Superstability of the d'Alembert functional equation in L_p^+ spaces
Abstract
Let (X,+,-,0,Σ,μ) be an abelian complete measurable group with μ(X)>0. Let f: X→ℂ be a function. We will show that if A(f)∈Lp+(X×X,ℂ) where
A{f)(x,y) = f{x+y) + f(x-y) - 2f(x)f(y), x,y∈X,
then f∈Lp+(X,ℂ) or there exists exactly one function g: X→ℂ with
g{x+y) + g(x-y) - 2g(x)g(y), x,y∈X
such that f is equal to g almost everywhere with respect to the measure μ.
Lp+ denotes the space of all functions for which the upper integral of ∥f∥p is finite.
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Instytut Matematyki, Politechnika Śląska Poland
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