Best Approximation in Inner Product SpacesThis 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. |
Contents
Inner Product Space | xviii |
Inner Product Spaces | xviii |
Orthogonality | 4 |
Topological Notions | 6 |
Hilbert Space | 10 |
Exercises | 11 |
Historical Notes | 14 |
Best Approximation | 17 |
Exercises | 144 |
Historical Notes | 147 |
Generalized Solutions of Linear Equations | 151 |
The Uniform Boundedness and Open Mapping Theorems | 160 |
The Closed Range and Bounded Inverse Theorems | 164 |
The Closed Graph Theorem | 165 |
Adjoint of a Linear Operator | 167 |
Generalized Solutions to Operator Equations | 173 |
Convex Sets | 18 |
Five Basic Problems Revisited | 23 |
Exercises | 26 |
Historical Notes | 28 |
Existence and Uniqueness of Best Approximations | 29 |
Uniqueness of Best Approximations | 31 |
Compactness Concepts | 34 |
Exercises | 35 |
Historical Notes | 36 |
Characterization of Best Approximations | 39 |
Dual Cones | 40 |
Characterizing Best Approximations from Subspaces | 46 |
GramSchmidt Orthonormalization | 47 |
Fourier Analysis | 50 |
Solutions to the First Three Basic Problems | 57 |
Exercises | 60 |
Historical Notes | 65 |
The Metric Projection | 67 |
Linear Metric Projections | 73 |
The Reduction Principle | 76 |
Exercises | 80 |
Historical Notes | 83 |
Bounded Linear Functionals and Best Approximation from Hyperplanes and HalfSpaces | 85 |
Representation of Bounded Linear Functionals | 89 |
Best Approximation from Hyperplanes | 93 |
Strong Separation Theorem | 98 |
Best Approximation from HalfSpaces | 103 |
Best Approximation from Polyhedra | 105 |
Exercises | 113 |
Historical Notes | 118 |
Error of Approximation | 121 |
Distance to FiniteDimensional Subspaces | 125 |
FiniteCodimensional Subspaces | 129 |
The Weierstrass Approximation Theorem | 135 |
Miintzs Theorem | 139 |
Generalized Inverse | 175 |
Exercises | 183 |
Historical Notes | 187 |
The Method of Alternating Projections | 189 |
Angle Between Two Subspaces | 193 |
Rate of Convergence for Alternating Projections Two Subspaces | 197 |
Weak Convergence | 199 |
Dykstras Algorithm | 203 |
The Case of Affine Sets | 211 |
Rate of Convergence for Alternating Projections | 213 |
Examples | 222 |
Exercises | 226 |
Historical Notes | 229 |
Constrained Interpolation from a Convex Set | 233 |
Stong Conical Hull Intersection Property strong CHIP | 234 |
Affine Sets | 243 |
Relative Interiors and a Separation Theorem | 247 |
Extremal Subsets of C | 257 |
Constrained Interpolation by Positive Functions | 266 |
Exercises | 273 |
Historical Notes | 279 |
Interpolation and Approximation | 283 |
Simultaneous Approximation and Interpolation | 288 |
Simultaneous Approximation Interpolation and Normpreservation | 290 |
Exercises | 291 |
Historical Notes | 294 |
Convexity of Chebyshev Sets | 297 |
Chebyshev Suns | 298 |
Exercises | 302 |
Historical Notes | 303 |
Zorns Lemma | 307 |
Every Hilbert Space Is l₂I | 308 |
References | 311 |
| 327 | |
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Common terms and phrases
affine set approximation theory attains its norm Banach best approximation bounded linear functional bounded linear operator C₁ Cauchy sequence Chapter Chebyshev set Chebyshev subspace closed convex set closed convex subset closed subspaces complete compute converges convex cone convex set convex subset Corollary deduce defined denote dense Deutsch dual cones Dykstra's algorithm equations example Exercise exists finite finite-dimensional subspace following statements H₁ half-spaces Hence Hilbert space Hint holds hyperplane implies inequality inner product space integer interpolation Lemma Let x1 linearly independent mapping matrix metric projection Moreover normed linear space obtain orthogonal orthonormal basis orthonormal set particular Pc(x PK(x PM(x polynomials problem Proof proves proximinal real numbers representer result ri A(C scalars Show span statements are equivalent strong CHIP Suppose Theorem 4.9 theorem Theorem unique vectors
Popular passages
Page vi - An approximate answer to the right problem is worth a good deal more than an exact answer to an approximate problem.
Page vi - What is written without effort is in general read without pleasure...
Page x - Z/2-norm (ie, the norm of a function is the square root of the integral of the square of the function).