Relating Fourier series and Fourier transforms

Fourier series and Fourier transforms may seem more different than they are because of the way they’re typically taught. Fourier series are presented more as a representation of a function, not a transformation. Here’s a function on an interval. We can write it as a sum of sines and cosines, just as we can write a function as a sum of powers in a power series. There’s not much emphasis on the coefficients per se. They appear inside a sum, but don’t get much attention on their own.

Fourier transforms, on the other hand, are presented as genuine transforms. Here’s a function, and here’s it’s transform, another function. One’s a function of time, the other a function of frequency. Or maybe both are presented as representations of the same function in two different domains, the time domain and the frequency domain.

You could think of the Fourier series as a kind of transform, taking a periodic function and mapping it to an infinite sequence, the Fourier series coefficients. And you could think of the Fourier transform as being a kind of continuous set of coefficients for representing a function, if you interpret the inversion theorem the right way.

Here are a couple connections between Fourier series and Fourier transforms. Start with a function f on an interval and compute its Fourier series. The Fourier series is periodic, so we could think of f as periodic, even though we only care about f on the interval. Instead, let’s think of extending f to be 0 everywhere outside the interval. Now we take the Fourier transform of f. The Fourier series coefficients are the Fourier transform of f evaluated at integer arguments.

Now let’s go back to thinking of f as a periodic function. What would it’s Fourier transform look like? In classical analysis, you can’t do that. Periodic functions have Fourier series but they don’t have Fourier transforms because the integral defining the latter does not converge. But by the magic of tempered distributions, we can indeed take the Fourier transform of a periodic function. The result is a weighted sum of delta distributions at each integer, and the coefficient of the delta distribution at n is the nth Fourier series coefficient.

The proof of the claim in the previous paragraph is simple once you understand the sha function Ш. Start with a function f defined on a unit interval and extended to be zero outside that interval. Convolving f with Ш make a periodic function f*Ш extending f. The Fourier transform of a convolution is the product of the convolutions. The Fourier transform of f is simply its classical Fourier transform F. The Ш function is it’s own Fourier transform, so the transform of f*Ш is FШ. Multiplying a function by Ш samples that function, and the samples of F are the Fourier coefficients of the Fourier series of f*Ш, the periodic extension of f.

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