Fourier transform vs Inner product

[Bipolarity]

So the complex exponential Fourier series form an orthonormal basis for the space of functions. A periodic function can be represented with countably many elements from the basis, and an aperiodic function requires uncountably many elements.

Given a signal, we can find the coefficients of the exponentials in two ways:
1) Fourier transform
2) Inner product with that complex exponential

Though these two formulas are similar, they are not identical. So how could they both possibly give us the coefficient of a complex exponential?

Thanks!

BiP

 

[jbunniii]

Can you please show the formulas you are comparing? There are several different conventions in use. Also, please clarify whether you are talking about Fourier series or Fourier transforms. You mentioned both.

 

[mathman]

[quotr]A periodic function can be represented with countably many elements from the basis, and an aperiodic function requires uncountably many elements.
[/quote]
What do have in mind? The basis has only a countable number of elements. Are you mixing Fourier series and Fourier transdforms?

 

[AlephZero]

an aperiodic function requires uncountably many elements.

But not necessarily uncountably many non-zero elements. For example $\cos t + \cos \pi t$.

But I agree with the other posters, it's hard to figure out exactly what your OP is asking.