Continuities of Metric Projection and GeometricConsequences

Huotari, Robert; Li, Wu

doi:10.1006/jath.1996.3018

Cited by 8 publications

(2 citation statements)

References 32 publications

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“…where x (p) is the l p -solution of Gx = b and x (st) is the strict Chebyshev solution. We formulate this property in the form (3.1) to underline that Gx (p) is the l p -approximation to b by vectors from a linear subspace V. The properties of the Pólya algorithm and the strict Chebyshev approximations are the subject of a series of papers by Huotari and his coauthors (see, for example, [28,29,30,46,47,48]). In [28] Egger and Huotari give two important examples for the approximation over closed convex sets in R m .…”

Section: The Chebyshev and L P -Solutions Of An Overdetermined Systemmentioning

confidence: 99%

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

Ziętak

2017

Banach Center Publ.

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The main aim of this review paper is approximation of a complex rectangular matrix, with respect to the unitarily invariant norms, by matrices from a linear subspace or from a convex closed subset of matrices. In particular, we focus on the properties and characterizations of the strict spectral approximant which is in some sense the best among all approximants to a given matrix with respect to the spectral norm. We formulate a conjecture that the strict spectral approximant to a matrix by matrices from a linear subspace is the limit of approximants with respect to the cp norms of Schatten when p tends to infinity. This conjecture corresponds to the conjecture stated by Rogers and Ward on approximation by positive semidefinite matrices and to the known analogous property of the strict Chebyshev approximation of a vector, proved by Descloux. We describe some special cases of an approximation of a matrix that confirm our conjecture and we discuss an attempt to prove the conjecture. Additionally, we focus on orthogonality of matrices in the sense of Birkhoff and James and approximation by matrices whose spectrum is in a strip. For the last case we recall a conjecture on characterization of the best approximation of a matrix in the spectral norm by matrices whose spectrum is in a strip. This kind of approximation is a generalization of the famous concept of Halmos of approximation by positive operators.

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Section: The Chebyshev and L P -Solutions Of An Overdetermined Systemmentioning

confidence: 99%

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

Ziętak

2017

Banach Center Publ.

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“…Let X = L p ( ) and M = span h G. Then X is strictly convex, and so is its dual X * . Hence, by a theorem of Vlasov (see [4]), if M is a Chebyshev set for which the best approximation of X by M is continuous, then M is convex. By theorem 2.1 if P M has a continuous selection, M is a Chebyshev set and thus is convex.…”

Section: Best Approximationmentioning

confidence: 99%

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Kainen

Kůrková

Vogt

2001

Annals of Operations Research

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Devices such as neural networks typically approximate the elements of some function space X by elements of a nontrivial finite union M of finite-dimensional spaces. It is shown that if X = L p ( ) (1 < p < ∞ and ⊂ R d ), then for any positive constant and any continuous function φ from X to M, f − φ(f ) > f − M + for some f in X. Thus, no continuous finite neural network approximation can be within any positive constant of a best approximation in the L p -norm.

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Geometry and Topology of Continuous Best and Near Best Approximations

Kainen

Krková

Vogt

2000

Journal of Approximation Theory

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The existence of a continuous best approximation or of near best approximations of a strictly convex space by a subset is shown to imply uniqueness of the best approximation under various assumptions on the approximating subset. For more general spaces, when continuous best or near best approximations exist, the set of best approximants to any given element is shown to satisfy connectivity and radius constraints. Academic Press

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Continuities of Metric Projection and GeometricConsequences

Cited by 8 publications

References 32 publications

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

Untitled

Geometry and Topology of Continuous Best and Near Best Approximations

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