Abstract
where ρX(x, M) = infz∈M ‖x− z‖X . A univalent mapping G : X → M is called a multiplicative (respectively, additive) e -selection, e ≥ 0 , from X into M if ∀x ∈ X ‖x−G(x)‖ ≤ ρ(x, M)(1 + e) (respectively, ‖x−G(x)‖ ≤ ρ(x, M) + e). If G is continuous, we say that G is a continuous selection. Tsar′kov proved that, for a closed set M , the existence, for any e > 0 , of continuous multiplicative e -selections into it, is equivalent to the existence, for all e > 0 , of continuous additive e -selections (into it). Let Pd[a, b] be the set of algebraic polynomials of degree ≤ d defined on [a, b] , and let Rm,n[a, b] be the set of algebraic rational fractions:
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