Kergin Approximation in Banach Spaces

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges

Set-valued Hermite interpolation

AbstractThe problem of interpolating a set-valued function with convex images is addressed by means of directed sets. A directed set will be visualised as a usually non-convex set in Rn consisting of three parts together with its normal directions: the convex, the concave and the mixed-type part. In the Banach space of the directed sets, a mapping resembling the Kergin map is established. The interpolating property and error estimates similar to the point-wise case are then shown; the representation of the interpolant through means of divided differences is given. A comparison to other set-valued approaches is presented. The method developed within the article is extended to the scope of the Hermite interpolation by using the derivative notion in the Banach space of directed sets. Finally, a numerical analysis of the explained technique corroborates the theoretical results

On the Convergence of Kergin and Hakopian Interpolants at Leja Sequences for the Disk

We prove that Kergin interpolation polynomials and Hakopian interpolation polynomials at the points of a Leja sequence for the unit disk $D$ of a sufficiently smooth function $f$ in a neighbourhood of $D$ converge uniformly to $f$ on $D$. Moreover, when $f$ is $C^\infty$ on $D$, all the derivatives of the interpolation polynomials converge uniformly to the corresponding derivatives of $f$

Contributions to polynomial interpolation in one and several variables

Cette thèse traite de l'interpolation polynomiale des fonctions d'une ou plusieurs variables. Nous nous intéresserons principalement à l'interpolation de Lagrange mais un de nos travaux concerne les interpolations de Kergin et d'Hakopian. Nous dénotons par K le corps de base qui sera toujours R ou C, Pd(KN) l'espace des polynômes de N variables et de degré au plus d à coefficients dans K. Un ensemble A dans KN contenant autant de points que la dimension de Pd(KN) est dit unisolvent s'il n'est pas contenu dans l'ensemble des zéros d'un polynôme de degré d. Pour toute fonction f définie sur A, il existe un unique L[A;f] dans Pd(KN) tel que L[A;f]=f sur A, appelé le polynôme d'interpolation de Lagrange de f en A. Les polynômes d'interpolation de Kergin et d'Hakopian sont deux généralisations naturelles en plusieurs variables de l'interpolation de Lagrange à une variable. La construction de ces polynômes nécessite le choix de points à partir desquels on construit certaines formes linéaires qui sont des moyennes intégrales et qui fournissent les conditions d'interpolation. La qualité des approximations fournies par les polynômes d'interpolation dépend pour une large mesure du choix des points d'interpolation. Cette qualité est mesurée par la croissance de la norme de l'opérateur linéaire qui à toute fonction continue associe son polynôme d'interpolation. Cette norme est appelée la constante de Lebesgue (associée au compact et aux points d'interpolation considérés). La majeure partie de cette thèse est consacrée à l'étude de cette constante. Nous donnons par exemples le premier exemple général explicite de familles de points possédant une constante de Lebesgue qui croit comme un polynôme. C'est une avancée significative dans ce domaine de recherche.This thesis deals with polynomial interpolation of functions in one and several variables. We shall be mostly concerned with Lagrange interpolation but one of our work deals with Kergin and Hakopian interpolants. We denote by K the field that may be either R or C, and Pd(KN) the vector space of all polynomials of N variables of degree at most d. The set A of KN is said to be an unisolvent set of degree d if it is not included in the zero set of a polynomial of degree not greater than d. For every function f defined on A, there exists a unique L[A; f ] in Pd(KN) such that L[A; f ] = f on A, which is called the Lagrange interpolation polynomial of a function f at A. Kergin and Hakopian interpolants are natural multivariate generalizations of univariate Lagrange interpolation. The construction of these interpolation polynomials requires the use of points with which one obtains a number of natural mean value linear forms which provide the interpolation conditions. The quality of approximation furnished by interpolation polynomials much depends on the choice of the interpolation points. In turn, the quality of the interpolation points is best measured by the growth of the norm of the linear linear operator that associates to a continuous function its interpolation polynomial. This norm is called the Lebesgue constant. Most of this thesis is dedicated to the study of such constant. We provide for instances the first general examples of multivariate points having a Lebesgue constant that grows like a polynomial. This is an important advance in the field

Об интерполяционном приближении дифференцируемых операторов в гильбертовом пространстве

У гільбертовому просторі побудовано інтерполяційне наближення полінома Тейлора для диференційовних операторів. За допомогою цього наближення отримано оцінки точності для аналітичних операторів, які підсилюють відомі раніше результати, та операторів, що мають скінченну кількість похідних Фреше.In a Hilbert space, we construct an interpolation approximation of the Taylor polynomial for differentiable operators. By using this approximation, we obtain estimates of accuracy for analytic operators that strengthen previously known results and for operators containing finitely many Fréchet derivatives