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What's the difference between Fourier transformations and Fourier Series?

Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?

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    $\begingroup$ Fourier series is an discrete analogue of Fourier transformation $\endgroup$ – M. Strochyk Oct 25 '12 at 21:22
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    $\begingroup$ @Dean Here is something that I found extremely useful for Fourier transforms. Link $\endgroup$ – dearN Oct 25 '12 at 22:36
  • $\begingroup$ @M.Strochyk: why would one even needed the discrete analogue if the continuous one seem to cover all types of functions? Whether I deal with periodic and non periodic function I should be able to apply the continuous FT? $\endgroup$ – Medan Jun 28 '16 at 21:33
  • $\begingroup$ See: math.stackexchange.com/q/1002/168764. $\endgroup$ – nbro Feb 28 '18 at 13:13
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The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials. The Fourier transform can be viewed as the limit of the Fourier series of a function with the period approaches to infinity, so the limits of integration change from one period to $(-\infty,\infty)$.

In a classical approach it would not be possible to use the Fourier transform for a periodic function which cannot be in $\mathbb{L}_1(-\infty,\infty)$. The use of generalized functions, however, frees us of that restriction and makes it possible to look at the Fourier transform of a periodic function. It can be shown that the Fourier series coefficients of a periodic function are sampled values of the Fourier transform of one period of the function.

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    $\begingroup$ +1. Thanks for pointing out the relations between FS and FT! In the last sentence, "the Fourier transform of one period of the function." Do you mean the function is truncated to be zero outside one period and still defined over $(-\infty, \infty)$, and then the FT applies to the truncated function over $(-\infty, \infty)$? $\endgroup$ – Tim Jan 1 '13 at 16:10
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    $\begingroup$ @Tim you are right. $\endgroup$ – chaohuang Jan 1 '13 at 17:20
  • $\begingroup$ Why would that be necessary, e.g. does that matter? Is it equivalent to say that our function is periodic with infinite period, so that we still integrate over a single period? $\endgroup$ – Felix Crazzolara May 31 '17 at 9:28
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Fourier transform and Fourier series are two manifestations of a similar idea, namely, to write general functions as "superpositions" (whether integrals or sums) of some special class of functions. Exponentials $x\rightarrow e^{itx}$ (or, equivalently, expressing the same thing in sines and cosines via Euler's identity $e^{iy}=\cos y+i\sin y$) have the virtue that they are eigenfunctions for differentiation, that is, differentiation just multiplies them: ${d\over dx}e^{itx}=it\cdot e^{itx}$. This makes exponentials very convenient for solving differential equations, for example.

A periodic function provably can be expressed as a "discrete" superposition of exponentials, that is, a sum. A non-periodic, but decaying, function does not admit an expression as discrete superposition of exponentials, but only a continuous superposition, namely, the integral that shows up in Fourier inversion for Fourier transforms.

In both case, there are several technical points that must be addressed, somewhat different in the two situations, but the issues are very similar in spirit.

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    $\begingroup$ +1! Thanks! This reply confirms my recent understanding. "A periodic function provably can be expressed as a "discrete" superposition of exponentials," should "A periodic function" be "A periodic function which is $L^1$ in a single period" instead? $\endgroup$ – Tim Jan 1 '13 at 16:03
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    $\begingroup$ @Tim... Oh, yes, certainly, "locally $L^1$" is necessary for the relevant integrals to be literal. But, also, suitable "generalized functions" (distributions) have similar expressions, convergent in a suitable topology. $\endgroup$ – paul garrett Jan 1 '13 at 17:44
  • $\begingroup$ do you know any resource on Fourier transform with mathematics focused? Like how does it connect with complex analysis, topology, group theory? $\endgroup$ – Ooker Sep 25 '17 at 7:07
  • $\begingroup$ Why not simply using the term "linear combination" instead of "superposition", which is a very vague and ambiguous term? This thread is part of a math website, not a physics one. $\endgroup$ – nbro Feb 28 '18 at 14:03
  • $\begingroup$ @nbro, I used "superposition" deliberately, because in the Fourier series case it is an infinite sums (i.e., limit of the finite partial sums), but in the Fourier transform case it is a (limit of) integral(s), not finite linear combinations. $\endgroup$ – paul garrett Feb 28 '18 at 14:05
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Fourier transform is used to transform periodic and non-periodic signals from time domain to frequency domain. It can also transform Fourier series into the frequency domain, as Fourier series is nothing but a simplified form of time domain periodic function.

Fourier series

  1. Periodic function => converts into a discrete exponential or sine and cosine function.

  2. Non-periodic function => not applicable

Fourier transform

  1. Periodic function => converts its Fourier series in the frequency domain.

  2. non-Periodic function => converts it into continuous frequency domain.

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If you have a locally compact Abelian group $G$ you can define a group called the Pontryagin dual group - $\widehat{G}$. You can define a Haar measure on $G$, $\mu$. We can define the Fourier transform of a function $f\in L^1(G)$:

$$\widehat{f(\chi)}=\int_Gf(x)\overline{\chi(x)}d\mu(x)$$

$\widehat{f(\chi)}$ is a bounded continuous function that vanishes at infinity on $\widehat{G}$.

If $G=\Bbb R$ then $\widehat{G}=\Bbb R$ and we have the regular Fourier transform.

If $G=S^1$ then $\widehat{G}=\Bbb Z$ and we have the Fourier series (an example of a Fourier transform).

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  • $\begingroup$ $\widehat{G}$ is a representation of G, its elements are called the "characters" of elements of G. $\endgroup$ – Yan King Yin Apr 14 at 16:05

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