Table Of ContentIntegration and
Cubature Methods
A Geomathematically
Oriented Course
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
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MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Integration and
Cubature Methods
A Geomathematically
Oriented Course
Willi Freeden
Martin Gutting
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———————————————————————————————————————–
Names:Freeden,W.,author.|Gutting,Martin,1978-author.
Title:Integrationandcubaturemethods:ageomathematicallyorientedcourse
/WilliFreedenandMartinGutting.
Description:BocaRaton:CRCPress,[2018]|Series:Chapman&Hall/CRC
monographsandresearchnotesinmathematics|Includesbibliographical
referencesandindex.
Identifiers:LCCN2017026232|ISBN9781138718821(hardcover:alk.paper)|
ISBN9781315195674(ebook)|ISBN9781351764766(ebook)|ISBN
9781351764759(ebook)|ISBN9781351764742(ebook)
Subjects:LCSH:Cubatureformulas.|Numericalintegration.|Differential
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Classification:LCCQA299.4.C83F742018|DDC515/.43–dc23
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Contents
Preface xiii
About the Authors xvii
List of Symbols xix
Introduction xxi
0.1 Necessity of Quadrature . . . . . . . . . . . . . . . . . . . . . xxi
0.2 Historical Roots of the Book . . . . . . . . . . . . . . . . . . xxiii
0.3 Own Roots and Concept of the Book . . . . . . . . . . . . . xxix
I Preparatory 1D-Integration 1
1 Algebraic Polynomial Integration 3
1.1 Interpolatory Integration Rules . . . . . . . . . . . . . . . . . 3
1.2 Peano’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Error Truncation . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Algebraic Spline Integration 11
2.1 Spline Integration Formulas . . . . . . . . . . . . . . . . . . . 11
2.2 Spline Interpolation . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Best Approximation and Spline Exact
Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Periodic Polynomial Integration 23
3.1 Integer Lattice and Periodic Polynomials . . . . . . . . . . . 23
3.2 Lattice Functions . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Euler Summation Formulas . . . . . . . . . . . . . . . . . . . 32
3.4 Euler Summation Formulas for Periodic Functions . . . . . . 35
4 Periodic Spline Integration 39
4.1 Best Approximate Integration in Sobolev Spaces . . . . . . . 39
4.2 Spline Lagrange Basis . . . . . . . . . . . . . . . . . . . . . . 42
vii
viii Contents
4.3 Peano’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Best Approximation and Spline Exact
Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 Smoothing Splines for Erroneous Data Points . . . . . . . . . 56
5 Trapezoidal Rules 63
5.1 Riemann Zeta Function and Lattice Function . . . . . . . . . 63
5.2 Classical Trapezoidal Sums for Finite Intervals . . . . . . . . 68
5.3 Romberg Integration . . . . . . . . . . . . . . . . . . . . . . 71
5.4 Poisson Summation Based Integration . . . . . . . . . . . . 73
5.5 Trapezoidal Sums over Dilated Lattices . . . . . . . . . . . . 79
6 Adaptive Trapezoidal Rules 83
6.1 Lattice Functions for Helmholtz Operators . . . . . . . . . . 84
6.2 Adaptive Trapezoidal Sums over Finite
Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3 Adaptive Poisson Summation Formula over Infinite Intervals 88
6.4 Adaptive Trapezoidal Sums over Infinite Intervals . . . . . . 92
6.5 Discontinuous Integrals of Hardy–Landau Type . . . . . . . 93
6.6 Periodic Polynomial Accuracy . . . . . . . . . . . . . . . . . 95
7 Legendre Polynomial Reflected Integration 99
7.1 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . 99
7.2 Legendre (Green’s) Functions . . . . . . . . . . . . . . . . . . 105
7.3 Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . . 108
8 Gaussian Integration 111
8.1 Gaussian Quadrature Formulas . . . . . . . . . . . . . . . . . 111
8.2 Adaptive Remainder Terms Involving Green’s
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.3 Convergence of Gaussian Quadrature . . . . . . . . . . . . . 118
II Integration on 2D-Spheres 121
9 Remainder Terms Involving Beltrami Operators 123
9.1 Spherical Framework . . . . . . . . . . . . . . . . . . . . . . 124
9.2 Sphere Functions Involving Beltrami Operators . . . . . . . . 133
9.3 Best Approximate Integration by Splines . . . . . . . . . . . 151
9.4 Integral Formulas under Boundary Conditions . . . . . . . . 158
9.5 Sphere Functions and Shannon Kernels . . . . . . . . . . . . 162
9.6 Peano’s Theorem Involving Beltrami Operators . . . . . . . 176
Contents ix
10 Integration Rules with Polynomial Accuracy 179
10.1 Lagrangian Integration . . . . . . . . . . . . . . . . . . . . . 179
10.2 Lebesgue Functions . . . . . . . . . . . . . . . . . . . . . . . 186
10.3 Spherical Geometry and Polynomial Cubature Rules . . . . . 192
10.4 Interpolatory Rules Based on Extremal Point Systems and
Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.5 Non-Existence of Spherical Gaussian Rules . . . . . . . . . . 196
11 Latitude-Longitude Cubature 199
11.1 Associated Legendre Functions . . . . . . . . . . . . . . . . . 199
11.2 Legendre Spherical Harmonics . . . . . . . . . . . . . . . . . 204
11.3 Latitude-Longitude Integration . . . . . . . . . . . . . . . . . 206
12 Remainder Terms Involving Pseudodifferential Operators 215
12.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 215
12.2 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . 217
12.3 Reproducing Kernels and Remainder Terms . . . . . . . . . 222
12.4 Particular Types of Kernel Functions . . . . . . . . . . . . . 228
12.5 Locally Supported Kernels . . . . . . . . . . . . . . . . . . . 238
12.6 Zonal Function Exact Integration . . . . . . . . . . . . . . . 247
13 Spline Exact Integration 251
13.1 Spline Interpolation . . . . . . . . . . . . . . . . . . . . . . . 251
13.2 Peano’s Theorem in Terms of Pseudodifferential Operators . 256
13.3 Best Approximations . . . . . . . . . . . . . . . . . . . . . . 258
13.4 Spline Exact Integration Formulas . . . . . . . . . . . . . . . 260
14 Equidistributions and Discrepancy Methods 265
14.1 Equidistributions . . . . . . . . . . . . . . . . . . . . . . . . 265
14.2 Discrepancy Variants . . . . . . . . . . . . . . . . . . . . . . 267
14.3 Examples of Equidistributions on the Sphere . . . . . . . . . 271
14.4 Sobolev Space Based Generalized Discrepancy . . . . . . . . 279
14.5 Statistics for Equidistributions . . . . . . . . . . . . . . . . . 289
15 Multiscale Approximate Integration 301
15.1 Singular Integrals and Approximate Identities . . . . . . . . 301
15.2 Locally Supported Scaling Functions . . . . . . . . . . . . . . 306
15.3 Locally Supported Difference Wavelets . . . . . . . . . . . . 310
15.4 Integration for Large Equidistributed Data . . . . . . . . . . 313
15.5 Error Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 320
