We know that a function(adsbygoogle = window.adsbygoogle || []).push({}); f(x) over an interval [a,b] can be written as an infinite weighted sum over some set of basis functions for that interval, e.g. sines and cosines:

[tex]f(x) = \alpha_0 + \sum_{k=1}^\infty \alpha_k\cos kx + \beta_k\sin kx.[/tex]

Hence, I could provide youeitherwith the functionf(x)orwith the complete set of Fourier coefficients {α,_{k}β}, and you would have all the same information over the interval [_{k}a,b].

But a function over an interval is anuncountablyinfinite amount of information: you need the value off(x) for every real numberx∈ [a,b]. Whereas the set of Fourier coefficients is acountablyinfinite amount of information: you need only the value ofαand_{k}βfor every natural number_{k}k.

So how can a countable set of Fourier coefficients encode all the same information as a function over an uncountable range of real numbers?

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# How can a Fourier expansion contain all the same info as original f'n?

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