Ebook: Best Approximation in Inner Product Spaces

This book evolved from notes originally developed for a graduate course, "Best Approximation in Normed Linear Spaces," that I began giving at Penn State Uni­ versity more than 25 years ago. It soon became evident. that many of the students who wanted to take the course (including engineers, computer scientists, and statis­ ticians, as well as mathematicians) did not have the necessary prerequisites such as a working knowledge of Lp-spaces and some basic functional analysis. (Today such material is typically contained in the first-year graduate course in analysis. ) To accommodate these students, I usually ended up spending nearly half the course on these prerequisites, and the last half was devoted to the "best approximation" part. I did this a few times and determined that it was not satisfactory: Too much time was being spent on the presumed prerequisites. To be able to devote most of the course to "best approximation," I decided to concentrate on the simplest of the normed linear spaces-the inner product spaces-since the theory in inner product spaces can be taught from first principles in much less time, and also since one can give a convincing argument that inner product spaces are the most important of all the normed linear spaces anyway. The success of this approach turned out to be even better than I had originally anticipated: One can develop a fairly complete theory of best approximation in inner product spaces from first principles, and such was my purpose in writing this book.

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This is the first systematic study of best approximation theory in inner product spaces and, in particular, in Hilbert space. Geometric considerations play a prominent role in developing and understanding the theory. The only prerequisites for reading the book is some knowledge of advanced calculus and linear algebra. Throughout the book, examples and applications have been interspersed with the theory. Each chapter concludes with numerous exercises and a section in which the author puts the results of that chapter into a historical perspective. The book is based on lecture notes for a graduate course on best approximation which the author has taught for over 25 years.

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This is the first systematic study of best approximation theory in inner product spaces and, in particular, in Hilbert space. Geometric considerations play a prominent role in developing and understanding the theory. The only prerequisites for reading the book is some knowledge of advanced calculus and linear algebra. Throughout the book, examples and applications have been interspersed with the theory. Each chapter concludes with numerous exercises and a section in which the author puts the results of that chapter into a historical perspective. The book is based on lecture notes for a graduate course on best approximation which the author has taught for over 25 years.
Content:
Front Matter....Pages i-xv
Inner Product Spaces....Pages 1-19
Best Approximation....Pages 21-32
Existence and Uniqueness of Best Approximations....Pages 33-41
Characterization of Best Approximations....Pages 43-70
The Metric Projection....Pages 71-87
Bounded Linear Functionals and Best Approximation from Hyperplanes and Half-Spaces....Pages 89-123
Error of Approximation....Pages 125-153
Generalized Solutions of Linear Equations....Pages 155-192
The Method of Alternating Projections....Pages 193-235
Constrained Interpolation from a Convex Set....Pages 237-285
Interpolation and Approximation....Pages 287-299
Convexity of Chebyshev Sets....Pages 301-309
Back Matter....Pages 311-338

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This is the first systematic study of best approximation theory in inner product spaces and, in particular, in Hilbert space. Geometric considerations play a prominent role in developing and understanding the theory. The only prerequisites for reading the book is some knowledge of advanced calculus and linear algebra. Throughout the book, examples and applications have been interspersed with the theory. Each chapter concludes with numerous exercises and a section in which the author puts the results of that chapter into a historical perspective. The book is based on lecture notes for a graduate course on best approximation which the author has taught for over 25 years.
Content:
Front Matter....Pages i-xv
Inner Product Spaces....Pages 1-19
Best Approximation....Pages 21-32
Existence and Uniqueness of Best Approximations....Pages 33-41
Characterization of Best Approximations....Pages 43-70
The Metric Projection....Pages 71-87
Bounded Linear Functionals and Best Approximation from Hyperplanes and Half-Spaces....Pages 89-123
Error of Approximation....Pages 125-153
Generalized Solutions of Linear Equations....Pages 155-192
The Method of Alternating Projections....Pages 193-235
Constrained Interpolation from a Convex Set....Pages 237-285
Interpolation and Approximation....Pages 287-299
Convexity of Chebyshev Sets....Pages 301-309
Back Matter....Pages 311-338
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