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Best approximation in inner product spaces. (English) Zbl 0980.41025
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. 7. New York, NY: Springer. xv, 338 p. (2001).
This is an interesting and intriguing book, and to that extent it is already a success. Its principal aim is pedagogical; it is ‘the book of the course’ which the author has offered at the Pennsylvania State University for a number of years, born out of the wish to provide a course in which the approximation theory does not appear initially as a distant range beyond a forbiddingly wide plain of prerequisites. The main text is almost entirely restricted to approximation in real inner product spaces and students are assumed to be familiar only with some “advanced calculus” and a little “linear algebra” (a first course being sufficient). In the first six chapters the minimum prerequisites and the basic approximation theory concepts are developed in tandem in a largely self-contained way. The author reports that the course had turned out to be even more successful than he had anticipated, and he hopes that the book will “be useful to mathematicians, statisticians, computer scientists, and to any others who need to use best approximation principles in their work”. In keeping with these hopes for a wide readership the author deliberately “err(s) on the side of including too much detail rather than too little”.
The book’s development flows from two decisions: the first, to work almost exclusively in the context of inner product spaces, and, the second, not to require that the spaces be complete. The second admits within the scope of the book the spaces \(C[a, b]\) of real continuous functions on an interval, with the inner product of the space \(L_2[a, b]\), without need for the theory of measure and integration.
Chapter 1 introduces inner product spaces, orthogonality, convergence of sequences and completeness. A normed linear space is defined but metric spaces are not. Compactness of a subset of an inner product space is defined sequentially in Chapter 3. Continuity of mappings between subsets of inner product spaces is defined in Chapter 5. The Baire category theorem and its applications to linear operators between inner product spaces are presented in Chapter 8. A simple form of the Hahn-Banach theorem for inner product spaces is given in Chapter 6. Thus the book contains an elementary introduction to the theory of Hilbert spaces. However, though the completion of an inner product space \(X\) does make an appearance, it is not named and no use of the completion is made in the text. (It may be that a development which obtained results first for Hilbert spaces, and then deduced more general results for not necessarily complete inner product spaces, would have been a little more efficient.)
Approximation theoretically the book is concerned mainly with best approximation to elements of an inner product space \(X\) from closed convex subsets \(K\) of \(X\). Some important results require that \(K\) be a cone, an affine subset or a linear subspace. Chapter 3 contains basic results concerning the existence and uniqueness of best approximations. Chapter 4 is concerned with characterisations of best approximations. Chapter 5 considers the metric projection onto a convex subset of an inner product space. Chapter 6 introduces linear functionals, hyperplanes and the dual space, and considers best approximation from polyhedra (finite intersections of closed half-spaces), obtaining calculable results. Chapter 7 centres upon the theorems of Weierstrass and Müntz concerning uniform approximation to a continuous function on an interval by polynomials. There are ‘Historical Notes’ to all chapters and in these the restriction to inner product spaces is relaxed. In the notes to Chapter 7 one of Jackson’s theorems is stated but there is no mention of Müntz-Jackson type theorems; the historical notes give tantalising glimpses of areas of approximation theory which are beyond the scope of the book. Chapter 8, entitled ‘Generalised solutions of linear equations’ defines the generalised inverse \(A^-\) of a bounded linear mapping \(A\) between Hilbert spaces, which has closed range, as that solution \(x_0 = A^-y\) of \[ \|Ax_0 - y\|= \inf_{x\in X}\|Ax - y\| \] which has minimum norm, thus placing it in an approximation theoretic context. The existence of \(A^-\) is then immediate, but its linearity is not; not until the equivalence of other definitions is established. Chapter 9, ‘The Method of Alternating Projections’ is a thorough discussion of the algorithms of von Neumann and Dykstra for calculating the metric projection onto the intersection of two (or of a finite set of) subspaces or, more generally, convex sets. The topic has been an entertainingly fertile one.
The book emphasizes applications of the theory. Chapter 1 begins by stating five basic problems concerning best least squares polynomial approximation, overdetermined systems of equations, a control problem and a problem of interpolation by positive functions; each of which is recast in Chapter 2 as a problem of best approximation. The problems are picked off in turn in the course of the book and the solution of the last is to be found in Chapter 10, ‘Constrained Interplation from a Convex Set’. Chapter II discusses simultaneous Interpolation and Approximation’.
A subset of a space is said to be Chebyshev if to each point of the space there exists a unique best approximation to it from the subset. The final Chapter 12 is an advertisement for the most outstanding unanswered question of abstract approximation theory: whether or not every Chebyshev subset of a Hilbert space is convex.
Each chapter is liberally supplied with exercises, the book contains a wealth of material and is a pleasure to read.

41A50 Best approximation, Chebyshev systems
41-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)