Best approximation theory and functional analysis (Q2781466)

This book is an introduction to approximation theory for those who had a first course in real analysis including elements of functional analysis. The following is a summary of the content.NEWLINENEWLINEChapter 1 contains the material necessary for reading the rest of the book. The definitions are given in full, but results from functional analysis are stated without proofs. In the subsequent chapters, all the results are proved.NEWLINENEWLINEIn Chapters 2--5, the author develops approximation theory in an abstract setting of normed linear spaces. Let \(X\) be a normed linear space, let \(M\) be a non-empty subset of \(X\) and let \(x\in X\). Then a best approximation to \(x\) in \(M\) is an element \(a\in M\) such that \(\|x-a\|= \roman{dist}(x,M)=\inf\{\|x-y\|\colon y\in M\}\). The set \(M\) is called an \(E\)-set if a best approximation exists in \(M\) to each \(x\in X\). The set \(M\) is said to be semi-Chebyshev if for each \(x\in X\) there is at most one best approximation to \(x\) in \(M\), and semi-Chebyshev \(E\)-sets are called Chebyshev sets.NEWLINENEWLINEIn Chapter 2, the characterizations for an element \(a\) of a linear subspace \(M\) to be a best approximation to \(x\in X\) are given from many points of view. The existence of best approximations is investigated in Chapter 3, where one finds sufficient conditions for closed convex sets \(M\) or closed linear subspaces \(M\) to be \(E\)-sets. The reflexive Banach spaces and uniformly convex Banach spaces are relevant here. The Chebyshev and semi-Chebyshev properties are treated in Chapter 4. The unique norm-preserving extension of linear functionals defined on linear subspaces plays an important role. A discussion of best approximations in Hilbert spaces concludes this chapter. In Chapter 5, sequences of best approximations are considered. To give an idea of the results, we quote one theorem here: Let \(X\) be a real Banach space, let \(\{e_ 1, e_ 2, \dots \}\) be a linearly independent sequence in \(X\), and let \(M_ n\) be the linear span of \(\{e_ 1, e_ 2, \dots, e_ n\}\). If \(\{\alpha_ n\}\) is a sequence of reals such that \(\alpha_ n \downarrow 0\) as \(n\rightarrow \infty\), then there is an \(x\in X\) such that \(\text{dist}(x, M_ n)=\alpha_ n\) for all \(n\).NEWLINENEWLINEThe above constitutes roughly the first half of the book. The second half (Chapters 6 and 7) treats more concrete problems of approximating functions. Here we encounter familiar spaces and many named theorems. Chapter 6 begins with a general theorem (Kolmogorov) characterizing a best approximation in a linear subspace \(M\) of \(C(\Omega)\) to an element \(f\in C(\Omega)\setminus\overline{M}\), where \(C(\Omega)\) is the Banach space of scalar-valued continuous functions on a compact Hausdorff space \(\Omega\) with the supremum norm, and immediately moves on to the case of finite-dimensional \(M\)'s. The standard topics such as the Haar condition, Chebyshev's alternation theorem, the de la Vallée-Poussin theorem and Chebyshev polynomials are all here. In connection with the case \(\Omega=\mathbb T=\{z\in\mathbb C: |z|=1\}\), uniform approximations of \(f\in C(\mathbb T)\) by convolutions \(k*f\) with Fejér or, more generally, de la Vallée-Poussin kernels \(k\) are studied and compared with the best approximations by trigonometric polynomials. A far-reaching generalization of the Korovkin theorem is proved and special cases are derived. The second part of this chapter investigates best approximations in \(L^ p\) spaces with \(1\leq p \leq\infty\). The relevant fact that \(L^ p\) is locally uniformly convex for \(1<p<\infty\) is proved here.NEWLINENEWLINEChapter 7 is perhaps the most technical part of this book and it deals exclusively with \(X=C(\mathbb T)\text{ or }L^ p(\mathbb T)\; (1\leq p<\infty)\). For each \(t\in\mathbb T\), let \(T_ t\colon X\rightarrow X\) be the linear isometry defined by \(T_ t(f)(z)=f(zt^{-1})\) for each \(f\in X\). Let \(\omega=\{0\}\cup\mathbb N\) and, for \(r\in\omega,\;t\in \mathbb T\), let \(\Delta^ r_ t\) be the bounded linear operator on \(X\) defined by \(\Delta^ r_ t=(T_ t - I)^ r\) where \(\Delta^ 0_ t=I\). Then, for \(r\in\omega,\;\delta\geq 0\), the \(r\)-th modulus of continuity \(\omega_ r(f,\delta)\) of \(f\in X\) is defined to be \(\sup\{\|\Delta^ r_ t(f)\|: |t|\leq\delta\}\). The first part of the chapter deals with approximations of \(f\in X\) with \(k*f\) where \(k\) is one of several named kernels introduced here. The norm \(\|k*f-f\|\) is estimated in terms of \(\omega_ 1(f',\delta)\) or \(\omega_ 2(f,\delta)\) with suitably chosen \(\delta\)'s. In the second part, Jackson-type theorems are proved. To be more specific, let \(M_ n\) be the subspace of \(X\) spanned by the family \(\{z^ m: |m|\leq n\}\) of functions. Then one of the Jackson-type theorems states that, for each \(r\in\mathbb N,\;f\in X\) and \(n\in\mathbb N\), \(\text{dist} (f,M_ n)\leq C_ r\omega_ r(f,1/n)\) where \(C_ r =2(\pi/2)^{2r+5}(r+4)^ r\). The last part of the chapter gives converses to Jackson-type theorems, where the modulus of continuity of a function \(f\) is estimated from the rate of convergence of \(\text{dist}(f, M_ n)\rightarrow 0\).NEWLINENEWLINEEach chapter is followed by a set of exercises, some of which seem highly nontrivial, and the outlines of the solutions are provided at the end of the book. The writing of the book is rather dry and relentless: one theorem comes after another in a quick succession with hardly a pause for motivation or historical remarks. For instance, this reviewer would have liked to see, in connection with Chebyshev sets, a discussion of the old problem of whether a Chebyshev set in a Hilbert space must be convex. For more than 40 years, there have been partial solutions, but the problem is still open. In summary then, this book is recommended to highly motivated students for self-study. For the rest, this is a good textbook for classes taught by instructors who can supply what is lacking here.