TITLE: Fourier transform vs Fourier series
QUESTION [8 upvotes]: Why are Fourier series considered to be the subset of Fourier transform? It should have been other way around. Because a non periodic pulse is a subset of periodic pulse with period infinite. So Fourier transform (non periodic signals) is a subset of Fourier series (periodic signals).

REPLY [2 votes]: You can zero-extend a period of a periodic function to $\mathbb{R}$ and calculate the Fourier transform of the extension. This is done within pointwise function definition $f:\mathbb{R} \rightarrow \mathbb{C}$ and doesn't need the use of distributions. The zero-extension method has also the property that Fourier coefficients of the original periodic function can be read directly from the values of the transform of the extension. With suitable assumptions on the function also the inverse transform gives correct values within the interior of the period. That's why Fourier series is considered as a subset of Fourier transform.
The other way round it is not as easy. That's because Fourier series handles periodic functions only. A mapping that maps an infinite period to finite nonzero period violates the algebraical structure of the domain and values of the transform of the original aperiodic function cannot be read from the coefficients of the series anymore in a straightforward way. The countable domain $\mathbb{Z}$ of the coefficients isn't the problem because it can be extended to $\mathbb{R}$ by $f(\omega) = \sum_{k=-\infty}^\infty c_k \textrm{sinc}(\omega-k)$,  where $\textrm{sinc}(x) = \bigg\{ \begin{eqnarray} & \frac{\sin(\pi x)}{\pi x} & , \ x\neq 0 \\ & 1 & , \ x = 0 \end{eqnarray}$. Now the obvious way to calculate the values of the transform of the original function is to form the Fourier series of the mapped version of the original function. Then it can be inverse mapped to obtain the original function. Values of the transform of the original function can be now obtained by just calculating all the values of the transform of the original function. The procedure consisted of inverse transformation, inverse mapping and transformation, that is not easy to do. However, that kind of procedure exists.