Invisible wrote:
> 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.)
> 

As has been mentioned, a function which is zero everywhere except for 
f(0)=1 is not what is meant by a delta function in this context.  The 
critical property of the delta function which your book is talking about 
is that it has an integral of 1 over any interval containing the 
`spike', whereas the function I described has an integral of zero 
everywhere.

There's another argument that can lead you to the same conclusion that 
you can't approximate any function over the reals with a Fourier 
transform, and that's to note that the cardinality of the number of 
possible Fourier representations is smaller than the cardinality of the 
number of possible functions on the reals, so there have to be some 
functions that you can't represent.

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