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 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 operatorAn 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. 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 . 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 :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 allIntegrable 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 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 . 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 : forSquare 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 forTempered 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
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
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 isProof
The proof uses a few facts, given and . # If and , then . # If and , then . # For , Fubini's theorem implies that . # Define ; then . # Define . Then with denotingNotes
References
* * {{cite book, last=Folland, first=G. B., authorlink=Gerald Folland, year=1995, title=Introduction to Partial Differential Equations, edition=2nd, publisher=Princeton Univ. Press, location=Princeton, USA, isbn=978-0-691-04361-6 Generalized functions Theorems in Fourier analysis