Fourier inversion theorem

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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.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

Fourier transform mathematical transform that expresses a mathematical function of time as a function of frequency

The Fourier transform (FT) decomposes a function of time into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude (modulus) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.

Phase (waves) position of a point in time (an instant) on a waveform cycle

In physics and mathematics, the phase of a periodic function of some real variable is the relative value of that variable within the span of each full period.

Contents

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 converge in an ordinary sense.

An absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite.

Lebesgue integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.

Statement

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 integral

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 theorem

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 operator

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 inverse

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

In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f ). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : XY and g : YZ are composed to yield a function that maps x in X to g(f ) in Z.

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 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

for almost every [1]

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 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. In one dimension (and one dimension only), 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 distributions

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 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.

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 .

Applications

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. Commutative diagram illustrating problem solving via the Fourier transform.svg
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 transform

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

Proof

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

Notes

  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.

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References

  1. "DMat0101, Notes 3: The Fourier transform on L^1". I Woke Up In A Strange Place. 2011-03-10. Retrieved 2018-02-12.