Abstract:
Two kinds of splines which interpolate the derivatives of some functions are studied,and formulas of approximation estimation are derived by constructing fundamental splines.

Abstract:
In this article, we study the problem of best $L_1$ approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces. We extend the Hobby-Rice theorem into an appropriate framework and prove the unicity of best $L_1$ approximation of Heaviside-type functions in an even-dimensional Chebyshev space under the condition that the dimension of the subspace composed of the even functions is half the dimension of the whole space. We also apply the results to compute best $L_1$ approximations of Heaviside-type functions by polynomials and Hermite polynomial splines with fixed knots.

Abstract:
We obtain strong and uniform asymptotics in every domain of the complex plane for the scaled polynomials $a (3nz)$, $b (3nz)$, and $c (3nz)$ where $a$, $b$, and $c$ are the type II Hermite-Pad\'e approximants to the exponential function of respective degrees $2n+2$, $2n$ and $2n$, defined by $a (z)e^{-z}-b (z)=\O (z^{3n+2})$ and $a (z)e^{z}-c (z)={\O}(z^{3n+2})$ as $z\to 0$. Our analysis relies on a characterization of these polynomials in terms of a $3\times 3$ matrix Riemann-Hilbert problem which, as a consequence of the famous Mahler relations, corresponds by a simple transformation to a similar Riemann-Hilbert problem for type I Hermite-Pad\'e approximants. Due to this relation, the study that was performed in previous work, based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems, can be reused to establish our present results.

Abstract:
This note is a continuation of our papers [1,2], devoted to $L$-approximation of characteristic function of $(-h, h)$ by trigonometric polynomials. In the paper [1] the sharp values of the best approximation for the special values of $h$ were found. In [2] we gave the complete solution of the problem for arbitrary values of $h$. In general case [2] the situation is more deep and results are not so simple as in [1]. For applications to the problem of optimal constants in the Jackson-type inequalities we need, however, results on $L$-approximation of $B$-splines and linear combinations of $B$-splines. Here we present some simple results about $L$-approximation of $B$-splines as well as give the the proof of its sharpness for the special values of $h$.

Abstract:
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 Hermite 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.

Abstract:
A survey of direct and inverse type results for row sequences of Pad\'e and Hermite-Pad\'e approximation is given. A conjecture is posed on an inverse type result for type II Hermite-Pad\'e approximation when it is known that the sequence of common denominators of the approximating vector rational functions has a limit. Some inverse type results are proved for the so called incomplete Pad\'e approximants which may lead to the proof of the conjecture and the connection is discussed.

Abstract:
En utilisant des approximants de Hermite-Pad\'e de fonctions exponentielles, ainsi que des d\'eterminants d'interpolation de Laurent, nous minorons la distance entre un nombre alg\'ebrique et l'exponentielle d'un nombre alg\'ebrique non nul. ----- We use Hermite-Pad\'e approximants of exponential functions along with Laurent's interpolation determinants to obtain lower bounds for the distance between an algebraic number and the exponential of another non-zero algebraic number.

Abstract:
Pad\'e approximation has two natural extensions to vector rational approximation through the so called type I and type II Hermite-Pad\'e approximants. The convergence properties of type II Hermite-Pad\'e approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite-Pad\'e approximants for Nikishin systems of functions.

Abstract:
We compare contemporary practices of global approximation using cubic B-splines in conjunction with double multiplicity of inner knots (-continuous) with older ideas of utilizing local Hermite interpolation of third degree. The study is conducted within the context of the Galerkin-Ritz formulation, which forms the background of the finite element structural analysis. Numerical results, concerning static and eigenvalue analysis of rectangular elastic structures in plane stress conditions, show that both interpolations lead to identical results, a finding that supports the view that they are mathematically equivalent. 1. Introduction Structural analysis is usually performed using commercial codes that include finite elements of low (usually first or second) degree, where the accuracy of the calculations increases by mesh refinement (-version). Alternatively, keeping the number and the position of the nodal points unaltered, the numerical solution improves using polynomials of higher degree (-version) [1]. As an extension of the above -version “philosophy,” tensor-product Lagrange polynomials as well as CAD-based (Gordon-Coons) macroelements—based on several interpolations—have been used [2–4]. The aforementioned macroelements integrate the solid modelling (CAD: computer-aided-design) with the analysis (CAE: computer-aided-engineering). In more detail, these macroelements use the same global approximation for both the geometry and the displacement vector. In order to avoid the undesired numerical oscillations caused by Lagrange polynomials of high degree, the next generation of CAD-based macroelements replaced them with tensor-product B-splines [5]. Since 2005, the nonuniform-rational-B-splines (NURBS) interpolation has started to prevail [6]. A careful study of literature reveals that most of recent papers referring to the so-called isogeometric analysis (IGA) start with some essentials on the definition of B-splines and relevant recursive formulas due to de Boor [7]. It should be recalled that NURBS is an extension of B-splines (nonuniform and rational) modified on the basis of weighting coefficients, thus producing fully controlled sculptured surfaces [8, 9]. In a B-splines expansion, the multiplicity of the inner knots plays a significant role in the continuity of the variables. In general, the multiplicity of inner knots per breakpoint in combination with a piecewise polynomial of degree ensures -continuity of the variable (here: displacement components) [7, 9]. Thus considering cubic B-splines () in conjunction with double inner knots (), -continuity