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Connectedness and solarity in problems of best and near-best approximation. (English. Russian original) Zbl 1350.41031
Russ. Math. Surv. 71, No. 1, 1-77 (2016); translation from Usp. Mat. Nauk 71, No. 1, 3-84 (2016).
This survey is mainly concerned with structural characteristics of suns in normed linear spaces, with special emphasis on connectedness and monotone path-connectedness, where consideration is given to both direct theorems in geometric approximation theory in which approximative properties of sets are derived from their structural characteristics, and converse theorems in which structural properties of sets are derived from their approximative characteristics.
In the 19th century, there arose a real need for solving an important practical problem involving the improvement of the design of a steam engine, the famous Watt parallelogram, which was used to convert the reciprocating motion of a piston into the rotational motion of the flywheel of the steam engine. The design of a steam engine looks surprisingly simple nowadays, but it was only through the efforts of many engineers that it acquired its definitive form. In the middle of the 19th century, Chebyshev introduced the important concept of the best approximation that is, the best uniform approximation and made systematic use of it in practical applications. Subsequently, the concepts of best approximation and an element of best approximation were extended to general normed linear spaces and became the starting point of geometric approximation theory. The best approximation, that is, the distance of a given element \(x\) in a normed linear space \(X\) from a given non-empty set \(M \subset X\) is defined by \(\rho (x,M) =\mathrm{inf}_{y\in {M}} ||x - y||\). The concepts and properties defined in terms of best approximation, and in particular the existence, uniqueness, and stability properties of elements of best approximation, are said to be approximative. First and foremost is the concept of an element of best approximation, or a nearest point, i.e., for a given \(x \in {X}\) a point \(y \in {M}\) such that \(||x - y||= \rho (x, M)\). For a given element \(x\), the set of all nearest points, i.e., elements of best approximation or, the best approximants in \(M\) is defined by \(P_{M}x = \{y \in{ M} \mid \rho (x,M) =||x - y||\}\) and denoted by \(P_{M}x\). The problem of stability of the operator of near best approximation arises not only in the theory of approximation and numerical methods, but also in ill-posed problems, optimal control, mathematical programming, and the stability of solutions of general extremal problems. Geometrical optics is a branch of optics in which it is assumed that the wavelength is negligibly small, and the eikonal equation is derived from the wave equation for a complex amplitude. The surfaces \(f =\mathrm{const}\) are called geometrical wave surfaces or geometrical wave fronts. The eikonal equation can also be regarded as the Hamilton-Jacobi equation for the variational problem \(\delta \int n ds = 0\), the optical counterpart of which goes back to Fermat’s principle, also known as the principle of the shortest optical path. One of the interesting challenges in geometric approximation theory involving solarity is connected with the Hamilton-Jacobi equations illustrated by considering the simple example of the eikonal equation, which is the fundamental equation of geometrical optics, \(| \nabla f(x)| = n\), where \(n = n(x)\) is the refractive index, and \(| . |\) is the standard Euclidean norm on \(R^{n}\). The potency of geometric approximation theory in problems connected with the eikonal equation is studied in the first section. The authors illustrate the geometric approximation theory involving the concept of solarity, which is connected with the Hamilton-Jacobi equations, by considering the eikonal equation, \(|\nabla f(x)| = n\). They employed geometric methods of approximation theory in solving the eikonal equation. They give a detailed study of structural characteristics of suns in normed linear spaces, with special emphasis on connectedness and monotone path-connectedness. Their consideration is given to both direct theorems in geometric approximation theory in which approximative properties of sets are derived from their structural characteristics, and converse theorems in which structural properties of sets are derived from their approximative characteristics. In the second section, they studied some other structural characteristics of suns, the most important of which are properties of geometric topological character in particular, connectedness and convexity. They deal with both direct theorems of geometric approximation theory, in which approximative properties of sets are derived from their structural characteristics, and with converse theorems, in which one derives their structural properties from approximative characteristics of sets. As approximative properties they consider the properties of uniqueness and existence of a best approximation, the Chebyshev property, approximative compactness, solarity, and the stability of the operators of best and near-best approximation. By structural characteristics of sets one usually understands properties of linearity, finite dimensionality, convexity, connectedness of various kinds, and smoothness of sets. For example, compactness of a non-empty set is well known to imply its proximinality. In terms of applications, converse theorems play an important role in the sense that once it is found that an object under study does not have good structural properties, converse theorems might be employed to infer that it also fails to have good approximative properties. One usually succeeds in showing that a certain object is not an existence or a uniqueness set. In the survey they touch upon the broad problem of convexity of Chebyshev sets and their generalizations. They are concerned with classical questions of geometric approximation theory, leaving behind a number of questions related, for example, to approximation with respect to Bregman distances. A set \(M\) is called an existence set, or a proximinal set, if for any point \(x\) the set \(P_{M}x\) of best approximations of \(x\) is non-empty. A set \(M\) is called a uniqueness set if for any point \(x\) the set \( P_{M}x \) is empty or a singleton. An existence set is always closed and non-empty. Indeed, if a limit point of \(M\) is not contained in \(M\), then this point clearly fails to have a nearest point in \(M\). The converse assertion clearly holds in any finite-dimensional space \(X\), i. e., any non-empty closed set is an existence set. Let \(\emptyset \neq M \subset X\). A point \( x \in {X\backslash M}\) is called a solar point if there exists a point \(y \in P_{M}x \neq \emptyset \) such that \(y \in P_{M}((1-\lambda ) y + \lambda x)\) for all \(\lambda > 0\), which geometrically means that there is a solar ray emanating from \(y\) and passing through \(x\) such that \(y\) is a nearest point in \(M\) for any point on the ray. A point \(x \in{ X \;M}\) is called a strict solar point if \(P_{M}x \neq \emptyset \) and \(y \in P_{M}(1-\lambda) y+ \lambda x)\) holds for any point \(y \in P_{M}x\). Further, if for \(x \in {X \;M}\) if the condition is satisfied for each \(y \in P_{M}x\), then \(x\) is called a strict protosolar point. The concept of a sun was introduced by N. V. Efimov and S. B. Stechkin [Dokl. Akad. Nauk SSSR 118, 17–19 (1958; Zbl 0081.16402)]. The term strict protosun was introduced by the authors in [“Connectedness and other geometric properties of suns and Chebyshev sets”, Fundam. Prikl. Mat. 19, No. 4, 21–91 (2014)]. The closed strict protosuns are exactly closed Kolmogorov sets that is, sets satisfying the Kolmogorov criterion for an element of best approximation. A closed set \(M \subset {X}\) is called a sun, respectively, a strict sun, if any point \(x\in{ X\;M}\) is a solar point, respectively, a strict solar point, for \(M\). A set \(M \subset X\) is called a strict protosun if any point \(x \in {X-M}\) is a strict protosolar point. A set \(M \subset {X}\) is called a protosun if for any \(x\notin{M}\) there exists a luminosity point in \(M\) such that \(P_{M}x \neq \emptyset\). A set \(M\) is said to be a sun relative to \(G\) if for any point \(x \in {G}\) there exists a point \(y\in P_{M}x\) which is a nearest point to any point \(z\) on the ray \( (1 - \lambda )y + \lambda x \lambda > 0\), in the equation, provided that \(z\in { G}\). Efimov and Stechkin in [loc. cit.] proposed the new term Chebyshev set, paying homage to Chebyshev as a founder of approximation theory, which was soon generally adopted among the different names for Chebyshev sets used before and sometimes after Efimov and Stechkin’s paper \(EU\)-set, Haar set, and Motzkin set are also used. A set \(M\) is called a Chebyshev set if for any \(x\) there is an element of best approximation in \(M\) and it is unique, in other words, the Chebyshev sets are precisely the sets of existence and uniqueness. A Chebyshev set that is a sun is called a Chebyshev sun. Chebyshev introduced the important concept of best approximation in particular, best uniform approximation, that made systematic use of it in practical applications, and developed its theoretical basis. Studying best approximation in \(C[a, b]\) by the sets \(P_{n}\) of polynomials of degree at most \(n\) and the sets
\[ R_{m,n}=\{\frac{p}{q}:p\{ P_{n}\}, q \{ P_{m}, q\neq 0 \} \] of rational fractions on \( [a, b]\), Chebyshev and his students introduced the concept of alternance (the term alternance was introduced much later by N. I. Akhiezer). Elaborating on the alternance idea of Chebyshev, Kirchberger, Borel, and Young for polynomials and Akhiezer and Walsh for rational fractions justified the uniqueness of best approximations and proved the existence theorem. Some weaker properties than convexity are frequently found to be useful in non-linear approximation theory. According to the geometric form of the Hahn-Banach theorem, the closed convex sets are characterized by the fact that any point outside such a set can be strictly separated from it by a closed hyperplane. A similar result holds for suns and strict protosuns with open half-spaces replaced by open support cones. From the geometric form of the Hahn-Banach theorem it follows that any convex set is a strict protosun and any convex existence set is a strict sun. A non-empty closed set \(M\) is said to be almost convex if, for any closed ball \(B(x, r)\) at a positive distance from \(M\), there exists a closed ball \(B(y, R) \supset {B(x, r)}\) of arbitrary large radius \(R\) which is also at a positive distance from \(M\). In the general setting a strict protosun, if it is not a strict sun, need not be convex nor even almost convex. A homology (cohomology) theory associates with any topological space \(X\) a sequence of abelian groups \(H_{k}(X)\), \(k = 0, 1, 2,\ldots \) (homology groups), and \(H_{k}(X)\), \(k = 0, 1, 2, \ldots \) (cohomology groups), which are homotopy invariants of the space if two spaces are homotopy equivalent, then the corresponding (co)homology groups are isomorphic. Let \(A\) be an arbitrary non-trivial abelian group. A metrizable space is said to be acyclic if its Čech cohomology group with coefficients in \(A\) is trivial. If a (co)homology has compact support (satisfies the compact-support axiom) and if the coefficients of the (co)homology group lie in a field, then the notions of homological and cohomological acyclicity coincide. A non-empty compact space is called an \(R_{\delta}\)-set if it is homeomorphic to the intersection of a countable decreasing sequence of compact absolute retracts. \(R_{\delta}\)-sets arise naturally as spaces of solutions to the Cauchy problem for autonomous and non-autonomous differential equations and inclusions. A compact space \(Y\) is said to be cell-like (or to have the shape of a point) if there exist an absolute neighbourhood retract \(Z\) and an embedding \(i : Y \rightarrow Z\) such that the image \(i(Y )\) is contractible in any neighbourhood \(U \subset Z\) of it, formula (82.4). A cell-like set need not to be contractible, where a space \(X\) having the homotopy type of a point is said to be contractible. A compact set with the shape of a point (a cell-like set) is contractible in each of its neighbourhoods in any ambient absolute neighbourhood retract. They also study the notions of infinitely connectedness, and \(B\)-infinitely connectedness, where a subset \(A\) of a semimetric space \((Y, d)\) is said to be infinitely connected if for any \( n\in {\mathbb{N}}\) any continuous map \(\varphi:\partial B \rightarrow A\) from the boundary of the closed unit ball \(B \subset {\mathbb R_{n}}\) to \(A\) has a continuous extension \(\overline{\varphi}: B \rightarrow A\), and a set \(M \subset {X}\) is said to be \(B\)-infinitely connected if the intersection of \(M\) with any open ball is either empty or infinitely connected. A \( \mathrm{ici }B\)-infinitely connected set need not be \(B\)-infinitely connected, and an intersection with some closed ball may fail to be infinitely connected and may even be disconnected. A set \(A \subset {X}\) is called a retract of \(X\) if there exists a continuous map \(r : X \rightarrow A\) such that \(r_{A} = 1_{A}\); that is, the identity map \(1_{A}\) has a continuous extension to the whole space \(X\). An asymmetric norm on \(X\) is defined to be a non-negative functional \(|| ||\) satisfying all norm conditions replacing absolute homogeneity condition with homogeneity condition for positive numbers. In general,\(||x|| \neq ||-x||\). Then the function \(||x||_{\mathrm{sym}} = \max\{||x||, ||-x||, x \in {X}\}\) is a norm. For a symmetrically or asymmetrically normed space \(X\), the points \(s_{1},\ldots , s_{n}\in {S}\) are said to be mutually far if for the bodies \(B_{I} = B\;s_{i}\) with \(i = 1,\ldots , n\) we have \(B_{i} \cap \operatorname{int} B_{j} = \emptyset\), for \(i \neq j, i, j = 1, \ldots , n\), that is, the translations \(B_{i}\) of the body \(B\) have pairwise disjoint interiors and have the common point \(0\). A subset \(M\) of \(X\) is said to be \(m\)-connected, or Menger-connected, if \(m(\{x, y\}) \cap M \neq \{x, y\}\) for any distinct points \(x, y \in {M}\). A segment or interval in an arbitrary normed linear space \(X\) is given by the formula \([[x, y]] = \{z \in{X} :\min\{\varphi (x), \varphi (y) \leq \varphi (z) \leq \max \varphi (x), \varphi (y) \forall \varphi \in\mathrm{ext }S^\ast\}\), where \(\mathrm{ext }S^{\ast}\) is the set of extreme points of the unit sphere \(S^{\ast}\) of the dual space \(X^{\ast}\).

MSC:
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
54C60 Set-valued maps in general topology
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