>> This isn't quite true over the reals, even assuming you're only looking 
>> for functions with a given period.  For example the function which is 
>> zero everywhere except being 1 at a single point will generate the same 
>> Fourier representation as the constant zero function since it will have 
>> the same integrals.
>
> O RLY?
>
> My DSP textbook says the Fourier transform of the delta function yields an 
> amplitude of 1 for all frequencies. (Whereas the Fourier transform of a 
> zero signal would be a zero signal.)

A function that is 0 everywhere except for f(0)=1 is not a delta funciton. 
A delta function has f(0)=infinity and when integrated it gives a non-zero 
value (ie it has area, unlike the function Kevin described).

> JPEG compression works by decomposing an image into a set of (2D) cosine 
> waves, and then quantinising the data. In my opinion, cosines are not 
> actually a particularly good choice for this. (Gibb's phenominon is very 
> ugly to look at.) So I'd like to try it with other functions - but first I 
> need to work out how...

Have you read:

http://en.wikipedia.org/wiki/Wavelet

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