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.

The theorem says that if we have a function 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.

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 make sense, or the theorem may hold for almost all rather than for all .

Contents

StatementEdit

In this section we assume that   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 integralEdit

The most common statement of the Fourier inversion theorem is to state the inverse transform as an integral. For any integrable function   and all   set

 

Then for all   we have

 

Fourier integral theoremEdit

The theorem can be restated as

 

If f is real valued then by taking the real part of each side of the above we obtain

 

Inverse transform in terms of flip operatorEdit

For any function   define the flip operator[note 1]   by

 

Then we may instead define

 

It is immediate from the definition of the Fourier transform and the flip operator that both   and   match the integral definition of  , and in particular are equal to each other and satisfy  .

Note also that since   we have   and

 

Two-sided inverseEdit

The form of the Fourier inversion theorem stated above, as is common, is that

 

In other words,   is a left inverse for the Fourier transform. However it is also a right inverse for the Fourier transform i.e.

 

Since   is so similar to  , this follows very easily from the Fourier inversion theorem (changing variables  ):

 

Alternatively, this can be seen from the relation between   and the flip operator and the associativity of function composition, since

 

Conditions on the functionEdit

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 functionsEdit

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 transformEdit

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 are absolutely integrable. Then   almost everywhere where g is a continuous function, and   for every  .

Integrable functions in one dimensionEdit

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

 

for almost every   [1]

Square integrable functionsEdit

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

 

in the mean squared norm. It can also be shown that it converges for almost every x∈ℝ- this is Carleson's theorem, but is much harder to prove than convergence in the mean squared norm.

Tempered distributionsEdit

The Fourier transform may be defined on the space of tempered distributions   by duality of the Fourier transform on the space of Schwartz functions. Specifically for   and for all test functions   we set

 

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 seriesEdit

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.

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

 
 
 

In particular, in one dimension   and the sum runs from   to  .

ApplicationsEdit

 
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  .

Properties of inverse transformEdit

The inverse Fourier transform is extremely similar to the original Fourier transform: as discussed above, it differs only in the application of a flip operator. For this reason the properties of the Fourier transform hold for the inverse Fourier transform, such as the Convolution theorem and the Riemann–Lebesgue lemma.

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

 

ProofEdit

The proof uses a few facts, given   and  .

  1. If   and  , then  .
  2. If   and  , then  .
  3. For  , Fubini's theorem implies that  .
  4. Define  ; then  .
  5. Define  . Then with   denoting convolution,   is an approximation to the identity: for any continuous   and point  ,   (where the convergence is pointwise).

First note that, since, by assumption,  , then it follows by the dominated convergence theorem that

 

Define  . Applying facts 1, 2 and 4, repeatedly for multiple integrals if necessary, we obtain

 

Using fact 3 on   and  , we have

 

the convolution of   with an approximate identity. But since  , fact 5 says that

 

Putting together the above we have shown that

 

NotesEdit

  1. ^ An operator is a transformation that maps functions to functions. The flip operator, the Fourier transform, the inverse Fourier transform and the identity transform are all examples of operators.

ReferencesEdit

  • Folland, G. B. (1992). Fourier Analysis and its Applications. Belmont, CA, USA: Wadsworth. ISBN 0-534-17094-3.
  • Folland, G. B. (1995). Introduction to Partial Differential Equations (2nd ed.). Princeton, USA: Princeton Univ. Press. ISBN 978-0-691-04361-6.
  • ^ "DMat0101, Notes 3: The Fourier transform on L^1". I Woke Up In A Strange Place. 2011-03-10. Retrieved 2018-02-12.