Devore / Scherer Quantitative Approximation

Proceedings of a Symposium on Quantitative Approximation Held in Bonn, West Germany, August 20-24, 1979
1. Auflage 2014
ISBN: 978-1-4832-6512-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 336 Seiten, Web PDF

Quantitative Approximation provides information pertinent to nonlinear approximation, including rational approximation and optimal knot spline approximation. This book discusses spline approximation with the most emphasis on multivariate and knot independent questions. Organized into 26 chapters, this book begins with an overview of the inequality for the sharp function in terms of the maximal rearrangement. This text then examines the best co-approximation in a Hilbert space wherein the existence ad uniqueness sets are the closed flats. Other chapters consider the inverse of the coefficient matrix for the system satisfied by the B-spline coefficients of the cubic spline interpolant at knots. This book discusses as well the relationship between the structural properties of a function and its degree of approximation by rational functions. The final chapter deals with the problem of existence of continuous selections for metric projections and provides a solution for this problem. This book is a valuable resource for mathematicians.

Autoren/Hrsg.

Devore, Ronald A.

Scherer, Karl

Inhaltsverzeichnis

1;Front Cover;1
2;Quantitative Approximation;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;10
6;Preface;12
7;CHAPTER 1. ON AN INEQUALITY FOR THE SHARP FUNCTION;14
7.1;REFERENCES;19
8;CHAPTER 2. ON THE BEST CO-APPROXIMATION IN A HILBERT SPACE;20
8.1;1. PRELIMINARY REMARKS;20
8.2;2. PROOF OF THE THEOREM;22
8.3;REFERENCES;23
9;CHAPTER 3. WHAT IS THE MAIN DIAGONAL OF A BIINFINITE BAND MATRIX ?;24
9.1;INTRODUCTION;24
9.2;1. MESH RATIO RESTRICTIONS IN EVEN-ORDER SPLINE INTERPOLATION AT KNOTS;25
9.3;2. THE INVERSE OF A BIINFINITE BAND MATRIX;27
9.4;3. CUBIC SPLINE INTERPOLATION AT KNOTS;33
9.5;REFERENCES;36
10;CHAPTER 4. RATIONAL APPROXIMATION AND EXOTIC LIPSCHITZ SPACES;38
10.1;REFERENCES;43
11;CHAPTER 5. SOME MATHEMATICAL PROPERTIES OF EMPIRICALGIBBS FUNCTIONS;44
11.1;§ 1 INTRODUCTION;44
11.2;§ 2 MINIMUM PROBLEM AND DUAL;45
11.3;§ 3 EMPIRICAL GIBBS FUNCTIONS BASED ON THE SOAVE-REDLICH-KWONG EQUATION OF STATE;49
11.4;§ 4 STABILITY AND LOCAL SOLVABILITY OF EQUILIBRIUM EQUATIONS;54
11.5;REFERENCES;60
12;CHAPTER 6. MULTIVARIATE INTERPOLATION AND THE RADON TRANSFORM,PART II: SOME FURTHER EXAMPLES;62
12.1;1. INTRODUCTION;62
12.2;2. MULTIVARIATE ABEL-GONCAROV INTERPOLATION;64
12.3;3. AREA MATCHING BY POLYNOMIALS;66
12.4;4. LIDSTONE TYPE INTERPOLATION ON m POINTS;69
12.5;REFERENCES;75
13;CHAPTER 7. LACUNARY TRIGONOMETRIC INTERPOLATION ON EQUIDISTANT NODES;76
13.1;1. INTRODUCTION;76
13.2;2. SOME DETERMINANTS;78
13.3;3. THE CASE n ODD: n = 2r+1;83
13.4;4. EXPLICIT FORMS FOR THE FUNDAMENTAL POLYNOMIALS;90
13.5;5. CONCLUSION;92
13.6;REFERENCES;93
14;CHAPTER 8. MONOTONE APPROXIMATION BY SPLINE FUNCTIONS;94
14.1;1. INTRODUCTION;94
14.2;2. MONOTONE APPROXIMATION BY PIECEWISE POLYNOMIALS;96
14.3;3. MONOTONE APPROXIMATION BY SPLINES;102
14.4;REFERENCES;111
15;CHAPTER 9. APPROXIMATION BY SMOOTH MULTIVARIATE SPLINES ONNON-UNI FORM GRIDS;112
15.1;1. INTRODUCTION;112
15.2;2. LOCAL REFINEMENTS OF UNIFORMCONFIGURATIONS OF KNOT SETS;113
15.3;3. LOCAL ERROR ESTIMATES;121
15.4;REFERENCES;127
16;CHAPTER 10. APPROXIMATION BY SMALL RANK TENSOR PRODUCTS OF SPLINES;128
16.1;REFERENCES;133
17;CHAPTER 11. VARIABLE KNOT, VARIABLE DEGREE SPLINE APPROXIMATION TO Xß;134
17.1;0. INTRODUCTION;134
17.2;1. POLYNOMIAL APPROXIMATION OF Xß;135
17.3;2. MAIN RESULTS;139
17.4;REFERENCES;144
17.5;ACKNOWLEDGEMENT;144
18;CHAPTER 12. ON SHARP NECESSARY CONDITIONS FOR RADIALFOURIER MULTIPLIERS;146
18.1;REFERENCES;154
19;CHAPTER 13. APPROXIMATING BIVARIATE FUNCTIONS AND MATRICESBY NOMOGRAPHIC FUNCTIONS;156
19.1;REFERENCES;164
20;CHAPTER 14. L8 -BOUNDS OF L2-- PROJECTIONS ON SPLINES;166
20.1;1. INTRODUCTION;166
20.2;2. PRELIMINARIES;167
20.3;3. SUFFICIENT CONDITIONS FOR UNIFORM BOUNDS;169
20.4;4. GALERKIN PROJECTIONS;173
20.5;REFERENCES;175
21;CHAPTER 15. DIAMETERS OF CLASSES OF SMOOTH FUNCTIONS;176
21.1;O. INTRODUCTION;176
21.2;1. DEFINITIONS, BASIC PROPERTIES OF SPLINE FUNCTIONS;177
21.3;2. ESTIMATES FOR an;179
21.4;3. n-WIDTH;182
21.5;4. APPROXIMATION NUMBERS;183
21.6;5. ENTROPY;184
21.7;REFERENCES;187
22;CHAPTER 16. ON ESTIMATES OF DIAMETERS;190
22.1;REFERENCES;197
23;CHAPTER 17. BOUNDS FOR THE ERROR IN TRIGONOMETRIC HERMITE INTERPOLATION;198
23.1;1. INTRODUCTION AND SUMMARY;198
23.2;2. TRIGONOMETRIC B-SPLINES;200
23.3;3. TRIGONOMETRIC INTERPOLATION I;205
23.4;4. TRIGONOMETRIC INTERPOLATION II;208
23.5;REFERENCES;209
24;CHAPTER 18. BIRKHOFF INTERPOLATION: SOME APPLICATIONS OF COALESCENCE;210
24.1;1. THE METHOD OF COALESCENCE;210
24.2;2. THE NUMBER C(L1,...,Lp;L1,...,Lp ') AND SIGNS OF DETERMINANTS;213
24.3;3. THE FUNCTION d (E);215
24.4;4. THE THEOREM OF FERGUSON;217
24.5;REFERENCES;221
25;CHAPTER 19. n-WIDTHS OF OCTAHEDRA;222
25.1;1. INTRODUCTION;222
25.2;2. EXACT VALUES;223
25.3;3. ESTIMATES;226
25.4;REFERENCES;229
26;CHAPTER 20. THE APPROXIMATION OF MULTIPLE INTEGRALS BY USING INTERPOLATORY CUBATURE FORMULAE;230
26.1;1. INTRODUCTION;230
26.2;2. LOWER BOUNDS FOR THE NUMBER OF KNOTS;232
26.3;3. COMMON ROOTS OF POLYNOMIALS AND CUBATURE FORMULAE;235
26.4;4. THE METHOD OF REPRODUCING KERNEL;239
26.5;5. INVARIANT CUBATURE FORMULAE;242
26.6;REFERENCES;253
27;CHAPTER 21. INTERPOLATORY CUBATURE FORMULAE AND REAL IDEALS;258
27.1;1. INTRODUCTION;258
27.2;2. REAL IDEALS;259
27.3;3. CHARACTERIZATION OF INTERPOLATORY CUBATURE FORMULAE;262
27.4;4. APPLICATIONS;263
27.5;REFERENCES;267
28;CHAPTER 22. STRONG UNIQUENESS OF BEST APPROXIMATIONS AND WEAK CHEBYSHEV SYSTEMS;268
28.1;1. INTRODUCTION;268
28.2;2. THE CHARACTERIZATION THEOREM;269
28.3;REFERENCES;279
29;CHAPTER 23. ON THE CONNECTION BETWEEN RATIONAL UNIFORM APPROXIMATION AND POLYNOMIAL LP APPROXIMATION OF FUNCTIONS;280
29.1;1. INTRODUCTION;280
29.2;2. MAIN RESULTS;281
29.3;3. PROOF OF THEOREM 3;283
29.4;REFERENCES;290
30;CHAPTER 24. BEST CHEBYSHEV APPROXIMATION BY SMOOTH FUNCTIONS;292
30.1;REFERENCES;302
31;CHAPTER 25. APPROXIMATIONS IN THE HARDY SPACE H1 (D) WITH RESPECT TO THE NORM TOPOLOGY;304
31.1;1. INTRODUCTION;304
31.2;2. TOEPLITZ OPERATORS WITH PERIODIC L1-SYMBOLS;306
31.3;3. PATIL TYPE APPROXIMATIONS WITH RESPECT TO I I ? I I H1 (D);311
31.4;REFERENCES;313
32;CHAPTER 26. CONTINUOUS SELECTIONS FOR METRIC PROJECTIONS;314
32.1;1. INTRODUCTION;314
32.2;2. PRELIMINARIES;316
32.3;3. DECOMPOSITION OF Wn INTO SUBCLASSES;317
32.4;4. THE CHARACTERIZATION THEOREM;322
32.5;REFERENCES;329
33;PROBLEMS;332