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Product Code:  GSM/101 
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Electronic ISBN:  9781470411657 
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Book DetailsGraduate Studies in MathematicsVolume: 101; 2000; 359 ppMSC: Primary 41;
This textbook is designed for graduate students in mathematics, physics, engineering, and computer science. Its purpose is to guide the reader in exploring contemporary approximation theory. The emphasis is on multivariable approximation theory, i.e., the approximation of functions in several variables, as opposed to the classical theory of functions in one variable.
Most of the topics in the book, heretofore accessible only through research papers, are treated here from the basics to the currently active research, often motivated by practical problems arising in diverse applications such as science, engineering, geophysics, and business and economics. Among these topics are projections, interpolation paradigms, positive definite functions, interpolation theorems of Schoenberg and Micchelli, tomography, artificial neural networks, wavelets, thinplate splines, box splines, ridge functions, and convolutions.
An important and valuable feature of the book is the bibliography of almost 600 items directing the reader to important books and research papers. There are 438 problems and exercises scattered through the book allowing the student reader to get a better understanding of the subject.
Originally published by Brooks Cole/Cengage Learning as ISBN: 9780534362249.ReadershipGraduate students and research mathematicians interested in approximation theory and applications.

Table of Contents

Chapters

Chapter 1. Introductory discussion of interpolation

Chapter 2. Linear interpolation operators

Chapter 3. Optimization of the Lagrange operator

Chapter 4. Multivariate polynomials

Chapter 5. Moving the nodes

Chapter 6. Projections

Chapter 7. Tensorproduct interpolation

Chapter 8. The Boolean algebra of projections

Chapter 9. The Newton paradigm for interpolation

Chapter 10. The Lagrange paradigm for interpolation

Chapter 11. Interpolation by translates of a single function

Chapter 12. Positive definite functions

Chapter 13. Strictly positive definite functions

Chapter 14. Completely monotone functions

Chapter 15. The Schoenberg interpolation theorem

Chapter 16. The Micchelli interpolation theorem

Chapter 17. Positive definite functions on spheres

Chapter 18. Approximation by positive definite functions

Chapter 19. Approximation reconstruction of functions and tomography

Chapter 20. Approximation by convolution

Chapter 21. The good kernels

Chapter 22. Ridge functions

Chapter 23. Ridge function approximation via convolutions

Chapter 24. Density of ridge functions

Chapter 25. Artificial neural networks

Chapter 26. Chebyshev centers

Chapter 27. Optimal reconstruction of functions

Chapter 28. Algorithmic orthogonal projections

Chapter 29. Cardinal Bsplines and the sinc function

Chapter 30. The GolombWeinberger theory

Chapter 31. Hilbert function spaces and reproducing kernels

Chapter 32. Spherical thinplate splines

Chapter 33. Box splines

Chapter 34. Wavelets, I

Chapter 35. Wavelets II

Chapter 36. Quasiinterpolation


Additional Material

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Working through this book provides an opportunity to review and apply areas of mathematics learned elsewhere, as well as to learn entirely new topics.
MAA Reviews 
The textbook, a clear and concise work written by worldrenowned experts in the field of approximation theory, will probe useful not only as a reference for professional mathematicians but also as a text for graduate students.
Mathematical Reviews


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 Book Details
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 Request Exam/Desk Copy
 Get Permissions
This textbook is designed for graduate students in mathematics, physics, engineering, and computer science. Its purpose is to guide the reader in exploring contemporary approximation theory. The emphasis is on multivariable approximation theory, i.e., the approximation of functions in several variables, as opposed to the classical theory of functions in one variable.
Most of the topics in the book, heretofore accessible only through research papers, are treated here from the basics to the currently active research, often motivated by practical problems arising in diverse applications such as science, engineering, geophysics, and business and economics. Among these topics are projections, interpolation paradigms, positive definite functions, interpolation theorems of Schoenberg and Micchelli, tomography, artificial neural networks, wavelets, thinplate splines, box splines, ridge functions, and convolutions.
An important and valuable feature of the book is the bibliography of almost 600 items directing the reader to important books and research papers. There are 438 problems and exercises scattered through the book allowing the student reader to get a better understanding of the subject.
Originally published by Brooks Cole/Cengage Learning as ISBN: 9780534362249.
Graduate students and research mathematicians interested in approximation theory and applications.

Chapters

Chapter 1. Introductory discussion of interpolation

Chapter 2. Linear interpolation operators

Chapter 3. Optimization of the Lagrange operator

Chapter 4. Multivariate polynomials

Chapter 5. Moving the nodes

Chapter 6. Projections

Chapter 7. Tensorproduct interpolation

Chapter 8. The Boolean algebra of projections

Chapter 9. The Newton paradigm for interpolation

Chapter 10. The Lagrange paradigm for interpolation

Chapter 11. Interpolation by translates of a single function

Chapter 12. Positive definite functions

Chapter 13. Strictly positive definite functions

Chapter 14. Completely monotone functions

Chapter 15. The Schoenberg interpolation theorem

Chapter 16. The Micchelli interpolation theorem

Chapter 17. Positive definite functions on spheres

Chapter 18. Approximation by positive definite functions

Chapter 19. Approximation reconstruction of functions and tomography

Chapter 20. Approximation by convolution

Chapter 21. The good kernels

Chapter 22. Ridge functions

Chapter 23. Ridge function approximation via convolutions

Chapter 24. Density of ridge functions

Chapter 25. Artificial neural networks

Chapter 26. Chebyshev centers

Chapter 27. Optimal reconstruction of functions

Chapter 28. Algorithmic orthogonal projections

Chapter 29. Cardinal Bsplines and the sinc function

Chapter 30. The GolombWeinberger theory

Chapter 31. Hilbert function spaces and reproducing kernels

Chapter 32. Spherical thinplate splines

Chapter 33. Box splines

Chapter 34. Wavelets, I

Chapter 35. Wavelets II

Chapter 36. Quasiinterpolation

Working through this book provides an opportunity to review and apply areas of mathematics learned elsewhere, as well as to learn entirely new topics.
MAA Reviews 
The textbook, a clear and concise work written by worldrenowned experts in the field of approximation theory, will probe useful not only as a reference for professional mathematicians but also as a text for graduate students.
Mathematical Reviews