On asymptotically almost periodic solutions of a convolution equation

Staffans, Olof J.

doi:10.1090/S0002-9947-1981-0617554-5

On asymptotically almost periodic solutions of a convolution equation
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Trans. Amer. Math. Soc. 266 (1981), 603-616 Request permission

Abstract:

We study questions related to asymptotic almost periodicity of solutions of the linear convolution equation $( \ast )\mu \ast x = f$. Here $\mu$ is a complex measure, and $x$ and $f$ are bounded functions. Basically we are interested in conditions which imply that bounded solutions of $( \ast )$ are asymptotically almost periodic. In particular, we show that a certain necessary condition on $f$ for this to happen is also sufficient, thereby strengthening earlier results. We also include a result on existence of bounded solutions, and indicate a generalization to a distribution equation.

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