Larry L. Schumaker - Böcker

Spline functions are universally recognized as highly effective tools in approximation theory, computer-aided geometric design, image analysis, and numerical analysis. The theory of univariate splines is well known but this text is the first comprehensive treatment of the analogous bivariate theory. A detailed mathematical treatment of polynomial splines on triangulations is outlined, providing a basis for developing practical methods for using splines in numerous application areas. The detailed treatment of the Bernstein-Bézier representation of polynomials will provide a valuable source for researchers and students in CAGD. Chapters on smooth macro-element spaces will allow engineers and scientists using the FEM method to solve partial differential equations numerically with new tools. Workers in the geosciences will find new tools for approximation and data fitting on the sphere. Ideal as a graduate text in approximation theory, and as a source book for courses in computer-aided geometric design or in finite-element methods.

These volumes contain carefully edited selections of papers that were presented at the Symposium on Trends in Approximation Theory, held in May 2000, and at the Oslo Conference on Mathematical Methods for Curves and Surfaces, held in July 2000. Both contain several invited surveys written by leading experts in the field, along with contributed research papers on the most current developments in approximation theory and in the theory and application of curves and surfaces. These books will be of great interest to mathematicians, engineers, and computer scientists.

This volume contains surveys written by leading experts, along with research papers on the developments in approximation theory and in the theory and application of curves and surfaces. Topics addressed include: approximation of eigenvalues, approximation with weights, Bernstein inequalities, Chebyshev centres, generalized polynomials, Harr sets, kernels on manifolds, linear operators, Markov inequalities, minimal projections, multivariate interpolation, nonlinear approximation, one-sided approximation, orthogonal polynomials, polynomial approximation, rational approximation, shift invariant spaces, uni-modular polynomials and zero-increasing transformations.

This volume contains surveys written by leading experts, along with research papers on the developments in approximation theory and in the theory and application of curves and surfaces. Topics addressed include: biharmonic splines, cardinal L-splines, discrete splines, exterior BVPs, financial optimization, framelets, FSI generators, Gabor frames, moving least squares, multiwavelets, nonstationary wavelets, non-MSF wavelets, orthogonal wavelets, penalized least squares, refinable functions, SAR interferometry, smoothing splines, spline bases, spline interpolation, Sturm-Liouville problems, subdivision, thin plate splines and wavelet packets.

This book is a continuation of the author's earlier book Spline Functions: Computational Methods, published in 2015 by SIAM. This new book focuses on computational methods developed in the last ten years that make use of splines to approximate functions and data and to solve boundary-value problems. The first half of the book works with bivariate spaces of splines defined on H-triangulations, T-meshes, and curved triangulations. Trivariate tensor-product splines and splines on tetrahedral partitions are also discussed. The second half of the book makes use of these spaces to solve boundary-value problems, with a special emphasis on elliptic PDEs defined on curved domains. The book contains numerous examples and figures to illustrate the methods and their performance.In addition to the included bibliography, a 125-page list of additional references can be downloaded from the SIAM website.All of the algorithms in the book have been coded in MATLAB and are included in a package that can also be downloaded from the website. It can be used to run all of the examples in the book. The package also provides an extensive toolbox of functions that readers can utilize to develop their own spline software.The book is designed for mathematicians, engineers, scientists, and anyone else wanting to make use of spline functions for numerical computation.

These proceedings are based on the international conference Approximation Theory XVI held on May 19–22, 2019 in Nashville, Tennessee. The conference was the sixteenth in a series of meetings in Approximation Theory held at various locations in the United States.  Over 130 mathematicians from 20 countries attended.  The book contains two longer survey papers on nonstationary subdivision and Prony’s method, along with 11 research papers on a variety of topics in approximation theory, including Balian-Low theorems, butterfly spline interpolation, cubature rules, Hankel and Toeplitz matrices, phase retrieval, positive definite kernels, quasi-interpolation operators, stochastic collocation, the gradient conjecture, time-variant systems, and trivariate finite elements. The book should be of interest to mathematicians, engineers, and computer scientists working in approximation theory, computer-aided geometric design, numerical analysis, and related approximation areas.

These proceedings are based on the international conference Approximation Theory XVI held on May 19–22, 2019 in Nashville, Tennessee. The conference was the sixteenth in a series of meetings in Approximation Theory held at various locations in the United States.  Over 130 mathematicians from 20 countries attended.  The book contains two longer survey papers on nonstationary subdivision and Prony’s method, along with 11 research papers on a variety of topics in approximation theory, including Balian-Low theorems, butterfly spline interpolation, cubature rules, Hankel and Toeplitz matrices, phase retrieval, positive definite kernels, quasi-interpolation operators, stochastic collocation, the gradient conjecture, time-variant systems, and trivariate finite elements. The book should be of interest to mathematicians, engineers, and computer scientists working in approximation theory, computer-aided geometric design, numerical analysis, and related approximation areas.

These proceedings were prepared in connection with the 14th International Conference on Approximation Theory, which was held April 7-10, 2013 in San Antonio, Texas. The conference was the fourteenth in a series of meetings in Approximation Theory held at various locations in the United States. The included invited and contributed papers cover diverse areas of approximation theory with a special emphasis on the most current and active areas such as compressed sensing, isogeometric analysis, anisotropic spaces, radial basis functions and splines. Classical and abstract approximation is also included. The book will be of interest to mathematicians, engineers\ and computer scientists working in approximation theory, computer-aided geometric design, numerical analysis and related application areas.

This volume constitutes the thoroughly refereed post-conference proceedings of the 8th International Conference on Curves and Surfaces, held in Paris, France, in June 2014. The conference had the overall theme: "Representation and Approximation of Curves and Surfaces and Applications". The 32 revised full papers presented were carefully reviewed and selected from 39 submissions. The scope of the conference was on following topics: approximation theory, computer-aided geometric design, computer graphics and visualization, computational geometry and topology, geometry processing, image and signal processing, interpolation and smoothing, mesh generation, finite elements and splines, scattered data processing and learning theory, sparse and high-dimensional approximation, subdivision, wavelets and multi-resolution method.

These proceedings were prepared in connection with the 14th International Conference on Approximation Theory, which was held April 7-10, 2013 in San Antonio, Texas. The conference was the fourteenth in a series of meetings in Approximation Theory held at various locations in the United States. The included invited and contributed papers cover diverse areas of approximation theory with a special emphasis on the most current and active areas such as compressed sensing, isogeometric analysis, anisotropic spaces, radial basis functions and splines. Classical and abstract approximation is also included. The book will be of interest to mathematicians, engineers\ and computer scientists working in approximation theory, computer-aided geometric design, numerical analysis and related application areas.

These proceedings are based on papers presented at the international conference Approximation Theory XV, which was held May 22–25, 2016 in San Antonio, Texas. The conference was the fifteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 146 participants. The book contains longer survey papers by some of the invited speakers covering topics such as compressive sensing, isogeometric analysis, and scaling limits of polynomials and entire functions of exponential type. The book also includes papers on a variety of current topics in Approximation Theory drawn from areas such as advances in kernel approximation with applications, approximation theory and algebraic geometry, multivariate splines for applications, practical function approximation, approximation of PDEs, wavelets and framelets with applications, approximation theory in signal processing, compressive sensing, rational interpolation, spline approximation in isogeometric analysis, approximation of fractional differential equations, numerical integration formulas, and trigonometric polynomial approximation.

This volume constitutes the thoroughly refereed post-conference proceedings of the 9th International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2016, held in Tønsberg, Norway, in June 2016. The topics range from mathematical theory to industrial applications.

TheSeventhInternationalConferenceonMathematicalMethodsforCurvesand SurfacestookplaceJune26-July 1,2008,inTonsberg,Norway. Theearlier conferences in the series took place in Oslo (1988),Biri (1991), Ulvik (1994), Lillehammer(1997),Oslo(2000),andTromso(2004). Theconferencegathered 165participants fromalmost30countries who presenteda total of129talks. Thisincludesnineinvitedtalksandsevenmini-symposia. Thisbookcontains28originalarticlesbasedontalkspresentedattheconf- ence. Thetopicsrangefrommathematicalanalysisofvariousmethodstoprac- calimplementationonmoderngraphicsprocessingunits. Thepapersre?ectthe newestdevelopmentsinthese?eldsandalsopointtothelatestliterature. The papershavebeensubjecttotheusualpeerreviewprocess,andwethankboth theauthorsandthereviewersfortheirhardworkandhelpfulcollaboration. Wewishtothankthosewhohavesupportedandhelpedorganizetheconf- ence. Firstandforemostitisapleasuretoacknowledgethegenerous?nancial support from the Department of Informatics and the Centre of Mathematics forApplications(CMA)attheUniversityofOslo,andtheResearchCouncilof Norway. WewouldalsoliketothankAndrewMcMurryforhishelpwithwith technicalmatters,andSaraMorkenforhelpwiththeregistration.November2009 Theeditors Organization Organizing Commitee and Editors MortenDaehlen UniversityofOslo,Norway MichaelFloater UniversityofOslo,Norway TomLyche UniversityofOslo,Norway Jean-LouisMerrien INSAdeRennes,France KnutMorken UniversityofOslo,Norway LarryL. Schumaker VanderbiltUniversity,USA Invited Speakers Jean-DanielBoissonnat,SophiaAntipolis,France MassimoFornasier,Linz,Austria TomHughes,Austin,USA JorgPeters,Gainesville,USA RagniPiene,Oslo,Norway RobertSchaback,Gottingen,Germany PeterSchroder,Caltech,USA JonathanShewchuk,Berkeley,USA JoachimWeickert,Saarland,Germany Mini-Symposia Organizers OlegDavydov,Glasgow,UK TorDokken,Oslo,Norway BinHan,Edmonton,Canada ChuckHansen,SaltLakeCity,USA RimvydasKrasauskas,Vilnius,Lithuania TrondKvamsdal,Trondheim,Norway CarlaManni,Rome,Italy Sponsoring Institutions DepartmentofInformatics,UniversityofOslo CentreofMathematicsforApplications,UniversityofOslo ResearchCouncilofNorway Table of Contents MMCS 2008 Partial Di?erential Equations for Interpolation and Compression of Surfaces ...1 Egil Bae and Joachim Weickert Construction of Rational Curves with Rational Rotation-Minimizing Frames via Mob .. ius Transformations...15 Michael Barton, ?Bert Juttl .. er, and Wenping Wang Fat Arcs for Implicitly De?ned Curves...26 Szilvia B' ela and Bert Juttl .. er Geometric Properties of the Adaptive Delaunay Tessellation...41 Tom Bobach, Alexandru Constantiniu, Paul Steinmann, and Georg Umlauf Quadrangular Parameterization for Reverse Engineering ...55 David Bommes, Tobias Vossemer, and Leif Kobbelt A Comparison of Three Commodity-Level Parallel Architectures: Multi-core CPU, Cell BE and GPU ...70 Andr' e Rigland Brodtkorb and Trond Runar Hagen Mean Distance from a Curve to Its Control Polygon...81 Jesu 's Carnicer and Jorge Delgado Compactly Supported Splines with Tension Properties on a Three-Direction Mesh ...

This volume constitutes the thoroughly refereed post-conference proceedings of the 8th International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2012, held in Oslo, Norway, in June/July 2012. The 28 revised full papers presented were carefully reviewed and selected from 135 submissions. The topics range from mathematical analysis of various methods to practical implementation on modern graphics processing units. The papers reflect the newest developments in these fields and also point to the latest literature.

This is the collection of the refereed and edited papers presented at the 8th Texas International Conference on Approximation Theory. It is interdisciplinary in nature and consists of two volumes. The central theme of Vol. I is the core of approximation theory. It includes such important areas as qualitative approximations, interpolation theory, rational approximations, radial-basis functions, and splines. The second volume focuses on topics related to wavelet analysis, including multiresolution and multi-level approximation, subdivision schemes in CAGD, and applications.