Best Approximation in Metric Spaces

Narang, T. D.; Gupta, Sahil

doi:10.1515/fascmath-2017-0008

Cited by 2 publications

(5 citation statements)

References 9 publications

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“…Concerning the graph of R G , we have the following theorem (a similar result for the metric projection P G was proved in [6]). …”

Section: Notementioning

confidence: 55%

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Proximinality and co-proximinality in metric linear spaces

Narang¹,

Gupta²

2015

Annales UMCS, Mathematica

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Abstract. As a counterpart to best approximation, the concept of best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi in 1972. Subsequently, this study was taken up by many researchers. In this paper, we discuss some results on the existence and uniqueness of best approximation and best coapproximation when the underlying spaces are metric linear spaces.A new kind of approximation, called best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi [2] to obtain some characterizations of real Hilbert spaces among real Banach spaces. This study was further taken up by many researchers in normed linear spaces and Hilbert spaces (see e.g. [3], [4], [9]). But only a few have taken up this study in more general abstract spaces. The theory of best coapproximation is much less developed as compared to the theory of best approximation in abstract spaces. The present paper is also a step in this direction. In this paper, we discuss the existence and uniqueness results on best approximation and best coapproximation in metric linear spaces thereby generalizing the various known results. We start with a few definitions.

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“…Concerning the graph of R G , we have the following theorem (a similar result for the metric projection P G was proved in [6]). …”

Section: Notementioning

confidence: 55%

“…It is known (see [6]) that if G is a subset of a convex metric space (X, d) and x ∈ P [6]) that if G is a Chebyshev subset of a convex metric space (X, d), then P G (z) = P G (x), where z ∈ X is any element between x and P G (x). Does such a result hold for best coapproximation?…”

Section: (7)mentioning

confidence: 99%

Proximinality and co-proximinality in metric linear spaces

Narang¹,

Gupta²

2015

Annales UMCS, Mathematica

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“…Donoho and Liu (1988) and Powell (1981) respectively). Note also that in this case the strong convexity of d * H ≡ g • Q H ∞ (used in Assumption 5.6.1) is immediately obtained since the later is by construction implied by the former (see Cheney (1974), Ahuja et al (1977) and Narang (1981)). This is also true of uniform convexity of power type p of • H ≡ g •Q H ∞ (Assumption 5.6.5) and L p representation of • H ≡ g • Q H ∞ (Assumption 5.6.4) in the case of least squares estimation (see Cheney (1974) or Cheney (1982, p.23)).…”

Section: Parameterization Mapping: Some Regression Modelsmentioning

confidence: 81%

“…Observe first the following useful definitions available e.g. in Cheney (1974), Ahuja et al (1977), Nurberger (1979) and Narang (1981). Let (B, d B ) be a linear metric space.…”

Section: Strong Unicity Of Best Approximationsmentioning

confidence: 99%

“…Consider a subset A ⊂ B. A projection mapping is a set valued map P A When a function g exists that satisfies the properties postulated in Assumption 5.4.3, then, the following lemmas adapted from Cheney (1974), Ahuja et al (1977), Powell (1981, p.4), Narang (1981) and Cheney (1982, p.4), are available to judge on the existence and uniqueness of a best approximation.…”

Section: Strong Unicity Of Best Approximationsmentioning

confidence: 99%

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Semi-nonparametric indirect inference

Blasques

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Best Approximation in Metric Spaces

Cited by 2 publications

References 9 publications

Proximinality and co-proximinality in metric linear spaces

Proximinality and co-proximinality in metric linear spaces

Semi-nonparametric indirect inference

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