I come up with the question when studying Fourier Analysis. I know that we have for sure in the Riemann sense,
Let $ f:\mathbb{R}\to\mathbb{R} $ be a real function. Let $ F $ be a primitive of $ f $ that is bounded on all of $ \mathbb{R}. $ Let $ f $ be periodic with period $ L. $ Then, $ F $ is also period with period $ L. $
So my thinking is that: if we have $f, g$ are two L-periodic Riemann integrable. Define their convolution as follows: $$(f*g)(x)=\frac{1}{L}\int_0^{L}f(y)g(x-y) dy,$$
then if their convolution is also periodic?
As mentioned above, somehow i can think that the primitive of a L-periodic function (this primitive is assumed to be bounded first) is also L-periodic function. Is there any connection between these.
Thanks in advance.