Is Fourier transform a generalisation of Fourier series?

Question

Is the Fourier transform a generalisation of a Fourier series or an a different concept?

I.e. Can Fourier transforms be used with periodic functions and will it reduce down to the Fourier series when this is done. Or can Fourier transforms not deal with Fourier series at all?

Answer 1

Fourier transform can represent periodic functions. If the Fourier series coefficient of $f(t)$ is $a_k$

$$F(\omega)=2\pi \sum _{k=-\infty}^\infty a_k \delta(\omega-k\omega _ 0)$$

Moreover, Fourier transform can be thought of as the limit of the fourier series as the period approaches infinity (a function that repeats once every infinity, so it happens only once)

Answer 2

From a more abstract point of view, Fourier transform and Fourier series are instances of harmonic analysis on locally compact groups. For the Fourier transform, the concerned group is $\mathbb{R}$ (where $f$ is defined) and the Fourier transform $\hat f$ is defined on its dual group, which happens to be also $\mathbb{R}$. In the case of Fourier series the groups are $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and the dual group is $\mathbb{Z}$.

Answer 3

From a moderately abstract point of view, the Fourier transform of a periodic function exists in the sense of distribution theory. It consists of a finite combination of Dirac deltas at the frequencies represented by the function. Similarly, for a square integrable function satisfying a "symmetric" boundary condition on a compact interval (e.g. homogeneous Dirichlet or homogeneous Neumann boundary conditions), the Fourier transform consists of a possibly infinite combination of Dirac deltas at the frequencies represented by the function. The weights on the Dirac deltas are exactly the Fourier coefficients (up to a normalization constant depending only on your choice of convention for the Fourier transform).