The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.
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In how far can this be generalized to other countable sets of functions? For example, if we keep our inner product, can we obtain another Schauder basis by an appropiate transform? What can we say about the bases in general?
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Does this generalize to other function spaces, say, periodic functions with one singularity?
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What do these thoughts lead to when considering the continouos FT?
Best Answer
It is not what you want, but may be worth mentioning. There is a huge branch of abstract harmonic analysis on (abelian) locally compact groups, which generalizes Fourier transformation on reals and circle. The main point about sin and cos (or rather complex exponent $e^{i n x}$) is that it is a character (continuous homomorphism from a group to a circle) and it is not hard to see that those are the only characters of the circle. That what makes Fourier transform so powerful. If you generalize it along the direction which drops characters, you'll probably get a much weaker theory.