Summer of Math Exposition

I liked the idea of approaching a technical detail about the Fourier transform by an experimental approach. And I also like your style of presenting. However, there are a few issues I would like to mention:

1. As you mentioned yourself, the video is really long. I found it hard to stay focused, since…
2. You take a lot of time to introduce ideas like the Gram Schmidt algorithm and unitary operators. I suspect that most people who are interested in a video like this, already know what the Gram Schmidt algorithm is, so I would skip it completely (and maybe refer to another video) in order to make your contribution shorter.
3. When I started the video, youtube gave me a German audio track that was (probably) read by an AI voice. This audio track was really horrible and it took me some time to realise that this wasn’t the default audio of your video. If it wouldn’t have been for the voting situation, I probably would have skipped the video after a few seconds.

I would recommend making the audio volume more consistent. Some parts were much quieter than others.

I like this approach! It’s nice that you made Hermite polynomials more hands-on and less magical.

Maybe it would’ve been nice to see the orthogonalization process completely by hand on your small 3x3 example (or at least talked-through/displayed/animated), to help make things more concrete.

Would have been great to have a quick refresher on the material from your previous videos just for completeness - even for a graduate audience, mentioning what the Hermite functions and L2/Hilbert spaces etc are could be a great way to ground your intro. Otherwise super comprehensive and easy to follow though!