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Fourier Inversion Theorem
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.
This last equation is called the Fourier integral theorem.
Another way to state the theorem is to note that if is the flip operator i.e. , then
The theorem holds if both and its Fourier transform are absolutely integrable (in the Lebesgue sense) and is continuous at the point . However, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals above may not converge in an ordinary sense.
When used in physics and engineering, the Fourier inversion theorem is often used under the assumption that everything "behaves nicely". In mathematics such heuristic arguments are not permitted, and the Fourier inversion theorem includes an explicit specification of what class of functions is being allowed. However, there is no "best" class of functions to consider so several variants of the Fourier inversion theorem exist, albeit with compatible conclusions.
The Fourier inversion theorem holds for all Schwartz functions (roughly speaking, smooth functions that decay quickly and whose derivatives all decay quickly). This condition has the benefit that it is an elementary direct statement about the function (as opposed to imposing a condition on its Fourier transform), and the integral that defines the Fourier transform and its inverse are absolutely integrable. This version of the theorem is used in the proof of the Fourier inversion theorem for tempered distributions (see below).
Integrable functions with integrable Fourier transform
The Fourier inversion theorem holds for all continuous functions that are absolutely integrable (i.e. ) with absolutely integrable Fourier transform. This includes all Schwartz functions, so is a strictly stronger form of the theorem than the previous one mentioned. This condition is the one used above in the statement section.
A slight variant is to drop the condition that the function be continuous but still require that it and its Fourier transform be absolutely integrable. Then almost everywhere where g is a continuous function, and for every .
Integrable functions in one dimension
Piecewise smooth; one dimension
If the function is absolutely integrable in one dimension (i.e. ) and is piecewise smooth then a version of the Fourier inversion theorem holds. In this case we define
Then for all
i.e. equals the average of the left and right limits of at . Note that at points where is continuous this simply equals .
A higher-dimensional analogue of this form of the theorem also holds, but according to Folland (1992) is "rather delicate and not terribly useful".
Piecewise continuous; one dimension
If the function is absolutely integrable in one dimension (i.e. ) but merely piecewise continuous then a version of the Fourier inversion theorem still holds. In this case the integral in the inverse Fourier transform is defined with the aid of a smooth rather than a sharp cut off function; specifically we define
The conclusion of the theorem is then the same as for the piecewise smooth case discussed above.
Continuous; any number of dimensions
If is continuous and absolutely integrable on then the Fourier inversion theorem still holds so long as we again define the inverse transform with a smooth cut off function i.e.
The conclusion is now simply that for all
No regularity condition; any number of dimensions
If we drop all assumptions about the (piecewise) continuity of and assume merely that it is absolutely integrable, then a version of the theorem still holds. The inverse transform is again defined with the smooth cut off, but with the conclusion that
In this case the Fourier transform cannot be defined directly as an integral since it may not be absolutely convergent, so it is instead defined by a density argument (see the Fourier transform article). For example, putting
we can set where the limit is taken in the -norm. The inverse transform may be defined by density in the same way or by defining it in terms of the Fourier transform and the flip operator. We then have
where is defined using the integral formula. If then this agrees with the usual definition. We may define the inverse transform , either by duality from the inverse transform on Schwartz functions in the same way, or by defining it in terms of the flip operator (where the flip operator is defined by duality). We then have
Relation to Fourier series
When considering the Fourier series of a function it is conventional to rescale it so that it acts on (or is -periodic). In this section we instead use the somewhat unusual convention taking to act on , since that matches the convention of the Fourier transform used here.
In particular, in one dimension and the sum runs from to .
Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform.
In applications of the Fourier transform the Fourier inversion theorem often plays a critical role. In many situations the basic strategy is to apply the Fourier transform, perform some operation or simplification, and then apply the inverse Fourier transform.
More abstractly, the Fourier inversion theorem is a statement about the Fourier transform as an operator (see Fourier transform on function spaces). For example, the Fourier inversion theorem on shows that the Fourier transform is a unitary operator on .
Tables of Fourier transforms may easily be used for the inverse Fourier transform by composing the looked-up function with the flip operator. For example, looking up the Fourier transform of the rect function we see that
so the corresponding fact for the inverse transform is