Les problèmes d'interpolation ont été largement étudiés dans la Conception Géométrique Assistée par Ordinateur. Ces problèmes consistent en la construction de courbes et de surfaces qui passent exactement par un ensemble de données. Dans ce cadre, l'objectif principal de cette thèse est de présenter des méthodes d'interpolation de données 2D et 3D au moyen de courbes Algébriques Trigonométriques à Hodographe Pythagorien (ATPH). Celles-ci sont utilisables pour la conception de modèles géométriques dans de nombreuses applications. En particulier, nous nous intéressons à la modélisation géométrique d'objets odontologiques. À cette fin, nous utilisons les courbes spatiales ATPH pour la construction de surfaces développables dans des volumes odontologiques. Initialement, nous considérons la construction de courbes planes ATPH avec continuité C² qui interpolent une séquence ordonnée de points. Nous employons deux méthodes pour résoudre ce problème et trouver la « bonne » solution. Nous étendons les courbes ATPH planes à l'espace tridimensionnel. Cette caractérisation 3D est utilisée pour résoudre le problème d'interpolation Hermite de premier ordre. Nous utilisons ces splines ATPH spatiales C¹ continues pour guider des facettes développables, qui sont déployées à l'intérieur de volumes tomodensitométriques odontologiques, afin de visualiser des informations d'intérêt pour le professionnel de santé. Cette information peut être utile dans l'évaluation clinique, diagnostic et/ou plan de traitement. / Interpolation problems have been widely studied in Computer Aided Geometric Design (CAGD). They consist in the construction of curves and surfaces that pass exactly through a given data set, such as point clouds, tangents, curvatures, lines/planes, etc. In general, these curves and surfaces are represented in a parametrized form. This representation is independent of the coordinate system, it adapts itself well to geometric transformations and the differential geometric properties of curves and surfaces are invariant under reparametrization. In this context, the main goal of this thesis is to present 2D and 3D data interpolation schemes by means of Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) curves. The latter are parametric curves defined in a mixed algebraic-trigonometric space, whose hodograph satisfies a Pythagorean condition. This representation allows to analytically calculate the curve's arc-length as well as the rational-trigonometric parametrization of the offsets curves. These properties are usable for the design of geometric models in many applications including manufacturing, architectural design, shipbuilding, computer graphics, and many more. In particular, we are interested in the geometric modeling of odontological objects. To this end, we use the spatial ATPH curves for the construction of developable patches within 3D odontological volumes. This may be a useful tool for extracting information of interest along dental structures. We give an overview of how some similar interpolating problems have been addressed by the scientific community. Then in chapter 2, we consider the construction of planar C2 ATPH spline curves that interpolate an ordered sequence of points. This problem has many solutions, its number depends on the number of interpolating points. Therefore, we employ two methods to find them. Firstly, we calculate all solutions by a homotopy method. However, it is empirically observed that only one solution does not have any self-intersections. Hence, the Newton-Raphson iteration method is used to directly compute this \good" solution. Note that C2 ATPH spline curves depend on several free parameters, which allow to obtain a diversity of interpolants. Thanks to these shape parameters, the ATPH curves prove to be more exible and versatile than their polynomial counterpart, the well known Pythagorean-Hodograph (PH) quintic curves and polynomial curves in general. These parameters are optimally chosen through a minimization process of fairness measures. We design ATPH curves that closely agree with well-known trigonometric curves by adjusting the shape parameters. We extend the planar ATPH curves to the case of spatial ATPH curves in chapter 3. This characterization is given in terms of quaternions, because this allows to properly analyze their properties and simplify the calculations. We employ the spatial ATPH curves to solve the first-order Hermite interpolation problem. The obtained ATPH interpolants depend on three free angular values. As in the planar case, we optimally choose these parameters by the minimization of integral shape measures. This process is also used to calculate the C1 interpolating ATPH curves that closely approximate well-known 3D parametric curves. To illustrate this performance, we present the process for some kind of helices. In chapter 4 we then use these C1 ATPH splines for guiding developable surface patches, which are deployed within odontological computed tomography (CT) volumes, in order to visualize information of interest for the medical professional. Particularly, we construct piecewise conical surfaces along smooth ATPH curves to display information related to the anatomical structure of human jawbones. This information may be useful in clinical assessment, diagnosis and/or treatment plan. Finally, the obtained results are analyzed and conclusions are drawn in chapter 5.

Objective: As the multiple trigonometric regressive spectral (MTRS) analysis is extraordinary in its ability to analyze short local data segments down to 12 s, we wanted to evaluate the impact of the data segment settings by applying the technique of MTRS analysis for baroreflex sensitivity (BRS) estimation using a standardized data pool. Methods: Spectral and baroreflex analyses were performed on the EuroBaVar dataset (42 recordings, including lying and standing positions). For this analysis, the technique of MTRS was used. We used different global and local data segment lengths, and chose the global data segments from different positions. Three global data segments of 1 and 2 min and three local data segments of 12, 20, and 30 s were used in MTRS analysis for BRS. Results: All the BRS-values calculated on the three global data segments were highly correlated, both in the supine and standing positions; the different global data segments provided similar BRS estimations. When using different local data segments, all the BRS-values were also highly correlated. However, in the supine position, using short local data segments of 12 s overestimated BRS compared with those using 20 and 30 s. In the standing position, the BRS estimations using different local data segments were comparable. There was no proportional bias for the comparisons between different BRS estimations. Conclusion: We demonstrate that BRS estimation by the MTRS technique is stable when using different global data segments, and MTRS is extraordinary in its ability to evaluate BRS in even short local data segments (20 and 30 s). Because of the non-stationary character of most biosignals, the MTRS technique would be preferable for BRS analysis especially in conditions when only short stationary data segments are available or when dynamic changes of BRS should be monitored.

We report a class of the mathematics in which an animation technology (calculating and plotting capabilities) of the software Mathematica is utilized. This class is taught for university students in a computer laboratory during a second semester. It is our purpose to make a student realize the usefulness and the importance of mathematics easily through visualization. In addition, we hope that students will acquire a new power of mathematics needed in the 21st century. For several years, we have continued this kind of class, and have continued to investigate the effectiveness that our teaching method (especially visualization) brings in the understanding of the mathematics. In this paper, we present some of this teaching method, which is performed in our class. From the questionnaire survey, it is found that our teaching method not only convinces students that the mathematics is useful or important but also deepens the mathematic understanding of students more.

Chapter 1 presents the definitions and basic properties of trigonometric functions including: Sum Identities, Difference Identities, Product-Sum Identities and Sum-Product Identities. These formulas provide effective tools to solve the problems in trigonometry. Chapter 2 handles the most important two theorems in trigonometry: The laws of sines and cosines and show how they can be applied to derive many well known theorems including: Ptolemy¡¦s theorem, Euler Triangle Formula, Ceva¡¦s theorem, Menelaus¡¦s Theorem, Parallelogram Law, Stewart¡¦s theorem and Brahmagupta¡¦s Formula. Moreover, the formulas of computing a triangle area like Heron¡¦s formula and Pick¡¦s theorem are also discussed. Chapter 3 deals with the method of superposition, inverse trigonometric functions, polar forms and De Moivre¡¦s Theorem.

For noiseless sampled data, we describe the close connections between Prony--like methods, namely the classical Prony method, the matrix pencil method and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

Cette thèse vise à la conception de nouveaux algorithmes rapides en calcul numérique via les matrices de Toeplitz. Tout d'abord, nous avons introduit un algorithme rapide sur le calcul de l'inverse d'une matrice triangulaire de Toeplitz en se basant sur des notions d'interpolation polynomiale. Cet algorithme nécessitant uniquement deux FFT(2n) est manifestement efficace par rapport à ses prédécésseurs. ensuite, nous avons introduit un algorithme rapide pour la résolution d'un système linéaire de Toeplitz bande. Cette approche est basée sur l'extension de la matrice donnée par plusieurs lignes en dessus, de plusieurs colonnes à droite et d'attribuer des zéros et des constantes non nulles dans chacune de ces lignes et de ces colonnes de telle façon que la matrice augmentée à la structure d'une matrice triangulaire inférieure de Toeplitz. La stabilité de l'algorithme a été discutée et son efficacité a été aussi justifiée. Finalement, nous avons abordé la résolution d'un système de Toeplitz bandes par blocs bandes de Toeplitz. Ceci étant primordial pour établir la connexion de nos algorithmes à des applications en restauration d'images, un domaine phare en mathématiques appliquées. / This thesis aims to design new fast algorithms for numerical computation via the Toeplitz matrices. First, we introduced a fast algorithm to compute the inverse of a triangular Toeplitz matrix with real and/or complex numbers based on polynomial interpolation techniques. This algorithm requires only two FFT (2n) is clearly effective compared to predecessors. A numerical accuracy and error analysis is also considered. Numerical examples are given to illustrate the effectiveness of our method. In addition, we introduced a fast algorithm for solving a linear banded Toeplitz system. This new approach is based on extending the given matrix with several rows on the top and several columns on the right and to assign zeros and some nonzero constants in each of these rows and columns in such a way that the augmented matrix has a lower triangular Toeplitz structure. Stability of the algorithm is discussed and its performance is showed by numerical experiments. This is essential to connect our algorithms to applications such as image restoration applications, a key area in applied mathematics.

For noiseless sampled data, we describe the close connections between Prony--like methods, namely the classical Prony method, the matrix pencil method and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

We consider multivariate trigonometric polynomials with frequencies supported on a fixed but arbitrary frequency index set I, which is a finite set of integer vectors of length d. Naturally, one is interested in spatial discretizations in the d-dimensional torus such that - the sampling values of the trigonometric polynomial at the nodes of this spatial discretization uniquely determines the trigonometric polynomial, - the corresponding discrete Fourier transform is fast realizable, and - the corresponding fast Fourier transform is stable. An algorithm that computes the discrete Fourier transform and that needs a computational complexity that is bounded from above by terms that are linear in the maximum of the number of input and output data up to some logarithmic factors is called fast Fourier transform. We call the fast Fourier transform stable if the Fourier matrix of the discrete Fourier transform has a condition number near one and the fast algorithm does not corrupt this theoretical stability. We suggest to use rank-1 lattices and a generalization as spatial discretizations in order to sample multivariate trigonometric polynomials and we develop construction methods in order to determine reconstructing sampling sets, i.e., sets of sampling nodes that allow for the unique, fast, and stable reconstruction of trigonometric polynomials. The methods for determining reconstructing rank-1 lattices are component{by{component constructions, similar to the seminal methods that are developed in the field of numerical integration. During this thesis we identify a component{by{component construction of reconstructing rank-1 lattices that allows for an estimate of the number of sampling nodes M |I|\le M\le \max\left(\frac{2}{3}|I|^2,\max\{3\|\mathbf{k}\|_\infty\colon\mathbf{k}\in I\}\right) that is sufficient in order to uniquely reconstruct each multivariate trigonometric polynomial with frequencies supported on the frequency index set I. We observe that the bounds on the number M only depends on the number of frequency indices contained in I and the expansion of I, but not on the spatial dimension d. Hence, rank-1 lattices are suitable spatial discretizations in arbitrarily high dimensional problems. Furthermore, we consider a generalization of the concept of rank-1 lattices, which we call generated sets. We use a quite different approach in order to determine suitable reconstructing generated sets. The corresponding construction method is based on a continuous optimization method. Besides the theoretical considerations, we focus on the practicability of the presented algorithms and illustrate the theoretical findings by means of several examples. In addition, we investigate the approximation properties of the considered sampling schemes. We apply the results to the most important structures of frequency indices in higher dimensions, so-called hyperbolic crosses and demonstrate the approximation properties by the means of several examples that include the solution of Poisson's equation as one representative of partial differential equations.

In this work, the fast evaluation and reconstruction of multivariate trigonometric polynomials with frequencies supported on arbitrary index sets of finite cardinality is considered, where rank-1 lattices are used as spatial discretizations. The approximation of multivariate smooth periodic functions by trigonometric polynomials is studied, based on a one-dimensional FFT applied to function samples. The smoothness of the functions is characterized via the decay of their Fourier coefficients, and various estimates for sampling errors are shown, complemented by numerical tests for up to 25 dimensions. In addition, the special case of perturbed rank-1 lattice nodes is considered, and a fast Taylor expansion based approximation method is developed. One main contribution is the transfer of the methods to the non-periodic case. Multivariate algebraic polynomials in Chebyshev form are used as ansatz functions and rank-1 Chebyshev lattices as spatial discretizations. This strategy allows for using fast algorithms based on a one-dimensional DCT. The smoothness of a function can be characterized via the decay of its Chebyshev coefficients. From this point of view, estimates for sampling errors are shown as well as numerical tests for up to 25 dimensions. A further main contribution is the development of a high-dimensional sparse FFT method based on rank-1 lattice sampling, which allows for determining unknown frequency locations belonging to the approximately largest Fourier or Chebyshev coefficients of a function. / In dieser Arbeit wird die schnelle Auswertung und Rekonstruktion multivariater trigonometrischer Polynome mit Frequenzen aus beliebigen Indexmengen endlicher Kardinalität betrachtet, wobei Rang-1-Gitter (rank-1 lattices) als Diskretisierung im Ortsbereich verwendet werden. Die Approximation multivariater glatter periodischer Funktionen durch trigonometrische Polynome wird untersucht, wobei Approximanten mittels einer eindimensionalen FFT (schnellen Fourier-Transformation) angewandt auf Funktionswerte ermittelt werden. Die Glattheit von Funktionen wird durch den Abfall ihrer Fourier-Koeffizienten charakterisiert und mehrere Abschätzungen für den Abtastfehler werden gezeigt, ergänzt durch numerische Tests für bis zu 25 Raumdimensionen. Zusätzlich wird der Spezialfall gestörter Rang-1-Gitter-Knoten betrachtet, und es wird eine schnelle Approximationsmethode basierend auf Taylorentwicklung vorgestellt. Ein wichtiger Beitrag dieser Arbeit ist die Übertragung der Methoden vom periodischen auf den nicht-periodischen Fall. Multivariate algebraische Polynome in Chebyshev-Form werden als Ansatzfunktionen verwendet und sogenannte Rang-1-Chebyshev-Gitter als Diskretisierungen im Ortsbereich. Diese Strategie ermöglicht die Verwendung schneller Algorithmen basierend auf einer eindimensionalen DCT (diskreten Kosinustransformation). Die Glattheit von Funktionen kann durch den Abfall ihrer Chebyshev-Koeffizienten charakterisiert werden. Unter diesem Gesichtspunkt werden Abschätzungen für Abtastfehler gezeigt sowie numerische Tests für bis zu 25 Raumdimensionen. Ein weiterer wichtiger Beitrag ist die Entwicklung einer Methode zur Berechnung einer hochdimensionalen dünnbesetzten FFT basierend auf Abtastwerten an Rang-1-Gittern, wobei diese Methode die Bestimmung unbekannter Frequenzen ermöglicht, welche zu den näherungsweise größten Fourier- oder Chebyshev-Koeffizienten einer Funktion gehören.

We consider multivariate trigonometric polynomials with frequencies supported on a fixed but arbitrary frequency index set I, which is a finite set of integer vectors of length d. Naturally, one is interested in spatial discretizations in the d-dimensional torus such that - the sampling values of the trigonometric polynomial at the nodes of this spatial discretization uniquely determines the trigonometric polynomial, - the corresponding discrete Fourier transform is fast realizable, and - the corresponding fast Fourier transform is stable. An algorithm that computes the discrete Fourier transform and that needs a computational complexity that is bounded from above by terms that are linear in the maximum of the number of input and output data up to some logarithmic factors is called fast Fourier transform. We call the fast Fourier transform stable if the Fourier matrix of the discrete Fourier transform has a condition number near one and the fast algorithm does not corrupt this theoretical stability. We suggest to use rank-1 lattices and a generalization as spatial discretizations in order to sample multivariate trigonometric polynomials and we develop construction methods in order to determine reconstructing sampling sets, i.e., sets of sampling nodes that allow for the unique, fast, and stable reconstruction of trigonometric polynomials. The methods for determining reconstructing rank-1 lattices are component{by{component constructions, similar to the seminal methods that are developed in the field of numerical integration. During this thesis we identify a component{by{component construction of reconstructing rank-1 lattices that allows for an estimate of the number of sampling nodes M |I|\le M\le \max\left(\frac{2}{3}|I|^2,\max\{3\|\mathbf{k}\|_\infty\colon\mathbf{k}\in I\}\right) that is sufficient in order to uniquely reconstruct each multivariate trigonometric polynomial with frequencies supported on the frequency index set I. We observe that the bounds on the number M only depends on the number of frequency indices contained in I and the expansion of I, but not on the spatial dimension d. Hence, rank-1 lattices are suitable spatial discretizations in arbitrarily high dimensional problems. Furthermore, we consider a generalization of the concept of rank-1 lattices, which we call generated sets. We use a quite different approach in order to determine suitable reconstructing generated sets. The corresponding construction method is based on a continuous optimization method. Besides the theoretical considerations, we focus on the practicability of the presented algorithms and illustrate the theoretical findings by means of several examples. In addition, we investigate the approximation properties of the considered sampling schemes. We apply the results to the most important structures of frequency indices in higher dimensions, so-called hyperbolic crosses and demonstrate the approximation properties by the means of several examples that include the solution of Poisson's equation as one representative of partial differential equations.