Some convergence results for the generalized Padé-type approximants

The aim of this paper is to give some convergence results for some sequences of generalized Padé-type approximants. We will consider two types of interpolatory functionals: one corresponding to Langrange and Hermite interpolation and the other corresponding to orthogonal expansions. For these two cases we will give sufficient conditions on the generating functionG(x, t) and on the linear functionalc in order to obtain the convergence of the corresponding sequence of generalized Padé-type approximants. Some examples are given.     

Dual Variational Principles and Pad?-type Approximants

Families of approximants for <?, f> are derived from dualvariational functionals associated with the linear equation(1+yL)?=f. Expansions in both ascending and descending powersof y are considered, and the approximants are identified eitheras one- or two-point Pad? approximants, or as approximants ofa closely related type. Compact formulae are obtained for theapproximants, and their duality and bounding properties areexhibited. Attention is paid to the special situations occurringwhen the non-negative self-adjoint operator L (i) has a zeroeigenvalue, (ii) is bounded.     

General order Newton-Padé approximants for multivariate functions

Summary Padé approximants are a frequently used tool for the solution of mathematical problems. One of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off(x ₁, ...,x _p ). Therefore multivariate Newton-Padé approximants are introduced; their computation will only use the value off at some points. In Sect. 1 we shall repeat the univariate Newton-Padé approximation problem which is a rational Hermite interpolation problem. In Sect. 2 we sketch some problems that can arise when dealing with multivariate interpolation. In Sect. 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate Newton-Padé approximants in Sect. 4. A numerical example is given in Sect. 5, together with the proof that forp=1 the classical Newton-Padé approximants for a univariate function are obtained.     

Oscillatory behavior of linear matrix Hamiltonian systems

We establish some new oscillation criteria for the matrix linear Hamiltonian system X ′ = A (t)X + B (t)Y, Y ′ = C (t)X –A *(t)Y by using a new function class X and monotone functionals on a suitable matrix space. In doing so, many existing results are generalized and improved. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation

Let X _t be a one-dimensional Harris recurrent diffusion, with a drift depending on an unknown parameter θ belonging to some metric compact Θ. We firstly show that all integrable additive functionals of X _t are asymptotically equivalent in probability to some deterministic process v _t . Then we use this result to study the behavior of the maximum likelihood estimator for the parameter θ. Under mild regularity assumptions, we find an upper rate of its convergence as a function of v _t , extending some recent results for ergodic diffusions.      

Exploring multivariate Padé approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions F _i , i = 1,...,4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the F _i onto ϰ=0 or y=0, namely, the Gauss function ₂ F ₁(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Padé approximants. In section 3 we prove that the table of homogeneous multivariate Padé approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F ₁(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Padé approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Padé approximants in this context and discussing directions for future work. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Toeplitz type quadratic functionals of stationary Gaussian processes

Summary A central limit theorem for Toeplitz type quadratic functionals of a stationary Gaussian processX(t),t, is proved, generalizing the result of Avram [1] for discrete time processes. The result is applied to the problem of nonparametric estimation of linear functionals of an unknown spectral density function. We give some upper bounds for the minimax mean square risk of the nonparametric estimators, similar to those by Ibragimov and Has'minskii [12] for a probability density function.     

Existence and stability of a summable generalized solution to a mathematical model of a catalytic fluidized bed process

For a system of first-order partial differential equations describing a catalytic process in a fluidized bed, we consider a mixed problem in the half-strip 0 ≤ x ≤ h, t ≥ 0. We prove the existence and uniqueness of a bounded summable generalized solution and study its stability. We prove the stabilization as t → ∞ of the values of some physically meaningful functionals of the solution.     

Some approximants in operator algebras

In this paper we extend to C*-algebras and to von Neumann algebras some results on approximants that have previously been found in the context of $ \mathcal{L} $ \mathcal{L} (H) and of the von Neumann-Schatten classesC _p , 1⩽ p <∞. We obtain results concerning positive approximants, unitary and partially isometric approximants and commutator approximants; and we study paranormality. Our main tools are the Gelfand-naimark Theorem and Berntzen’s results on normal spectral approximation.     

On the noncommutative residue and the heat trace expansion on conic manifolds

 Given a cone pseudodifferential operator P we give a full asymptotic expansion as t → 0 ⁺ of the trace Tr Pe ^-tA , where A is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains log t and new (log t)² terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals. Received: 12 November 2001 / Revised version: 26 June 2002 Mathematics Subject Classification (2000): Primary 58J35; Secondary 35C20, 58J42     

Are regularly estimable functionals always differentiable?

Summary. According to van der Vaart (1991), regularly estimable functionals k \kappa are necessarily differentiable, if lim_{t? 0}t^-1(k(P_t)-k(P₀)) \lim\limits_{t\to 0}t^{-1}(\kappa(P_t)-\kappa(P_0)) exists for every differentiable path. In the present paper, a comparable result is obtained under slightly weaker conditions. A counterexample shows that these conditions are minimal.     

Minimum relative error approximations for 1/t

Summary We give explicit solutions to the problem of minimizing the relative error for polynomial approximations to 1/t on arbitrary finite subintervals of (0, ). We give a simple algorithm, using synthetic division, for computing practical representations of the best approximating polynomials. The resulting polynomials also minimize the absolute error in a related functional equation. We show that, for any continuous function with no zeros on the interval of interest, the geometric convergence rates for best absolute error and best relative error approximants must be equal. The approximation polynomials for 1/t are useful for finding suitably precise initial approximations in iterative methods for computing reciprocals on computers.     

Representation of martingale additive functionals on Banach spaces

Assume thatB is a separable real Banach space andX(t) is a diffusion process onB. In this paper, we will establish the representation theorem of martingale additive functionals ofX(t).     

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

An Algorithm for the Computation of Hermite–Padé Approximations to the Exponential Function: Divided Differences and Hermite–Padé Forms

We present the first of two different algorithms for the explicit computation of Hermite–Padé forms (HPF) associated with the exponential function. Some roots of the algebraic equation associated with a given HPF are good approximants to the exponential in some subsets of the complex plane: they are called Hermite–Padé approximants (HPA) to this function. Our algorithm is recursive and based upon the expression of HPF as divided differences of the function texp(xt) at multiple integer nodes. Using this algorithm, we find again the results obtained by Borwein and Driver for quadratic HPF. As an example, we give an interesting family of quadratic HPA to the exponential.     

Limiting process for integral functionals of a wiener process on a cylinder

We prove that integral functionals, whose integrands are bounded functions of a Wiener process on a cylinder, weakly converge to the processw ₁(τ(t)), τ(t) = β₁ t + (β₂ − β₁)mes {s:w ₂(s)≥0,s<t}, wherew ₁(t andw ₂(t) are independent one-dimensional Wiener processes, β₁ and β₂ are nonrandom values, and β₂≥β₁≥0. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 765–768, June, 1994.     

Approximants de pade-hermite. 1ère partie: theorie

Summary We present in this first paper a generalization of Padé approximants which gives us as particular cases Shafer's and Baker'sD-log approximants.First we define these approximants following an old idea of Hermite, then we prove some fundamental properties for their constructions.

     

Some small ball probabilities for Gaussian processes under nonuniform norms

We establish lower and upper bounds for the small ball probability of a centered Gaussian process(X(t)) _t[0,1] ^N under Hölder-type norms as well as upper bounds for some more general functionals. This extends recently established results for the uniform norm. In addition, our proof of the lower bound is considerably simpler. In the special caseN=1 we establish precise estimates under a wider class of norms including in particular the Besov norms.     

Transfer theorems and asymptotic distributional results for m‐ary search trees

We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m‐ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so‐called shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that: (i) for small toll sequences (t_n) [roughly, t_n = O(n^1/2)] we have asymptotic normality if m ≤ 26 and typically periodic behavior if m ≥ 27; (ii) for moderate toll sequences [roughly, t_n = ω(n^1/2) but t_n = o(n)] we have convergence to nonnormal distributions if m ≤ m₀ (where m₀ ≥ 26) and typically periodic behavior if m ≥ m₀ + 1; and (iii) for large toll sequences [roughly, t_n = ω(n)] we have convergence to nonnormal distributions for all values of m. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Existence of positive solutions for a class of second-order two-point boundary value problem

By using different convex functionals to compute fixed point index, the existence of positive solutions for a class of second-order two-point boundary value problem