L'approximation de fonctions et de données discrètes est fondamentale dans des domaines tels que la planification de trajectoire ou le traitement du signal (données issues de capteurs). Dans ces domaines, il est important d'obtenir des courbes conservant la forme initiale des données. L'utilisation des splines L1 semble être une bonne solution au regard des résultats obtenus pour le problème d'interpolation de données discrètes par de telles splines. Ces splines permettent notamment de conserver les alignements dans les données et de ne pas introduire d'oscillations résiduelles comme c'est le cas pour les splines d'interpolation L2. Nous proposons dans cette thèse une étude du problème de meilleure approximation au sens de la norme L1. Cette étude comprend des développements théoriques sur la meilleure approximation L1 de fonctions présentant une discontinuité de type saut dans des espaces fonctionnels généraux appelés espace de Chebyshev et faiblement Chebyshev. Les splines polynomiales entrent dans ce cadre. Des algorithmes d'approximation de données discrètes au sens de la norme L1 par procédé de fenêtre glissante sont développés en se basant sur les travaux existants sur les splines de lissage et d'ajustement. Les méthodes présentées dans la littérature pour ces types de splines peuvent être relativement couteuse en temps de calcul. Les algorithmes par fenêtre glissante permettent d'obtenir une complexité linéaire en le nombre de données. De plus, une parallélisation est possible. Enfin, une approche originale d'approximation, appelée interpolation à delta près, est développée. Nous proposons un algorithme algébrique avec une complexité linéaire et qui peut être utilisé pour des applications temps réel. / Data and function approximation is fundamental in application domains like path planning or signal processing (sensor data). In such domains, it is important to obtain curves that preserve the shape of the data. Considering the results obtained for the problem of data interpolation, L1 splines appear to be a good solution. Contrary to classical L2 splines, these splines enable to preserve linearities in the data and to not introduce extraneous oscillations when applied on data sets with abrupt changes. We propose in this dissertation a study of the problem of best L1 approximation. This study includes developments on best L1 approximation of functions with a jump discontinuity in general spaces called Chebyshev and weak-Chebyshev spaces. Polynomial splines fit in this framework. Approximation algorithms by smoothing splines and spline fits based on a sliding window process are introduced. The methods previously proposed in the littérature can be relatively time consuming when applied on large datasets. Sliding window algorithm enables to obtain algorithms with linear complexity. Moreover, these algorithms can be parallelized. Finally, a new approximation approach with prescribed error is introduced. A pure algebraic algorithm with linear complexity is introduced. This algorithm is then applicable to real-time application.

<p>Impressive examples of the generation of hierarchically-patterned, three-dimensional (3-D) structures for the control of light can be found throughout nature. <i>Morpho rhetenor</i> butterflies, for example, possess scales with periodic parallel ridges, each of which consists of a stack of thin (nanoscale) layers (lamellae). The bright blue color of <i>Morpho</i> butterflies has been attributed to controlled scattering of the incident light by the lamellae of the wing scales. Another stunning example is the frustule (microshell) of the <i>Coscinodiscus wailesii</i> diatom, which is capable of focusing red light without possessing a traditional lens morphology. The photonic structures and the optical behaviors of <i>Morpho</i> butterflies and <i>Coscinodiscus wailesii</i> diatoms have been extensively studied. However, no work has been conducted to shift such light manipulation from the visible to the infrared (IR) range via shape-preserving conversion of such biogenic structures. Controlling IR radiation (i.e., heat) utilizing biogenic or biomimetic structures can be of significant utility for the development of energy-harvesting devices. In order to enhance the optical interaction in the IR range, inorganic replicas of biogenic structures comprised of high-refractive-index materials have been generated in this work. Such replicas of <i>Morpho</i> <i>rhetenor</i> scales were fabricated via a combination of sol-gel solution coating, organic pyrolysis, and gas/solid reaction methods. Diatomimetic structures have also been generated via sol-gel coating, gas/solid reaction, and then patterning of pore arrays using focused ion beam (FIB) milling.</p> Throughout the sol-gel solution coating and chemical conversion steps of the processes developed in this study, attention was paid to preserve the starting shapes of the nanopatterned, microscale biogenic or biomimetic structures. Factors affecting such shape preservation included the thicknesses and uniformities of coatings applied to the biogenic or biomimetic templates, nano/microstructural evolution during thermal treatment, and reaction-induced volume changes. A conformal surface sol-gel (SSG) coating process was developed in this work to generate oxide replicas of <i>Morpho rhetenor</i> butterfly scales with precisely-controlled coating thicknesses. The adsorption kinetics and relevant adsorption isotherm of the SSG process were investigated utilizing a quartz crystal microbalance. Analyses of thermodynamic driving forces, rate-limiting kinetic steps, and volume changes associated with various chemical reactions were used to tailor processing parameters for optimized shape preservation.