On studying about Fourier series, I encountered 2 doubts:
1] How is it that a non-periodic function have a Fourier series?
2] When expressing a periodic function as summation of sinusoids, why is the fundamental (minimum) frequency taken to be that of the periodic signal? Why cannot one of the sinusoids have a frequency less than the fundamental?
Is there any mathematical logic for the second point?
Some insight would be much helpful.
To your questions:
1) Non-periodic functions will be restricted to some interval of the variable(s), and then this function will be periodically repeated (if required, in odd or even manner) outside of this interval. The Fourier series then holds only in the given interval of the variable(s).
2a) Well there is the sinusoid with frequency zero, i.e. the mean, which is less than the fundamental frequency.
2b) So you have a genuine periodical function (unlike the construction in 1) $f(x)$ with period length $L$. We will use the "normal" Fourier series representation to show that there are no subharmonics.
We know that this function can be written by a Fourier series $f(x) = \sum_{k=0}^{\infty} (a_k \sin(2 \pi kx/L) + b_k \cos(2 \pi kx/L))$
Now suppose you want to represent $f(x)$ in a subharmonic way. The easiest example would be to use period length $L/2$, so you write $f(x) = \sum_{n=0}^{\infty} (A_n \sin(\pi nx/L) + B_n \cos(\pi nx/L))$ with unknown coefficients $A_n$, $B_n$. Now lets compute these coefficients. We have $$A_n = \frac{2}{L} \int_{-L}^L f(x) \sin(\pi nx/L)\\ = \frac{2}{L} \sum_{k=0}^{\infty} (a_k \int_{-L}^L \sin(2 \pi kx/L) \sin(\pi nx/L) dx + b_k \int_{-L}^L \cos(2 \pi kx/L) \sin(\pi nx/L)dx ) $$
Due to the orthonormality of the $\sin$-functions,the only surviving term is the $\sin$-term with $n= 2k$, i.e. $A_n = a_{2k}$. This means that the subharmonic component you were looking for, represented by $A_1$, does not exist. The argument for the $B_n$ works analogously.
If you were trying out other lengths $L \cdot p/k$, you would have to ensure that the periodicity is obtained in your integration interval.