1)w(z â 1) = R(z,w), where R is rational in w with meromorphic coefficients.