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lwan

The fourier transform was a really cool example of this. What's another example of these kinds bases, and how would we find find them?

ehsun

If I understand correctly, the Legendre polynomials are another basis, at least for a certain set of functions. It's kind of cool that you can use the Gram-Schmidt process (used for finding bases for vector spaces) with very few modifications to find that.

keenan

Yeah, totally.

A very natural generalization of basic Fourier modes (sines and cosines) are eigenfunctions of the Laplace-Beltrami operator - a fun place this shows up is in spectral geometry processing. If you want to understand this viewpoint in greater generality, the place to start is the spectral theorem.

aabhagwa

Is there a nice description of some basis for the vector space of all real-valued functions? I've always seen this question dodged, but it would be cool to read about such a basis.

keenan

Perhaps you should start with the question, "what does it mean to have a basis for R^1?", i.e., just the real numbers. One starting point is the Hamel Basis (though perhaps there is a better source than the Mathworld page...)

hophop

Here's a good definition (that is slightly more abstract than R^1) of a Hamel Basis. Actually the whole PDF has pretty interesting content.