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.

Blending Approximations with Sine Functions.- Quasi-interpolation in the Absence of Polynomial Reproduction.- Estimating the Condition Number for Multivariate Interpolation Problems.- Wavelets on a Bounded Interval.- Quasi-Kernel Polynomials and Convergence Results for Quasi-Minimal Residual Iterations.- Rate of Approximation of Weighted Derivatives by Linear Combinations of SMD Operators.- Approximation by Multivariate Splines: an Application of Boolean Methods.- Lm, ?, s-Splines in ?d.- Constructive Multivariate Approximation via Sigmoidal Functions with Applications to Neural Networks.- Spline-Wavelets of Minimal Support.- Necessary Conditions for Local Best Chebyshev Approximations by Splines with Free Knots.- C1 Interpolation on Higher-Dimensional Analogs of the 4-Direction Mesh.- Tabulation of Thin Plate Splines on a Very Fine Two-Dimensional Grid.- The L2-Approximation Orders of Principal Shift-Invariant Spaces Generated by a Radial Basis Function.- A Multi-Parameter Method for Nonlinear Least-Squares Approximation.- Analog VLSI Networks.- Converse Theorems for Approximation on Discrete Sets II.- A Dual Method for Smoothing Histograms using Nonnegative C1-Splines.- Segment Approximation By Using Linear Functionals.- Construction of Monotone Extensions to Boundary Function

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.

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 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 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.

for example, the so-called Lp approximation, the Bernstein approxima tion problem (approximation on the real line by certain entire functions), and the highly interesting studies of J. L. WALSH on approximation in the complex plane. I would like to extend sincere thanks to Professor L. COLLATZ for his many encouragements for the writing of this book. Thanks are equally due to Springer-Verlag for their ready agreement to my wishes, and for the excellent and competent composition of the book. In addition, I would like to thank Dr. W. KRABS, Dr. A. -G. MEYER and D. SCHWEDT for their very careful reading of the manuscript. Hamburg, March 1964 GUNTER MEINARDUS Preface to the English Edition This English edition was translated by Dr. LARRY SCHUMAKER, Mathematics Research Center, United States Army, The University of Wisconsin, Madison, from a supplemented version of the German edition. Apart from a number of minor additions and corrections and a few new proofs (e. g., the new proof of JACKSON'S Theorem), it differs in detail from the first edition by the inclusion of a discussion of new work on comparison theorems in the case of so-called regular Haar systems ( 6) and on Segment Approximation ( 11). I want to thank the many readers who provided comments and helpful suggestions. My special thanks are due to the translator, to Springer-Verlag for their ready compliance with all my wishes, to Mr."

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-B??zier 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.

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

"In truth, it is not knowledge, but learning, not possessing, but production, not being there, but travelling there, which provides the greatest pleasure. When I have completely understood something, then I turn away and move on into the dark; indeed, so curious is the insatiable man, that when he has completed one house, rather than living in it peacefully, he starts to build another. " Letter from C. F. Gauss to W. Bolyai on Sept. 2, 1808 This textbook adds a book devoted to applied mathematics to the series "Grundwissen Mathematik. " Our goals, like those of the other books in the series, are to explain connections and common viewpoints between various mathematical areas, to emphasize the motivation for studying certain prob lem areas, and to present the historical development of our subject. Our aim in this book is to discuss some of the central problems which arise in applications of mathematics, to develop constructive methods for the numerical solution of these problems, and to study the associated questions of accuracy. In doing so, we also present some theoretical results needed for our development, especially when they involve material which is beyond the scope of the usual beginning courses in calculus and linear algebra. This book is based on lectures given over many years at the Universities of Freiburg, Munich, Berlin and Augsburg."

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 Tonsberg, Norway, in June 2016. The 17 revised full papers presented were carefully reviewed and selected from 115 submissions. The topics range from mathematical theory to industrial applications.

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.

Survey papers written by experts in the fields of Abstract and Classical Analysis, along with contributed research papers.

Topics 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
wavelet packets

Survey papers written by experts in the fields of Wavelets, Splines, and Applications, along with contributed research papers.

Topics include:

approximation of eigenvalues
approximation with weights
Bernstein inequalities
Chebyshev centers
generalized polynomials
Haar 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
zero-increasing transformations

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.

The series of books edited or co-edited by Professor Larry One of Two new volumes of carefully refereed and edited papers on the most current developments in the theory and applications of curves and surfaces.
These two volumes contain a selection of papers presented at the Saint-Malo Conference on Approximation Theory in July 1999. Each contains several invited survey papers written by experts in the field, along with contributed research papers. They will be of great interest to mathematicians, engineers, and computer scientists working in the fields of Approximation Theory, Computer-Aided Geometric Design (CAGD), Computer Graphics, Numerical Analysis, CAD/CAM, and application areas.
"Curve and Surface Design" includes the following topics:
B-bases
canal surfaces
conics
curvature
discrete fairing
free form surfaces
G2-splines
highlight lines
involute curves
multisided patches
NURBs and NURPs
offsets
ray tracing
ruled surfaces
sculptured surfaces
segmentation methods
shape preservation
texture
voronoi diagrams

This meticulously edited selection of papers comes out of the Ninth International Symposium on Approximation Theory held in Nashville, Tennessee, in January, 1998. Each volume contains several invited survey papers written by experts in the field, along with contributed research papers.

This book should be of great interest to mathematicians, engineers, and computer scientists working in approximation theory, wavelets, computer-aided geometric design (CAGD), and numerical analysis.

Among the topics included in the books are the following:

adaptive approximation
approximation by harmonic functions
approximation by radial basis functions
approximation by ridge functions
approximation in the complex plane
Bernstein polynomials
bivariate splines
constructions of multiresolution analyses
convex approximation
frames and frame bases
Fourier methods
generalized moduli of smoothness
interpolation and approximation by splines on triangulations
multiwavelet bases
neural networks
nonlinear approximation
quadrature and cubature
rational approximation
refinable functions
subdivision schemes
thin plate splines
wavelets and wavelet systems

This meticulously edited selection of papers comes out of the Ninth International Symposium on Approximation Theory held in Nashville, Tennessee, in January, 1998. Each volume contains several invited survey papers written by experts in the field, along with contributed research papers.

This book should be of great interest to mathematicians, engineers, and computer scientists working in approximation theory, wavelets, computer-aided geometric design (CAGD), and numerical analysis.

Among the topics included in the books are the following:

adaptive approximation
approximation by harmonic functions
approximation by radial basis functions
approximation by ridge functions
approximation in the complex plane
Bernstein polynomials
bivariate splines
constructions of multiresolution analyses
convex approximation
frames and frame bases
Fourier methods
generalized moduli of smoothness
interpolation and approximation by splines on triangulations
multiwavelet bases
neural networks
nonlinear approximation
quadrature and cubature
rational approximation
refinable functions
subdivision schemes
thin plate splines
wavelets and wavelet systems

Refereed and edited papers on the most current developments in the theory and applications of curves and surfaces.