A note on simultaneous polynomial approximation of exponential functions

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Abstract
Let α1, …, αmbe distinct complex numbers and τ(1), …, τ(m) be non-negative integers. We obtain conditions under which the functions form a perfect system, that is, for every set ρ(1), …, ρ(m) of non-negative integers, there are polynomialsa1(z), …,am(z) with respective degrees exactly ρ(1)−1, …, ρ(m)−1, such that the function has a zero of order at least ρ(1) + … + ρ(m)−1 at the origin. Moreover, subject to the evaluation of certain determinants, we give explicit formulae for the approximating polynomialsa1(z), …,am(z).

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