Fourier inversion theorem
Encyclopedia
In mathematics
, Fourier inversion recovers a function
from its Fourier transform
. Several different Fourier inversion theorems exist.
Sometimes the following expression is used as the definition of the Fourier transform:
Then it is asserted that
In this way, one recovers a function from its Fourier transform.
However, this way of stating a Fourier inversion theorem sweeps some more subtle issues under the carpet. One Fourier inversion theorem assumes that f is Lebesgue-integrable, i.e., the integral of its absolute value
is finite:
In that case, the Fourier transform is not necessarily Lebesgue-integrable. For example, the function f(x) = 1 if −a < x < a and f(x) = 0 otherwise has Fourier transform
In such a case, Fourier inversion theorems usually investigate the convergence of the integral
By contrast, if we take f to be a tempered distribution
-- a type of generalized function -- then its Fourier transform is another tempered distribution; and the Fourier inversion formula is then more simple to prove.
such that, for any multi-indices 
and 
,
These functions are clearly seen to be absolutely integrable, and the Fourier transform of a Schwarz function is also a Schwartz function. An example is the Gaussian function
, which we will actually use in proving the inversion formula. We will use the convention that 
, and the claim is that for a Schwartz function 
,
To do this, we will need a few facts.
We can now prove the inversion formula. First, note that by the dominated convergence theorem
Define
. Applying the second and then third fact from above, 
With 
as before, we can push the Fourier transform onto 
in the last integral to get
the convolution of ƒ with an approximate identity. Hence by the last fact
This establishes that the Fourier transform is an invertible map of the Schwartz space to itself. In particular, it is an isometry in the
norm, and Schwartz functions are dense in 
. The Fourier transform and its inverse then extend to unitary operators 
on all of 
for which 
, with 
the identity map.
While the integral defining the Fourier transform or its inverse may not make sense for general
functions, one can always integrate over a symmetric rectangle and take the limits as its length tends to infinity. What one is really doing here is taking an increasing sequence 
of relatively compact sets growing to 
, and taking the limit of 
, where 
denotes the indicator function of a set. Since 
is compactly supported, the integral defining its Fourier transform exists. But clearly 
in 
, hence 
as well.
, one can also define the Fourier transform of a square-integrable function, i.e., one satisfying
Then the Fourier transform is another quadratically integrable function.
In case f is a square-integrable periodic function on the interval
,
it has a Fourier series
whose coefficients are
The Fourier inversion theorem might then say that
What kind of convergence is right? "Convergence in mean square" can be proved fairly easily:
What about convergence almost everywhere
? That would say that if f is square-integrable, then for "almost every" value of x between 0 and 2π we have
This was not proved until 1966 in (Carleson, 1966).
For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Fourier inversion recovers a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
from its Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
. Several different Fourier inversion theorems exist.
Sometimes the following expression is used as the definition of the Fourier transform:
Then it is asserted that
In this way, one recovers a function from its Fourier transform.
However, this way of stating a Fourier inversion theorem sweeps some more subtle issues under the carpet. One Fourier inversion theorem assumes that f is Lebesgue-integrable, i.e., the integral of its absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
is finite:
In that case, the Fourier transform is not necessarily Lebesgue-integrable. For example, the function f(x) = 1 if −a < x < a and f(x) = 0 otherwise has Fourier transform
In such a case, Fourier inversion theorems usually investigate the convergence of the integral
By contrast, if we take f to be a tempered distribution
Tempered distribution
*Distribution *Tempered representation...
-- a type of generalized function -- then its Fourier transform is another tempered distribution; and the Fourier inversion formula is then more simple to prove.
Proof of the inversion theorem
First we will consider Fourier transforms of functions in the Schwartz space; these are smooth functions such that, for any multi-indices and ,These functions are clearly seen to be absolutely integrable, and the Fourier transform of a Schwarz function is also a Schwartz function. An example is the Gaussian function
, which we will actually use in proving the inversion formula. We will use the convention that , and the claim is that for a Schwartz function ,To do this, we will need a few facts.
- For
and 
Schwartz functions, Fubini's theorem implies that 
. - If
and 
, then 
. - If
and 
, then 
- Define
; then 
- Set
. Then with 
denoting convolutionConvolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
,
is an approximation to the identityApproximation to the identityIn mathematics, an approximation to the identity refers to a sequence or net that converges to the identity in some algebra. Specifically, it can mean:* Nascent delta function, most commonly* Mollifier, more narrowly* Approximate identity, more abstractly...
:
, where the convergence is uniform on bounded sets for 
bounded and uniformly continuous and the convergence is in the p-norm for 
.
We can now prove the inversion formula. First, note that by the dominated convergence theorem
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
Define
. Applying the second and then third fact from above, With as before, we can push the Fourier transform onto in the last integral to getthe convolution of ƒ with an approximate identity. Hence by the last fact
This establishes that the Fourier transform is an invertible map of the Schwartz space to itself. In particular, it is an isometry in the
norm, and Schwartz functions are dense in . The Fourier transform and its inverse then extend to unitary operators on all of for which , with the identity map.While the integral defining the Fourier transform or its inverse may not make sense for general
functions, one can always integrate over a symmetric rectangle and take the limits as its length tends to infinity. What one is really doing here is taking an increasing sequence of relatively compact sets growing to , and taking the limit of , where denotes the indicator function of a set. Since is compactly supported, the integral defining its Fourier transform exists. But clearly in , hence as well.Fourier transforms of square-integrable functions
Via the Plancherel theoremPlancherel theorem
In mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....
, one can also define the Fourier transform of a square-integrable function, i.e., one satisfying
Then the Fourier transform is another quadratically integrable function.
In case f is a square-integrable periodic function on the interval
,it has a Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
whose coefficients are
The Fourier inversion theorem might then say that
What kind of convergence is right? "Convergence in mean square" can be proved fairly easily:
What about convergence almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
? That would say that if f is square-integrable, then for "almost every" value of x between 0 and 2π we have
This was not proved until 1966 in (Carleson, 1966).
For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.