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Fourier integral and Fourier Transform

  1. May 11, 2014 #1
    Which is the difference between the Fourier integral and Fourier transform? Or they are the same thing!?

    Fourier integral:
    image.png
     
    Last edited by a moderator: Sep 25, 2014
  2. jcsd
  3. May 11, 2014 #2

    Mark44

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  4. May 11, 2014 #3

    jbunniii

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    The Fourier integral is one way to calculate the Fourier transform of a function:
    $$\hat{f}(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx$$
    This definition only makes sense for certain kinds of functions. Since ##|f(x) e^{-i \omega x}| = |f(x)|##, the Fourier integral is defined if and only if ##|f|## is integrable. (In the case of Lebesgue integration, we say that ##f \in L^1##.)

    But the Fourier transform can be defined for a larger class of functions, and even some objects that are not functions.

    For example, we can define the Fourier transform of a function which is square-integrable but not integrable (i.e. ##f \in L^2 \setminus L^1##). To do this, we approximate ##f## by a sequence of functions ##f_n \in L^1 \cap L^2## and define ##\hat{f} = \lim \hat{f_n}##. Of course there are lot of details to check, such as the existence of such a sequence, and the fact that the limit does not depend on a particular choice for the sequence.

    It is also possible to define the Fourier transform of certain types of distributions, which can be thought of as generalized functions. See here for example:

    http://en.wikipedia.org/wiki/Fourier_transform#Tempered_distributions

    We can also define a Fourier transform on other types of objects, such as groups (a special case of this is the discrete Fourier transform):

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

    So, the Fourier transform is the more general concept, and the Fourier integral is how it is defined/computed in the case of integrable functions.
     
    Last edited: May 11, 2014
  5. May 11, 2014 #4

    lurflurf

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    ^I agree. The Fourier integral is a method of calculating the Fourier transform. In many cases it is not useful to distinguish between the two. Be aware that there are different Fourier transforms and using a slightly different one can cause confusion.
     
  6. May 14, 2014 #5

    TheDemx27

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    Is there a good way to make FFTs without premade modules in programming? Just an aside...
     
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