Superstability of the d'Alembert functional equation in L_p^+ spaces


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,yX,
then fLp+(X,ℂ) or there exists exactly one function g: X→ℂ with
g{x+y) + g(x-y) - 2g(x)g(y),   x,yX
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 ∥fp is finite.

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Published : 2004-09-30

PrzybyłaM. J. (2004). Superstability of the d’Alembert functional equation in L_p^+ spaces. Annales Mathematicae Silesianae, 18, 39-47. Retrieved from

Maciej J. Przybyła
Instytut Matematyki, Politechnika Śląska  Poland

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