Padé approximation of Stieltjes series

Using Nuttall's compact formula for the [n, n − 1] Pad'e approximant, the authors show that there is a natural connection between the Padé approximants of a series of Stieltjes and orthogonal polynomials. In particular, we obtain the precise error formulas. The [n, n − 1] Padé approximant in this case is just a Gaussian quadrature of the Stieltjes integral. Hence, analysis of the error is now possible and under very mild conditions it is shown that the [n, n + j], j − 1, Padé approximants converge to the Stieltjes integral.     

B-splines and Hermite–Padé approximants to the exponential function

This paper is the continuation of a work initiated in [P. Sablonnière, An algorithm for the computation of Hermite–Padé approximations to the exponential function: divided differences and Hermite–Padé forms. Numer. Algorithms 33 (2003) 443–452] about the computation of Hermite–Padé forms (HPF) and associated Hermite–Padé approximants (HPA) to the exponential function. We present an alternative algorithm for their computation, based on the representation of HPF in terms of integral remainders with B-splines as Peano kernels. Using the good properties of discrete B-splines, this algorithm gives rise to a great variety of representations of HPF of higher orders in terms of HPF of lower orders, and in particular of classical Padé forms. We give some examples illustrating this algorithm, in particular, another way of constructing quadratic HPF already described by different authors. Finally, we briefly study a family of cubic HPF.     

One-step multiderivative methods for first order ordinary differential equations

A family of one-step multiderivative methods based on Padé approximants to the exponential function is developed. The methods are extrapolated and analysed for use inPECE mode.Error constants, stability intervals and stability regions are given in two associated Technical Reports.Comparisons are made with well-known linear multi-step combinations and combinations using high accuracy Newton-Cotes quadrature formulas as correctors.     

Legendre modified moments for Euler's constant

Polynomial moments are often used for the computation of Gauss quadrature to stabilize the numerical calculation of the orthogonal polynomials, see [W. Gautschi, Computational aspects of orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials-Theory and Practice, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 294. Kluwer, Dordrecht, 1990, pp. 181–216 [6]; W. Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48(4) (1986) 369–382 [5]; W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3(3) (1982) 289–317 [4]] or numerical resolution of linear systems [C. Brezinski, Padé-type approximation and general orthogonal polynomials, ISNM, vol. 50, Basel, Boston, Stuttgart, Birkhäuser, 1980 [3]]. These modified moments can also be used to accelerate the convergence of sequences to a real or complex numbers if the error satisfies some properties as done in [C. Brezinski, Accélération de la convergence en analyse numérique, Lecture Notes in Mathematics, vol. 584. Springer, Berlin, New York, 1977; M. Prévost, Padé-type approximants with orthogonal generating polynomials, J. Comput. Appl. Math. 9(4) (1983) 333–346]. In this paper, we use Legendre modified moments to accelerate the convergence of the sequence H_n-log(n+1) to the Euler's constant γ. A formula for the error is given. It is proved that it is a totally monotonic sequence. At last, we give applications to the arithmetic property of γ.     

On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator

We study connections between continued fractions of type J and spectral properties of second order difference operators with complex coefficients. It is known that the convergents of a bounded J-fraction are diagonal Padé approximants of the Weyl function of the corresponding difference operator and that a bounded J-fraction converges uniformly to the Weyl function in some neighborhood of infinity. In this paper we establish convergence in capacity in the unbounded connected component of the resolvent set of the difference operator and specify the rate of convergence. Furthermore, we show that the absence of poles of Padé approximants in some subdomain implies already local uniform convergence. This enables us to verify the Baker–Gammel–Wills conjecture for a subclass of Weyl functions. For establishing these convergence results, we study the ratio and the nth root asymptotic behavior of Padé denominators of bounded J-fractions and give relations with the Green function of the unbounded connected component of the resolvent set. In addition, we show that the number of “spurious” Padé poles in this set may be bounded.     

Reduction of the Gibbs phenomenon for smooth functions with jumps by the -algorithm

Recently, Brezinski has proposed to use Wynn's ε-algorithm in order to reduce the Gibbs phenomenon for partial Fourier sums of smooth functions with jumps, by displaying very convincing numerical experiments. In the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions, and obtain the rate of column convergence for such functions, possibly perturbed by another sufficiently differentiable function. We also analyze the connection to Padé–Fourier and Padé–Chebyshev approximants, including those recently studied by Kaber and Maday.     

Convergence of the Nested Multivariate Padé Approximants

The nested multivariate Padé approximants were recently introduced. In the case of two variablesxandy, they consist in applying the Padé approximation with respect to y to the coefficients of the Padé approximation with respect tox. The principal advantage of the method is that the computation only involves univariate Padé approximation. This allows us to obtain uniform convergence where the classical multivariate Padé approximants fail to converge.     

Quadrature Formulas for the Wiener Measure

We present a new method for the approximation of Wiener integrals and provide an explicit error bound for a class of smooth integrands. The purely deterministic algorithm is a sequence of quadrature formulas for the Wiener measure, where the knots are piecewise linear functions. It uses ideas of Smolyak, as well as the multiscale decomposition of the Wiener measure due to Lévy and Ciesielski. For the class we obtain n()max(1, 2⁻⁴), where n() is the number of integrand evaluations needed to guarantee an error at most for f .     

On the distribution of poles of Padé approximants to the Z-transform of complex Gaussian white noise

In the application of Padé methods to signal processing a basic problem is to take into account the effect of measurement noise on the computed approximants. Qualitative deterministic noise models have been proposed which are consistent with experimental results. In this paper the Padé approximants to the Z-transform of a complex Gaussian discrete white noise process are considered. Properties of the condensed density of the Padé poles such as circular symmetry, asymptotic concentration on the unit circle and independence on the noise variance are proved. An analytic model of the condensed density of the Padé poles for all orders of the approximants is also computed. Some Monte Carlo simulations are provided.     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

Gaussian quadrature formulae on the unit circle

Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form

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For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szegö polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Padé approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szegö quadrature formulae is presented through some illustrative numerical examples.     

A-stable block implicit one-step methods

A class of methods for solving the initial value problem for ordinary differential equations is studied. We developr-block implicit one-step methods which compute a block ofr new values simultaneously with each step of application. These methods are examined for the property ofA-stability. A sub-class of formulas is derived which is related to Newton-Cotes quadrature and it is shown that for block sizesr=1,2,..., 8 these methods areA-stable while those forr=9,10 are not. We constructA-stable formulas having arbitrarily high orders of accuracy, even stiffly (strongly)A-stable formulas.     

On Some Problems of P. Turán Concerning Lm Extremal Polynomials and Quadrature Formulas

The L_m extremal polynomials in an explicit form with respect to the weights (1−x)^−1/2 (1+x)^(m−1)/2 and (1−x)^(m−1)/2 (1+x)^−1/2 for even m are given. Also, an explicit representation for the Cotes numbers of the corresponding Turán quadrature formulas and their asymptotic behavior is provided.     

Error estimates for generalized compound quadrature formulas

We define quadrature formulas for integrals with weight functionsby applying a given approximation method locally. This allowsthe generalisation of different quadrature formulas, e.g., thecompound Newton-Cotes formulas, Gauss summation formulas, orGregory's formulas, to the case of weighted integrals, as wellas to construct new quadrature formulas, and to derive errorestimates for all these quadrature formulas. The estimates consideredhere are mainly of the form |R[f]|c||f^(r)||, provided the underlyingapproximation method is exact for polynomials of degree <r(R[f] is the quadrature error). Explicit, asymptotically sharperror estimates are obtained for arbitrary integrable weightfunctions. Further, estimates are obtained for the case thatthe quadrature error is of higher order than the approximationerror.     

Noyaux potentiels associés à une filtration

We study from the point of view of potential theory some operators V which are “integrals of martingales” and noteworthy the formula (I + V)⁻¹ = I − N where N is a submarkovian kernel. We give an explicit expression of N when the filtration is finite and get the general case with an usual approximation procedure. Some links are made with the matrix theory (ultrametric and Stieltjes matrices) and the graph theory (flows and capacities) when the space is finite.

Résumé

On étudie, du point de vue de la théorie du potentiel, des opérateurs V du type “intégrales de martingale”, et notamment la formule (I + V)⁻¹ = I − N où N est un noyau sous-markovien. On donne une expression explicite de N dans le cas d'une filtration finie, et on traite le cas général par un procédé d'approximation usuel. On fait le lien avec la théorie des matrices (matrices ultramétriques et de Stieltjes) et la théorie des graphes (flots et capacités) quand l'espace est fini.     

Matrices dont deux lignes quelconque coïncident dans un nombre donne de positions communes

Le nombre maximal de lignes de matrices seront désignées par:

1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;

2. (b) S⁰(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y₁, y₂, y₃, ou y₁ = y₂ ≠ y₃;

3. (c) T⁰(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.

La fonction T⁰(k, λ) était introduite par Chvátal et dans les articles de Deza, Mullin, van Lint, Vanstone, on montrait que T⁰(k, λ) max(λ + 2, (k − λ)² + k − λ + 1). La fonction S⁰(k, λ) est introduite ici et dans le Théorème 1 elle est étudiée analogiquement; dans les remarques 4, 5, 6, 7 on donne les généralisations de problèmes concernant T⁰(k, λ), S⁰(k, λ), dans la remarque 9 on généralise le problème concernant R(k, λ). La fonction R(k, λ) était introduite et étudiée par Bolton. Ci-après, on montre que R(k, λ) S⁰(k, λ) T⁰(k, λ) d'où découle en particulier: R(k, λ) λ + 2 pour λ k + 1 − (k + 2)^1/2; R(k, λ) = 0(k²) pour k − λ = 0(k); R(k, λ) (k − 1)² − (k + 2) pour k 1191.     

Front Propagation Problems with Nonlocal Terms, II

We study the propagation of hyper surfaces Σ_t of ^N satisfying the equation V = h(x, Ω_t), where V is the normal velocity of Σ_t at x, Ω_t is the interior of Σ_t, and h is a given evolution law. We prove the existence of generalized solutions and an inclusion principle for these generalized solutions. Copyright 2001 Academic Press.Dans cet article nous étudions certains problèmes de propagation d'hypersurfaces Σ_t de ^N suivant des lois de la forme: V = h(x, Ω_t), où V est la vitesse normale à Σ_t en x, Ω_t est l'intérieur de Σ_t et h est une loi d'évolution. On définit une notion de solutions généralisées, et on montre l'existence de telles solutions. De plus on démontre un principe d'inclusion pour ces solutions.     

Hermite–Padé approximation and simultaneous quadrature formulas

We study Hermite–Padé approximation of the so-called Nikishin systems of functions. In particular, the set of multi-indices for which normality is known to take place is considerably enlarged as well as the sequences of multi-indices for which convergence of the corresponding simultaneous rational approximants takes place. These results are applied to the study of the convergence properties of simultaneous quadrature rules of a given function with respect to different weights.     

Functional inequalities on arbitrary Riemannian manifolds

For any connected (not necessarily complete) Riemannian manifold, we construct a probability measure of type , where dx is the Riemannian volume measure and V is a function C^∞-smooth outside a closed set of zero volume, satisfying Poincaré–Sobolev type functional inequalities. In particular, V is C^∞-smooth on the whole manifold when the Poincaré and the super-Poincaré inequalities are considered. The Sobolev inequality for infinite measures are also studied.