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

On the Continuity of Multivariate Lagrange Interpolation at Chung-Yao Lattices

We give a natural geometric condition that ensures that sequences of Chung-Yao interpolation polynomials (of fixed degree) of sufficiently differentiable functions converge to a Taylor polynomial

Automatic Crack Detection in Built Infrastructure Using Unmanned Aerial Vehicles

This paper addresses the problem of crack detection which is essential for health monitoring of built infrastructure. Our approach includes two stages, data collection using unmanned aerial vehicles (UAVs) and crack detection using histogram analysis. For the data collection, a 3D model of the structure is first created by using laser scanners. Based on the model, geometric properties are extracted to generate way points necessary for navigating the UAV to take images of the structure. Then, our next step is to stick together those obtained images from the overlapped field of view. The resulting image is then clustered by histogram analysis and peak detection. Potential cracks are finally identified by using locally adaptive thresholds. The whole process is automatically carried out so that the inspection time is significantly improved while safety hazards can be minimised. A prototypical system has been developed for evaluation and experimental results are included.Comment: In proceeding of The 34th International Symposium on Automation and Robotics in Construction (ISARC), pp. 823-829, Taipei, Taiwan, 201

Lagrange interpolation at real projections of Leja sequences for the unit disk

We show that the Lebesgue constant of the real projection of Leja sequences for the unit disk grows like a polynomial. The main application is the first construction of explicit multivariate interpolation points in $[-1,1]^N$ whose Lebesgue constant also grows like a polynomial.Comment: 12 pages, 2 figure

Vitali's theorem without uniform boundedness

Let {fm}m≥1 be a sequence of holomorphic functions defined on a bounded domain D ⊂ Cn or a sequence of rational functions (1 ≤ deg rm ≤ m) defined on Cn. We are interested infinding sufficient conditions to ensure the convergence of {fm}m≥1 on a large set provided the convergence holds pointwise on a not too small set. This type of result is inspired from a theorem of Vitali which gives a positive answer for uniformly bounded sequence

Determinants Influencing Tax Audit Services: The Case of Vietnam

Unlike former researches on tax audit activities in Vietnam which often use qualitative method to analyze and give recommendations, this research uses quantitative method to identify and verify determinants influencing tax audit activities, by assessing reliability and suitability of measuring scales; verifying research model and research hypothesis; determining impact levels of different drivers of tax audit activities (regarding tax audit conclusions) over 268 tax auditors through questionnaires. The research results are recommendations for Vietnamese tax authorities to consider enhancing the supervision of tax audit activities; build a database to serve tax audit activities and restructure procedures, finalize tax audit methods to increase tax audit performance. Keywords: tax audit, tax audit conclusions, drivers, audit supervision, audit process, database about taxpayers

Allometric relationships among tree-size variables under tropical forest stages in Gia Lai, Vietnam

Allometric models play an undeniable role for estimating hard-to-measure quantities such as volume, biomass and carbon stock in forests. However, so far there has been limited model development for native forests in Vietnam. Therefore, this study was conducted to build and analyze the effectiveness of nonlinear and mixed models for secondary and old-growth forests in Gia Lai, Vietnam. The study measured diameter at breast height, total height, commercial height and crown width of forest trees in 20 plots (10 plots for each forest stage). The results showed that diameter had the strongest relationship with height. In the secondary forest, the Power, Korf and Ratskowky models were the best for pairs of variables, while Prodan, Weibull and Power models were the best fit in the old-growth forest. The nonlinear mixed-effect models were better than classic nonlinear models in both forest stages. Fixed and mixed models developed in this study are very valuable for estimating difficult-to-measure quantities and contribute to effective forest management in the study region