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I am working on my understanding of various transforms and I have been thinking about the Fourier transform, what i "does" to the function it is applied to.

The way I see it:

The function $f$ that is transformed is multiplied with $\exp(icx)$ which essentially describes a rotating vector in the complex plane.

  • If $f$ is periodic cosine and the period of $f$ does not match the period of $\exp(icx)$ the "terms" in the integral will vary and cancel each other leading to a value of zero for the transform.

  • If $f$ is periodic cosine and the period of $f$ matches the period of $\exp(icx)$ the "terms" in the integral will be constant and the value of the transform will be $\infty$.

  • If $f$ is periodic but not a cosine it can be decomposed to a sum of cosines and the different cosine terms of this sum will work as above resulting in a spectra for $f$.

If this is somewhat correct I wonder:

  1. What about aperiodic functions?

  2. Is there a similar way of thinking about the Laplace transform?

Please forgive the non mathematical language, I'm neither a math major nor is English my mother tongue.

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    I really think that http://en.wikipedia.org/wiki/Fourier_transform will answer all your questions.2012-12-12
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    You're not completely wrong but I don't think that's the best way to think at it either. Periodic functions are to be represented with Fourier series, period (pun intended :-) ). Now there clearly is relationship between Fourier series and Fourier integrals, but that's another story.2013-01-28

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