This question about the intuition behind the scaling property of the Fourier transform made me wonder about the corresponding notion for a Fourier series.
The Fourier transform of $f(ax)$ is $\frac{1}{|a|} \mathcal{F(\frac{u}{a})}$. If $a>1$ then the graph of $f(ax)$ is $f$ compressed and so its Fourier transform has frequencies that are higher. However they are scaled down in magnitude.
On the other hand take $f(x)= \cos x$. Its fourier series is trivially itself, with the coefficient of $\cos x$ being $1$. Scaling it to $f(ax) = \cos ax$ where $a>1$ is an integer still has a trivial fourier series, and the coefficient of $\cos ax$ is $1$. This is unlike the Fourier transform where the "coefficients" of each frequency get scaled as the formula shows: $\frac{1}{|a|} \mathcal{F(\frac{u}{a})}$.
Is there a conceptual way to explain this discrepancy?