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Approximation with constraints. (Approksimatsiya s ogranicheniyami). (Russian) Zbl 0531.41001

Kiev: ”Naukova Dumka”. 252 p. R. 2.30 (1982).
The three basic problems of best approximation theory in a normed space X by elements belonging to a subspace \({\mathcal N}\subseteq X\), are treated here under certain restrictions given by inequalities. An example of such problems consists in unilateral approximation, the function \(u\in {\mathcal N}\) satisfying a restriction of the form u(t)\(\geq x(t)\) or u(t)\(\leq x(t)\). An important part in problems of this kind is played by the notion of a cone K of the space X. The problem of best approximation under the restriction K of elements \(x\in X\) by elements of \({\mathcal N}\) consists in the determination of the value \(E_ K(x,{\mathcal N})=\inf_{u\in {\mathcal N}(K,x)}\| x-u\|,\) in the metric of X, where \({\mathcal N}(K,x)=\{u\in {\mathcal N} | u-x\in K\}\). The element \(u_ 0\in {\mathcal N}\) with the property \(E_ K(x,{\mathcal N})=\| x-u_ 0\|\) is called the best approximation element under the restriction K of the element \(x\in X\). By the best approximation in the set \({\mathcal N}\) under the restriction K of the set \({\mathcal M}\subset X\), in the metric of the space X, is understood the value \(E_ K({\mathcal M},{\mathcal N})=\sup_{x\in {\mathcal M}}E_ K(x,{\mathcal N})\), and the element \(x_ 0\in {\mathcal M}\) with the property \(E_ K({\mathcal M},{\mathcal N})=E_ K(x_ 0,{\mathcal N})\) is called an extremal element under the restriction K, relative to \({\mathcal N}\). Finally relative to the best choice of the approximation set one poses the problem of determination of the value \(\inf_{{\mathcal N}\in {\mathcal A}}E_ K({\mathcal M},{\mathcal N})\) where \({\mathcal A}\) is the class of the sets \({\mathcal N}\) in X.
One gives general properties of the best approximation under the restrictions relative to the lower semicontinuity, or to the convexity of the functional \(E_ K(x,{\mathcal N})\) and so on. One poses the problem of the existence and unicity of the element of best approximation under the restriction K. In the case of the restrictions given by the convex cone with vertex x(t), there are established connections between the problem of the best approximation under the restrictions and the extremal problem in the dual space. This connection permits the determination of some exact solutions on concrete classes of functions, in a series of problems of approximation under restrictions. In a chapter are treated the best approximation and the best unilateral approximation on the class \(W^ r_ p=\{x\in L^ r_ p\), \(\| x^{(r)}\|_ p\leq 1\), \(r=1,2,...\}\) and in another one, the best approximation on the class \(W^ rH_{\omega}\) of the 2\(\pi\)-periodic functions with \(x\in C^ r\), with the property \(\omega(x^{(r)},t)\leq \omega(t)\), \(\omega\) (t) being the continuity module.
In this book the connection between best approximation under restrictions and the problem of spline interpolation or other extremal problems of Mathematical Analysis is studied. The end of the book is composed of extensive comments and bibliographic indications on the problems treated.
Reviewer: L.Ciobanu

MSC:

41-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)