Abstract
Let Ω be a metric space, Δ⊂Ω and B Ω a linear space of real functions defined on Ω. The distance r A (f>g), f, gεB Ω is an A-distance in B Ω on Δ if:
-
1)
$${r_A}\left( {f,g} \right) = {r_A}\left( {g,f} \right)o$$
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2)
$${r_A}\left( {f,g} \right) \leqslant {r_A}\left( {f,h} \right) + {r_A}\left( {h,g} \right)$$
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3)
if for every xεΔ
\( \varphi \left( x \right) \leqslant f\left( x \right) \leqslant \psi \left( x \right) \) and \( \varphi \left( x \right) - C \leqslant g\left( x \right) \leqslant \psi \left( x \right) + C \) where C is a constant, then
$${r_A}\left( {f,g} \right) \leqslant {r_A}\left( {\varphi ,\psi } \right) + \left| C \right|$$ -
4)
if C is a constant, then
$${r_A}\left( {f,f + C} \right) = 0 \Leftrightarrow C = 0$$
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References
Korovkin, P.P., Axiomatic approach to some problems in approximation theory. (Russian). In: Constructive function theory (Proceedings of the Int.Conf. on Constructive Function Theory, Golden Sands (Varna), May 19–25, 1970.) Sofia, 1972, 55–63.
Sendov, Bl., Convergence of sequences of monotonie operators in A-distance. C.R.Acad.Bulgare Sci., 30 (1977), No 5, 657–660.
Sendov, Bl., Some problems in the theory of approximation of functions and sets in the Hausdorff metric. (Russian). Uspehi Mat. Nauk. 24 (1969), 143–180.
Vesselinov, V.M., Approximation of non-bounded functions with linear positive operators in Hausdorff distance. (Russian). C.R. Acad. Bulgare Sci., 22 (1969), No 5, 499–502.
Hsu, L.S., Approximation of non-bounded continuous functions by certain sequences of linear positive operators or polynomials. Studia Math., 21 (1961), 37–43.
Müller, M.W., Approximation unbeschränkter Funktionen bezüglich einer Korovkin-Metrik. Theory of approximation of functions. Moskva, 1977, 269–272.
Schmid, G., Approximation unbeschränkter Funktionen. Diss., Stuttgart, 1972.
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© 1978 Birkhäuser Verlag Basel
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Sendov, B. (1978). Approximation With Monotonic Operators in A-Distance. In: Butzer, P.L., Szökefalvi-Nagy, B. (eds) Linear Spaces and Approximation / Lineare Räume und Approximation. International Series of Numerical Mathematics / Intermationale Schriftenreihe zur Numberischen Mathematik / Sùrie Internationale D’analyse Numùruque, vol 40. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7180-8_29
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DOI: https://doi.org/10.1007/978-3-0348-7180-8_29
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