Concrete algebraic cohomology for the group (ℝ, +) or how to solve the functional equation 𝑓(𝑥+𝑦) - 𝑓(𝑥) - 𝑓(𝑦) = 𝑔(𝑥, 𝑦)

Abstract

The functional equation ð‘“(ð‘¥+ð‘¦) - ð‘“(ð‘¥) - ð‘“(ð‘¦) = ð‘”(ð‘¥, ð‘¦) has a solution ð‘“ that belongs to C⁰(â„) if and only if the symmetric cocycle ð‘” belongs to C⁰(Ⅎ). If the symmetric cocyle ð‘” is recursively approximable, there exists a solution ð‘“ which is recursively approximable also. If ð‘” belongs to C¹(Ⅎ) then there exists an integral expression in ð‘” for a solution ð‘“ that belongs to C¹(â„), and the same happens for the classes C^k, C^∞, analytic and polynomial.