Fourier integral

The non-discrete analogue of a Fourier series. The representation of a function given on a finite interval of the real axis by a Fourier series is very important. An analogous role is played by the representation of a function $ f $ given on the whole axis by a Fourier integral:

$$ \tag{1 } f ( x) = \ \int\limits _ { 0 } ^ \infty [ A ( \lambda ) \cos \lambda x + B ( \lambda ) \sin \lambda x] \ d \lambda , $$

where

$$ \tag{2 } A ( \lambda ) = { \frac{1} \pi } \int\limits _ {- \infty } ^ { {+ } \infty } f ( \xi ) \cos \lambda \xi \ d \xi , $$

$$ B ( \lambda ) = { \frac{1} \pi } \int\limits _ {- \infty } ^ { {+ } \infty } f ( \xi ) \sin \lambda \xi d \xi . $$

The representation (1) can be constructed formally by assuming the existence of the described integrals. It is valid, for example, for a smooth function $ f $ of compact support. There are a great number of tests guaranteeing equation (1) in some sense or other. The substitution of (2) into (1) gives the so-called Fourier integral formula

$$ \tag{3 } f ( x) = { \frac{1} \pi } \int\limits _ { 0 } ^ \infty \int\limits _ {- \infty } ^ { {+ } \infty } f ( \xi ) \cos \lambda ( x - \xi ) d \xi d \lambda , $$

whose proof reduces to the tests mentioned above. It can be of great advantage to use the representation of $ f $ by the simple Fourier integral

$$ f ( x) = \ \lim\limits _ {N \rightarrow \infty } \ { \frac{1} \pi } \int\limits _ {- \infty } ^ { {+ } \infty } f ( \xi ) \frac{\sin N ( x - \xi ) }{x - \xi } d \xi , $$

which is obtained from (3) by writing the inner integral as a limit over the interval $ ( 0, N) $ and changing the order of integration. In applied sciences the representation (1) is often interpreted as expansion into harmonics: If

$$ D ( \lambda ) = \ \sqrt {| A ( \lambda ) | ^ {2} + | B ( \lambda ) | ^ {2} } , $$

$$ \cos \phi ( \lambda ) = \frac{B ( \lambda ) }{D ( \lambda ) } ,\ \sin \phi ( \lambda ) = \frac{A ( \lambda ) }{D ( \lambda ) } , $$

then (1) takes the form:

$$ f ( x) = \ \int\limits _ { 0 } ^ \infty D ( \lambda ) \sin [ \lambda x + \phi ( \lambda )] \ d \lambda , $$

and thus $ f $ is represented by a superposition of harmonics with frequencies $ \lambda $ which continuously fill the real semi-axis $ ( 0, \infty ) $, while the amplitude $ D $ and the initial phase $ \phi $ depend on $ \lambda $.

In many cases (in particular for complex-valued functions $ f $) it is more convenient to present (1) in exponential form:

$$ \tag{4 } f ( x) = \ \int\limits _ {- \infty } ^ { {+ } \infty } C ( \lambda ) e ^ {i \lambda x } \ d \lambda ,\ \ C ( \lambda ) = \ { \frac{1}{\sqrt {2 \pi } } } \widetilde{f} ( \lambda ), $$

where

$$ \tag{5 } \widetilde{f} ( \lambda ) = \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \infty } ^ { {+ } \infty } f ( x) e ^ {- i \lambda x } dx. $$

The function $ \widetilde{f} $ is called the Fourier transform of $ f $( in applied sciences $ C ( \lambda ) $ is called the frequency characteristic or the spectrum of $ f $).

Under the condition that the function $ f $ is summable $ ( f \in L _ {1} (- \infty , + \infty )) $, the function $ \widetilde{f} $ is bounded, uniformly continuous on the real axis and $ \widetilde{f} ( \lambda ) \rightarrow 0 $ as $ | \lambda | \rightarrow \infty $. The function $ \widetilde{f} $ need not be integrable and so the integral (4) need not exist. However, (4) admits a reasonable interpretation if methods of summing integrals are used (here one can consider not only pointwise convergence but also convergence in the mean). For example, the arithmetical means of the truncated Fourier integrals

$$ { \frac{1}{N} } \int\limits _ { 0 } ^ { N } { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \omega } ^ { {+ } \omega } \widetilde{f} ( \lambda ) e ^ {i \lambda x } \ d \lambda d \omega \equiv $$

$$ \equiv \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ { - } N ^ { N } \left ( 1 - { \frac{| \lambda | }{N} } \right ) \widetilde{f} ( \lambda ) e ^ {i \lambda x } d \lambda \equiv $$

$$ \equiv \ { \frac{1} \pi } \int\limits _ {- \infty } ^ { {+ } \infty } f ( \xi ) \frac{2 \sin ^ {2} N ( x - \xi )/2 }{N ( x - \xi ) ^ {2} } d \xi $$

of a summable function $ f $ converge in the mean to $ f $ almost-everywhere as $ N \rightarrow \infty $. With additional restrictions on $ f $ one can obtain more specific assertions. For example, if $ f \in L _ {1} $ and has bounded variation in a neighbourhood of $ x $, then

$$ \tag{6 } { \frac{f ( x + 0) + f ( x - 0) }{2} } = \ \lim\limits _ {\omega \rightarrow \infty } \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \omega } ^ { {+ } \omega } \widetilde{f} ( \lambda ) e ^ {i \lambda x } \ d \lambda . $$

In applications one often uses the representation

$$ { \frac{f ( x + 0) + f ( x - 0) }{2} } = \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \infty } ^ { {+ } \infty } \widetilde{f} ( \lambda ) e ^ {i \lambda x } \ d \lambda , $$

which is true for an absolutely-integrable function $ f $ that is piecewise smooth in every finite interval, where the integral on the right-hand side is taken to mean the principal value (6). The Fourier integral is also studied when $ f $ is assumed to be locally summable and with some hypotheses imposing restrictions on the behaviour of $ f $ at $ \infty $. For example, let $ f \in L _ {p} $, $ 1 \leq p \leq 2 $, then

$$ \tag{7 } \widetilde{f} ( \lambda ) = \ \lim\limits _ {A \rightarrow \infty } {} ^ {p ^ \prime } \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ { - } A ^ { + } A f ( x) e ^ {- i \lambda x } dx, $$

where the limit is understood to be convergence in the mean of order $ p ^ \prime $, $ 1/p + 1/p ^ \prime = 1 $( however, the limit in (7) also exists in the sense of convergence almost-everywhere). This result acquires a simple form when $ p = 2 $( see Plancherel theorem).

The theory of multiple Fourier integrals is constructed analogously when one discusses the expansion of a function given on an $ n $- dimensional space. The concept of the Fourier integral has been extended also to generalized functions.

References

[1]	E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)
[2]	S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)
[3]	A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)