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Continuous Fourier transform

In mathematics, the continuous Fourier transform is a linear operator which maps functions to other functions. Loosely, the Fourier transform decomposes a function into a continuous spectrum of the frequencies that comprise that function. In mathematical physics, the Fourier transform of a signal <math>f(t)</math> can be thought of as that signal in the "frequency domain". This is similar to the basic idea of the various other Fourier transforms inclucing the Fourier series of a periodic function.

A number of slightly different but essentially equivalent definitions are used in the literature. Suppose f : R -> C is a Lebesgue integrable function. We then define its continuous Fourier transform F : R -> C as

<math> F(s) = \int f(t) e^{-2\pi ist} dt </math>

for every real number <math>s</math>. (Here, <math>\pi</math> is pi and <math>i</math> is the imaginary unit). We think of <math>s</math> as a frequency and <math>F(s)</math> as the complex number which encodes amplitude and phase of the signal <math>f(t)</math> at that frequency.

The Fourier transform is close to self-inverse: if <math>F(s)</math> is defined as above, and <math>f</math> is sufficiently smooth, then

<math> f(t) = \int F(s) e^{2\pi ist} ds </math>

for every real number t.

As a rule of thumb: the more concentrated <math>f(t)</math> is, the more spread out is <math>F(s)</math>. The only functions which coincide with their own Fourier transforms are the constant multiples of the function <math>f(t) = \exp(- \pi t^2)</math>. In a certain sense, this function therefore strikes the precise balance between being concentrated and being spread out. The Fourier transform also translates between smoothness and decay: if <math>f(t)</math> is several times differentiable, then <math>F(s)</math> decays rapidly towards zero for s→±∞.

Fourier transforms, and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with differentiation in the following sense: if <math>f(t)</math> is a differentiable function with Fourier transform <math>F(s)</math>, then the Fourier transform of its derivative is given by <math>2 \pi is F(s)</math>. This can be used to transform differential equations into algebraic equations. Note that this technique only applies to problems whose domain is the whole set of real numbers. By extending the Fourier transform to functions of several variables (as outlined below), partial differential equations with domain Rn can also be translated into algebraic equations.

Furthermore, the Fourier transform translates between convolution and multiplication of functions: if <math>f(t)</math> and <math>g(t)</math> are integrable functions with Fourier transforms F(s) and G(s) respectively, and if the convolution of f and g exists and is integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms <math>F(s) G(s)</math>. If the the product <math>f(t) g(t)</math> is integrable, then the Fourier transform of this product is given by the convolution of F(s) and G(s).

The most general and natural context for studying the continuous Fourier transform is given by the tempered distributions; these include all the integrable functions mentioned above and have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution and the rule for the inverse of the Fourier transform is universally valid. Furthermore, the useful Dirac delta is a tempered distribution but not a function; its Fourier transform is the constant function 1. Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.

The following table records some important Fourier transforms. F(s) and G(s) denote the Fourier transforms of f(t) and g(t), respectively. f and g may be integrable functions or tempered distributions.

  Signal Fourier transform Remarks
1. <math>a f(t) + b g(t)</math> <math>a F(s) + b G(s)</math> Linearity
2. <math>f(t - a)</math> <math>e^{-2\pi ias} F(s)</math> Shift in time domain
3. <math>e^{2 \pi iat} f(t)</math> <math>F(s - a)</math> Shift in frequency domain
4. <math>f(a t)</math> <math>|a| F\left(s \over a\right)</math> If a is large, then f(at) is concentrated around 0 and F(s/a)/|a| spreads out and flattens
5. <math>f'(t)</math> <math>2 \pi is F(s)</math> <math>f'(t)</math> is the (distribution) derivative of <math>f(t)</math>
6. <math>t f(t)</math> <math>{1 \over 2 \pi i} F'(-s)</math> This is the inverse rule to 5.
7. <math>(f * g)(t)</math> <math>F(s) G(s)</math> <math>f * g</math> denotes the convolution of <math>f</math> and <math>g</math>
8. <math>f(t) g(t)</math> <math>(F * G)(-s)</math> This is the inverse of 7.
9. <math>\delta(t)</math> 1 <math>\delta(t)</math> denotes the Dirac delta distribution.
10. 1 <math>\delta(s)</math> Inverse of 9. This rule shows why the Dirac delta is important: it shows up as the Fourier transform of everyday functions.
11. <math>t^n</math> <math>{(-1)^{n (n + 1) / 2} \over (2 \pi i)^n} \delta^{(n)} \left( (-1)^n s \right)</math> Here, <math>n</math> is a natural number. δ(n)(s) is the n-th distribution derivative of the Dirac delta. This rule follows from rules 6. and 10. Combining this rule with 1., we can transform all polynomials.
12. <math>e^{2 \pi iat}</math> <math>\delta(s - a)</math> This follows from and 3. and 10.
13. <math>\cos(2 \pi at)</math> <math>{\delta(s - a) + \delta(s + a) \over 2}</math> Follows from rules 1 and 12 using cos(2πat) = 1/2 ( eiat + e-2πiat ) (Euler's formula)
14. <math>\sin(2 \pi at)</math> <math>{\delta(s - a) - \delta(s + a) \over 2i} </math> Also from 1 and 12.
15. <math>\exp(-a t^2)</math> <math>\sqrt{\frac{\pi}{a}} \exp(\frac{-\pi^2 s^2}{a})</math> Shows that the Gaussian function exp(-π t2) is its own Fourier-transform

If a function f : R -> C is square-integrable, that is

<math>\int |f|^2 dt < \infty,</math>
then it can be viewed as a tempered distribution and hence has a Fourier transform. This transform is again square integrable. Furthermore, if <math>f(t)</math> and <math>g(t)</math> are square integrable and <math>F(s)</math> and <math>G(s)</math> are their Fourier transforms, then we have
<math>\int f(t) g(t)^* dt = \int F(s) G(s)^* ds</math>
(where the star denotes complex conjugation). Therefore, the Fourier transformation yields an isometric automorphism of the Hilbert space L2(R).

The Fourier transform can also be defined for functions (and distributions) f : Rn -> C. In the definition, the product st is then to be interpreted as the inner product of the two vectors s and t. All the above properties and formulas remain valid.

See also: Fourier transform, Fourier series, Laplace transform, Discrete Fourier transform

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