Wikipedia / Fourier inversion theorem
Wikipedia / 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.
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.
is the flip operator i.e. , then
and its Fourier transform are absolutely integrable (in the Lebesgue sense 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.
Statement
is an integrable continuous function. Use the convention for the Fourier transform that
Furthermore, we assume that the Fourier transform is also integrable.
Inverse Fourier transform as an integral
set
we have
and , the proof uses the following facts:
and , then .
and , then .
For , Fubini's theorem implies .
such that .
Define ; an approximation to the identity. That is, and point .
Since, by assumption, , it follows by the dominated convergence theorem Define and , for each the convolution with an approximate identity. But since
Fourier integral theorem
The theorem can be restated as
By taking the real part of each side of the above we obtain
Inverse transform in terms of flip operator
define the flip operator by
Then we may instead define
match the integral definition of , and in particular are equal to each other and satisfy .
and .
Two-sided inverse
The form of the Fourier inversion theorem stated above, as is common, is that
is a left inverse for the Fourier transform. However it is also a right inverse for the Fourier transform i.e.
is so similar to , this follows very easily from the Fourier inversion theorem (changing variables ):
and the flip operator and the associativity of function composition, since
Conditions on the function
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.
Schwartz functions
The Fourier inversion theorem holds for all Schwartz functions (roughly speaking, smooth functions that decay quickly and whose derivatives of all orders 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.
almost everywhere where 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
at 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
then the Fourier inversion theorem still holds so long as we again define the inverse transform with a smooth cut off function i.e.
No regularity condition; any number of dimensions
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 .
Square-integrable functions
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 ). For example, putting
-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
in the mean squared norm. In one dimension (and one dimension only), it can also be shown that it converges for almost every . This is Carleson's theorem, but is much harder to prove than convergence in the mean squared norm.
Tempered distributions
The Fourier transform may be defined on the space of tempered distributions we set
is defined using the integral formula. 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
The Fourier inversion theorem is analogous to the convergence of Fourier series. In the Fourier transform case we have
In the Fourier series case we instead have
to .
Applications
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 shows that the Fourier transform is a unitary operator on .
See also
Notes
References