Table Of ContentInterpolation and Extrapolation Optimal Designs 1
Series Editor
Nikolaos Limnios
Interpolation and Extrapolation
Optimal Designs 1
Polynomial Regression and
Approximation Theory
Giorgio Celant
Michel Broniatowski
First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Library of Congress Control Number: 2016933881
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ISBN 978-1-84821-995-3
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Part1.ElementsfromApproximationTheory . . . . . . . . . . . . . . . 1
Chapter1.UniformApproximation . . . . . . . . . . . . . . . . . . . . . . 3
1.1.Canonicalpolynomialsanduniformapproximation . . . . . . . . . . . 3
1.2.Existenceofthebestapproximation . . . . . . . . . . . . . . . . . . . . 4
1.3.Characterizationanduniquenessofthebestapproximation . . . . . . . 5
1.3.1.ProofoftheBorel–Chebyshevtheorem . . . . . . . . . . . . . . . 7
1.3.2.Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter2.ConvergenceRatesfortheUniformApproximation
andAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.TheBorel–Chebyshevtheoremandstandardfunctions . . . . . . . . . 15
2.3.Convergenceoftheminimaxapproximation . . . . . . . . . . . . . . . 20
2.3.1.Rateofconvergenceoftheminimaxapproximation . . . . . . . . . 21
2.4.ProofofthedelaValléePoussintheorem . . . . . . . . . . . . . . . . . 24
2.5.TheYevgenyYakovlevichRemezalgorithm . . . . . . . . . . . . . . . 28
2.5.1.TheRemezalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.2.ConvergenceoftheRemezalgorithm . . . . . . . . . . . . . . . . . 33
Chapter3.ConstrainedPolynomialApproximation . . . . . . . . . . . 43
3.1.Introductionandexamples . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.Lagrangepolynomialinterpolation . . . . . . . . . . . . . . . . . . . . 47
3.3.Theinterpolationerror . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
vi InterpolationandExtrapolationOptimalDesigns1
3.3.1. Aqualitativeresult . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2.Aquantitativeresult . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.Theroleofthenodesandtheminimization
oftheinterpolationerror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.Convergenceoftheinterpolationapproximation . . . . . . . . . . . . . 56
3.6.Rungephenomenonandlackofconvergence . . . . . . . . . . . . . . . 57
3.7.UniformapproximationforC(∞)([a,b])functions . . . . . . . . . . . 62
3.8.Numericalinstability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.9.Convergence,choiceofthedistributionofthe
nodes,Lagrangeinterpolationandsplines . . . . . . . . . . . . . . . . . . . 67
Part2.OptimalDesignsforPolynomialModels . . . . . . . . . . . . . 69
Chapter 4. Interpolation and Extrapolation Designs for the
PolynomialRegression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.Definitionofthemodelandoftheestimators . . . . . . . . . . . . . . . 71
4.2.Optimalextrapolationdesigns: Hoel–LevineorChebyshevdesigns . . 75
4.2.1.Uniformoptimalinterpolationdesigns(accordingtoGuest) . . . . 85
4.2.2.TheinterplaybetweentheHoel–LevineandtheGuestdesigns . . . 95
4.2.3.Confidenceboundforinterpolation/extrapolationdesigns. . . . . . 98
4.3. AnapplicationoftheHoel–Levinedesign . . . . . . . . . . . . . . . . 100
4.4.Multivariateoptimaldesigns: aspecialcase . . . . . . . . . . . . . . . 103
Chapter5.AnIntroductiontoExtrapolationProblems
BasedonObservationsonaCollectionofIntervals . . . . . . . . . . 113
5.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2.Themodel,theestimatorandthecriterion
forthechoiceofthedesign . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.1.Criterionfortheoptimaldesign . . . . . . . . . . . . . . . . . . . . 121
5.3.AconstrainedBorel–Chebyshevtheorem . . . . . . . . . . . . . . . . . 122
5.3.1.ExistenceofsolutionstothePg−1(0,1)problem . . . . . . . . . . 122
5.3.2.Aqualitativediscussiononsome
constrainedBorel–Chebyshevtheorem . . . . . . . . . . . . . . . . . . . . 123
5.3.3.Borel–Chebyshevtheoremon[a,b]∪[d,e] . . . . . . . . . . . . . . 125
5.3.4.FromtheconstrainedBorel–Chebyshev
theoremtothesupportoftheoptimaldesign . . . . . . . . . . . . . . . . 126
5.4.Qualitativepropertiesofthepolynomialwhich
determinestheoptimalnodes . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4.1.Thelinearcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4.2.Thegeneralpolynomialcase . . . . . . . . . . . . . . . . . . . . . . 128
5.5.Identificationofthepolynomialwhich
characterizestheoptimalnodes . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.5.1.Thedifferentialequation . . . . . . . . . . . . . . . . . . . . . . . . 130
5.5.2.Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.6.Theoptimaldesigninfavorablecases . . . . . . . . . . . . . . . . . . . 134
Contents vii
5.6.1.Someexplicitoptimaldesigns . . . . . . . . . . . . . . . . . . . . . 136
5.7.Theoptimaldesigninthegeneralcase . . . . . . . . . . . . . . . . . . 137
5.7.1.Theextremepointsofalinearfunctional . . . . . . . . . . . . . . . 138
5.7.2.Someresultsontherepresentationoftheextremepoints . . . . . . 138
5.7.3.ThespecificcaseoftheDiracfunctionalatpoint0 . . . . . . . . . 142
5.7.4.Remezalgorithmfortheextreme
polynomial: theoptimaldesigningeneralcases. . . . . . . . . . . . . . . 145
5.8.Spruilltheorem: theoptimaldesign . . . . . . . . . . . . . . . . . . . . 146
Chapter6.InstabilityoftheLagrangeInterpolation
SchemeWithRespecttoMeasurementErrors . . . . . . . . . . . . . . 147
6.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.2.Theerrorsthatcannotbeavoided . . . . . . . . . . . . . . . . . . . . . 147
6.2.1.Theroleoftheerrors: interpolationdesignswith
minimalpropagationoftheerrors . . . . . . . . . . . . . . . . . . . . . . . 150
6.2.2.Optimizingonthenodes . . . . . . . . . . . . . . . . . . . . . . . . 153
6.3.Controloftherelativeerrors . . . . . . . . . . . . . . . . . . . . . . . . 157
6.3.1.ImplementationoftheRemezalgorithm
fortherelativeerrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.4.Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.5.Someinequalitiesforthederivativesofpolynomials. . . . . . . . . . . 167
6.6.Concentrationinequalities . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.7.Upperboundsoftheextrapolationerrorduetorandomness,and
theresultingsizeofthedesignforrealanalyticregressionfunctions . . . . 172
6.7.1.Case1: therangeoftheobservationsisbounded . . . . . . . . . . 177
6.7.2.Case2: therangeoftheobservationsisunbounded . . . . . . . . . 183
Part3.MathematicalMaterial . . . . . . . . . . . . . . . . . . . . . . . . . 185
Appendix1.NormedLinearSpaces . . . . . . . . . . . . . . . . . . . . . 187
Appendix2.ChebyshevPolynomials . . . . . . . . . . . . . . . . . . . . 217
Appendix3.SomeUsefulInequalitiesforPolynomials . . . . . . . . 221
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Description:This book is the first of a series which focuses on the interpolation and extrapolation of optimal designs, an area with significant applications in engineering, physics, chemistry and most experimental fields. In this volume, the authors emphasize the importance of problems associated with the cons
