Hacker News Reader

Comment by yagyu | Hacker News Reader

There is this hard question that is often glossed over in a first course on Fourier analysis: We learn that the Fourier transform of f(t) is an integral of exp(iwt)f(t) over all t. But when we look at a table of Fourier transform pairs, we find things like the Fourier transform of a constant function (is the Dirac delta). Integral does not converge? WTF?

I really think Prof. Osgood hits a sweet spot of not sweeping this under the carpet, and not becoming a course on functional/integration theory. I'm a postdoc in theoretical physics, and use these transforms regularly, and I have yet to find a more honest but still useful introduction (although I understand that real mathematicians cringe at my abuse of mathematics.. :)

edit: typo

View on HN

As someone who has gone down the rabbit hole of measure theory, functional analysis, and the like, I completely agree with the wonderful balance that Prof. Osgood strikes.

For a concrete instance, he brings up the distributional viewpoint of Laurent Schwartz and talks about the class of decaying functions for which one can operate nicely. This was something I next saw primarily in mathematics texts, with very few engineering books caring to talk about this even though it is not very hard to introduce and gives significant clarity to the process IMHO.

Studying Prof. Osgood's stuff off Stanford SEE during my summer vacation at the end of high school was one of the best learning experiences I have had till date.

the other somewhat confusing thing is that algorithms like discrete fourier transform/fft are actually fourier series.

There are three of them.

1. Fourier transform: unbounded real domain <-> unbounded real domain

2. Fourier series: bounded real domain <-> unbounded integer domain

3. Discrete Fourier transform (DFT): bounded integer domain <-> bounded integer domain

FFT is an algorithm for quick calculation of DFT, which is not Fourier series.

you are missing the discrete time fourier transform, which is the dual of your point (2).

Together, these eight domains can be arranged on a commutative cube, all of whose edges have beautiful meanings: sampling, interpolation, etc. The fact that the cube is commutative gives a lot of theorems relating each transform and these operations.

The point being that by your definition of not, DFT is also NOT a Fourier transform. It's only a "transform" in that the two domains are similar (identical, mod complexity). That's quite confusing to engineers who only care about the results. Computationally, the first two are strictly speaking impossible, but effectively, your data come in being modeled as finite real-valued arrays that represent function values, (close enough to "bounded real domain") and need to come out as finite real-valued arrays, represented integer-indexed frequencies, and a DFT on your data will operationally give you that. This is much more realistically described as a discretized pseudo-fourier series.

The discrete fourier transform is actually finite-dimensional linear algebra.

No infinite series nor convergence problems here.

No, it really is a Fourier series.

More precisely take a set of points, do a DFT, then make it infinite by making the rest of the terms 0. You now have the Fourier series of a periodic function that takes hits all of those points. Furthermore it is the "smallest" possible such periodic function in the L2 norm on periodic functions.

> The discrete fourier transform is actually finite-dimensional linear algebra.

Not only that, it's the most interesting kind of linear algebra: it shows that, though every (finite-dimensional if you don't like choice) vector space has a basis—so that we can pretend that all n-dimensional vector spaces are the same for a fixed n—the simple act of changing that basis (which is all the DFT is doing) can have a profound impact.

No, this is not what people mean when they say "Fourier series" which is something like "the Fourier series expansion of a periodic function".

It's not about whether the values are continuous or discrete, but instead about the domain over which the values are defined. Fourier series maps a periodic function over a continuous domain into a countably infinite number of coefficients.

Reply on news.ycombinator.com