The direct and inverse theorems of the approximation theory for solutions of differential equations in a Banach space
DOI:
https://doi.org/10.15407/dopovidi2016.07.007Keywords:
Banach space, differential equation in a Banach space, weak solution, C0-group of linear operators, entire vector-valued function, entire vector-valued function of exponential type, the best approximation, continuity moduleAbstract
An equation of the form y ′ (t) = Ay(t), t ∈ (−∞,∞), where A is the generator of a C0-group of linear operators on a Banach space, is considered. The direct and inverse theorems of the theory of approximation of weak solutions of this equation by entire solutions of exponential type, which establish the one-to-one correspondence between the order of convergence to zero of the best approximation of a solution and its smoothness degree, are presented.
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