Continuous Fourier transform

In mathematics, the continuous Fourier transform is a certain linear operator that 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 f(t) can be thought of as that signal in the "frequency domain". This is similar to the basic idea of the various other Fourier transforms including the Fourier series of a periodic function.

A number of slightly different but essentially equivalent definitions are used in the literature. These are due to minor differences in normalizations. Suppose f is a complex-valued Lebesgue integrable function. We then define its continuous Fourier transform F to be also a complex-valued function:

for every real number ω. (Here, i is the imaginary unit). We think of ω as an angular frequency and F(ω) as the complex number which is the amplitude and phase of the component of the signal f(t) at that frequency.

The Fourier transform is close to a self-inverse mapping: if F(ω) is defined as above, and f is sufficiently smooth, then

for every real number t.

If we define the Fourier transform   in this way on the set of complex-valued functions on the line with compact support and extend by continuity to the Hilbert space of square-integrable functions, then it is a unitary operator

Moreover,

Parseval's theorem, a special case of the Plancherel theorem, states that

This theorem is usually interpreted as asserting the unitary property of the Fourier transform. See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.

As a rule of thumb: the more concentrated f(t) is, the more spread out is F(ω). The only functions which coincide with their own Fourier transforms are the constant multiples of the function f(t) = exp(−t²/2). 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 f(t) is several times differentiable, then F(ω) 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 f(t) is a differentiable function with Fourier transform F(ω), then the Fourier transform of its derivative is given by iω F(ω). 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 f(t) and g(t) are integrable functions with Fourier transforms F(ω) and G(ω) 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 F(ω')G(ω). If the product f(t)g(t) is integrable, then the Fourier transform of this product is given by the convolution of F(ω) and G(ω).

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.

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

then it can be viewed as a tempered distribution and hence has a Fourier transform. This transform is again square integrable. Furthermore, if f(t) and g(t) are square-integrable and F(ω) and G(ω) are their Fourier transforms, then we have the Plancherel theorem:

(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 ωt is then to be interpreted as the inner product of the two vectors ω and t. All the above properties and formulas remain valid.

Table of important Fourier transforms

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

See also

• Fourier transform,
• Fourier series,
• Laplace transform,
• Discrete Fourier transform.

References

• Fourier Transforms from eFunda - includes tables
• Dym & McKean, Fourier Series and Integrals. (For readers with a background in mathematical analysis.)
• K. Yosida, Functional Analysis, Springer-Verlag, 1968.
• L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, 1976. (Somewhat terse.)