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It's very interesting but why isn't there any practical example? What are the equations of three periodic functions that would add up to y = x^2 for instance?
by simias 13y ago
It's very interesting but why isn't there any practical example? What are the equations of three periodic functions that would add up to y = x^2 for instance?
How would one even construct those three equations? If I'm not mistaken TFA demonstrates that these functions exist, not how to find them.
I wish I hadn't stopped with maths in high school...
- impendia 13y agoBrief answer: the author invokes the Axiom of Choice, so the three equations will be really screwball and there will be no way to write them down explicitly. This is, in some sense, regarded as cheating. (The Axiom of Choice is generally accepted as a legitimate mathematical axiom, but not universally.) http://en.wikipedia.org/wiki/Axiom_of_choice http://en.wikipedia.org/wiki/Axiom_of_choice
- nmc 13y agoYou are right, the demonstration is only partially constructive. The functions are well defined: they are the canonical projectors of the Hamel basis. The trick is, the basis itself is not constructible.
- ufo 13y agoThere are no practical examples. The solutions are unbounded, discontinuous functions (that wont have nice equations) and since the proof uses the axiom of choice it is fundamentally non constructive (meaning you would need a totally different proof technique to be able to build a concrete solution)
- thaumasiotes 13y agoI can't quite follow your thinking here? > If I'm not mistaken TFA demonstrates that these functions exist, not how to find them. Completely correct. > why isn't there any practical example? You answer this yourself... the article demonstrates that the functions exist. It can't demonstrate the functions themselves; no one can do that.