The reason why the Fourier basis works so well is because it forms an orthogonal basis for all smooth real-valued functions. We'd need a similar way to define square waves which form an orthogonal basis. Found one paper doing that: A REVIEW OF ORTHOGONAL SQUARE-WAVE FUNCTIONS AND THEIR APPLICATION TO LINEAR NETWORKS [1]

Give it a try!

[1] https://www.sciencedirect.com/science/article/abs/pii/001600...

If I'm not mistaken, the square wave case should in essence be the limiting case of the general case you get when looking at "trigonometric" functions which don't arise as the parametrisation of xˆ2 + yˆ2 = 1, but xˆn + yˆn = 1 for n > 2. It turns out that people have looked at that, and the study of those functions have been given the funny name "squigonometry". There's a Springer textbook out there.

Typically mathematical treatment of DSP is based on the discrete-time Fourier transform and the Z transform, which differ from respectively the Fourier transform and the Laplace transform not in using square waves, but in using an infinite series rather than an integral to capture the idea of discrete sampling.

Yes those exist. It is called a Haar wavelet. https://en.m.wikipedia.org/wiki/Haar_wavelet

Huh. In all of those years when people were talking about wavelets being the ultimate solution to JPG artifacts (on square boundaries) , nobody ever mentioned that they were built on a square wave function. They just talked about being able to move the positions and adjust the diameter arbitrarily. I have to say I'm a little annoyed.

For jpg2000 other wavelets are used not the square ones. Regular jpg is cosines.