Fourier inversion theorem
satisfying certain conditions, and we use the convention for the Fourier transform that then In other words, the theorem says that This last equation is called the Fourier integral theorem.
and its Fourier transform are absolutely integrable (in the Lebesgue sense) and
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.
Applying facts 1, 2 and 4, repeatedly for multiple integrals if necessary, we obtain
The theorem can be restated as By taking the real part[1] of each side of the above we obtain For any function
by Then we may instead define It is immediate from the definition of the Fourier transform and the flip operator that both
and The form of the Fourier inversion theorem stated above, as is common, is that In other words,
, this follows very easily from the Fourier inversion theorem (changing variables
and the flip operator and the associativity of function composition, since 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.
The Fourier inversion theorem holds for all continuous functions that are absolutely integrable (i.e.
This includes all Schwartz functions, so is a strictly stronger form of the theorem than the previous one mentioned.
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.
) and is piecewise smooth then a version of the Fourier inversion theorem holds.
A higher-dimensional analogue of this form of the theorem also holds, but according to Folland (1992) is "rather delicate and not terribly useful".
) 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.
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 for almost every
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).
The Fourier transform may be defined on the space of tempered distributions
by duality of the Fourier transform on the space of Schwartz functions.
, 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).
In many situations the basic strategy is to apply the Fourier transform, perform some operation or simplification, and then apply the inverse Fourier transform.
shows that the Fourier transform is a unitary operator on
