convolution equation — 1 results found.

We show that any Lebesgue measurable function \(f \colon \mathbb{R} \to [0,\infty)\) satisfying \( f(x) = \int_0^{\infty} f(x+y) f(y) \,dy \) has the form \( f(x) = 2 \lambda e^{-\lambda x} \) with a \(\lambda \in [0,\infty)\).

Nous démontrons que toute fonction mesurable au sens de Lebesgue \(f \colon \mathbb{R} \to [0,\infty)\) satisfaisant à \( f(x) = \int_0^{\infty} f(x+y) f(y) \,dy \) est de la forme \( f(x) = 2 \lambda e^{-\lambda x} \) avec un \(\lambda \in [0,\infty)\).