Approximation With Monotonic Operators in A-Distance

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)
  $${r_A}\left( {f,g} \right) = {r_A}\left( {g,f} \right)o$$
• 2)
  $${r_A}\left( {f,g} \right) \leqslant {r_A}\left( {f,h} \right) + {r_A}\left( {h,g} \right)$$

• 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

  $${r_A}\left( {f,f + C} \right) = 0 \Leftrightarrow C = 0$$