I have an almost periodic function $f_{t}(x,y)$ in the Besocovitch sense. In particular, $f_{t}(x,y):= \sum_{i}a_{i}e^{it(x-y)\lambda_{i}}$, where $a_{i}\geq 0$ for all $i$, $\sum_{i}a_{i}^{2}<\infty$ and the $\lambda_{i}$ are all real. Let g(x,y) be a compactly supported integrable function with the property $g(x,y)\geq 0$ for all (x,y) in its support. I am wondering if the integral $\int\int |f_{t}(x,y)|g(x,y)dxdy$ is also an almost periodic function.
I have tried to write $f_{t}(x,y)$ as a series but this has led me nowhere. I think that there should be a more straightforward argument.
Thank you very much for your help.