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Quasi-Interpolation [Kõva köide]

(Justus-Liebig-Universität Giessen, Germany), (Justus-Liebig-Universität Giessen, Germany)
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Püsilink: https://www.kriso.ee/db/9781107072633.html
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Quasi-interpolation is one of the most useful and often applied methods for the approximation of functions and data in mathematics and practice. This book provides an introduction into the field for graduate students and researchers, discussing both the theory and applications of quasi-interpolants.

Quasi-interpolation is one of the most useful and often applied methods for the approximation of functions and data in mathematics and applications. Its advantages are manifold: quasi-interpolants are able to approximate in any number of dimensions, they are efficient and relatively easy to formulate for scattered and meshed nodes and for any number of data. This book provides an introduction into the field for graduate students and researchers, outlining all the mathematical background and methods of implementation. The mathematical analysis of quasi-interpolation is given in three directions, namely on the basis (spline spaces, radial basis functions) from which the approximation is taken, on the form and computation of the quasi-interpolants (point evaluations, averages, least squares), and on the mathematical properties (existence, locality, convergence questions, precision). Learn which type of quasi-interpolation to use in different contexts and how to optimise its features to suit applications in physics and engineering.

Arvustused

' the overall exposition and references make this book a potentially useful reference and an appropriate starting point for an advanced graduate student or researcher interested in studying the subject.' Edward J. Fuselier, MathSciNet

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Delve into an in-depth description and analysis of quasi-interpolation, starting from various areas of approximation theory.
Preface xi
1 Introduction
1(5)
2 Generalities on Quasi-Interpolation
6(20)
2.1 Approximation Properties
9(4)
2.2 Main Examples of Quasi-Interpolants
13(11)
2.3 Some Applications Connected with Different Forms of Quasi-Interpolants
24(2)
3 Univariate RBF Quasi-Interpolants
26(12)
3.1 Univariate Quasi-Interpolants
27(2)
3.2 Examples of Convergence Results
29(4)
3.3 Further Examples
33(4)
3.4 Notes
37(1)
4 Spline Quasi-Interpolants
38(38)
4.1 Spaces of Piecewise Polynomials
38(3)
4.2 General Form of Spline Quasi-Interpolants
41(2)
4.3 Marsden's Identity and de Boor-Fix Differential Quasi-Interpolants
43(7)
4.4 Approximation Order and Error Estimates
50(6)
4.5 Discrete Quasi-Interpolants
56(7)
4.6 Integral Quasi-Interpolants
63(3)
4.7 Local Spline Projectors
66(1)
4.8 Near-Minimally Normed Quasi-Interpolants
67(6)
4.9 Schoenberg's Quasi-Interpolation and Discrete Spline Quasi-Interpolants
73(1)
4.10 Notes
74(2)
5 Quasi-Interpolants for Periodic Functions
76(30)
5.1 Trigonometric Spline Quasi-interpolants
77(13)
5.2 Multiquadric Trigonometric Spline Quasi-Interpolants
90(3)
5.3 De la Vallee Poussin Quasi-Interpolants
93(11)
5.4 Notes
104(2)
6 Multivariate Spline Quasi-Interpolants
106(37)
6.1 Quasi-Interpolants from Tensor Product Splines
107(11)
6.2 Spline Spaces Constructed from Shifts of One Spline
118(23)
6.3 Notes
141(2)
7 Multivariate Quasi-Interpolants: Construction in n Dimensions
143(28)
7.1 Introduction and Further Notation
143(3)
7.2 Polynomial Reproduction
146(8)
7.3 Examples
154(7)
7.4 Convergence Theorems
161(4)
7.5 Quasi-Interpolation via Approximate Approximation
165(4)
7.6 Notes
169(2)
8 Quasi-Interpolation on the Sphere
171(17)
8.1 Spherical Quasi-Interpolation using Tensor Products
172(4)
8.2 Quasi-Interpolation on the Sphere Using Approximate Fourier Coefficients
176(8)
8.3 Quasi-Interpolants using Shepard's Method for the Sphere
184(2)
8.4 Notes
186(2)
9 Other Quasi-Interpolants and Wavelets
188(38)
9.1 Prewavelets with Radial Basis Functions and Quasi-Lagrange Bases
188(15)
9.2 Least-Squares Approximations as Quasi-Interpolants
203(12)
9.3 Weierstrass Operators
215(10)
9.4 Notes
225(1)
10 Special Cases and Applications
226(40)
10.1 Even- and Odd-Dimensional Spaces and Equally Spaced Centres
226(14)
10.2 A Look at the Inverse Multiquadric Radial Function in One Dimension
240(8)
10.3 Applications: Fredholm Integral Equations
248(4)
10.4 Applications: Numerical Solution of Partial Differential Equations
252(7)
10.5 Compression in the Space of Continuous Functions
259(6)
10.6 Notes
265(1)
References 266(8)
Index 274
Martin D. Buhmann is Professor in the Mathematics Department at Justus Liebig University Giessen. He is the author of over 100 papers in numerical analysis, approximation theory, optimisation and differential equations, and of the monograph Radial Basis Functions: Theory and Implementations (Cambridge, 2003). Janin Jäger is Postdoctoral Fellow in the Mathematics Department at Justus Liebig University Giessen. Her research focuses on approximation theory using radial basis functions and their application to spherical data and neurophysiology.