The Fourier Integral Theorem

4.6.5 The Fourier Integral Theorem

We have seen how Fourier methods can be used on functions defined on a finite interval usually taken to be [−π, π]. Fourier found that expansion of an arbitrary function in a

Fourier series remains possible even if the function is defined on an interval that extends on both sides to infinity. In this case the fundamental frequency converges to zero and the summation process changes into one of integration.

Suppose that the function f (x) is defined on the entire real axis, and that it satisfies the regularity properties which we required in Theorem 4.6.2. Set

ϕ(ξ ) = f (x), ξ = 2πx/L ∈ [−π, π],

and continue ϕ(ξ) outside [−π, π] so that it has period 2π. By Theorem 4.6.2, if

1 L/ 2

f (x)e −2πixj/L dx, (4.6.32) 2π −π

ϕ(ξ )e −ijξ dξ =

L −L/2

then ϕ(ξ) = ij ξ

j =−∞ c j e , ξ ∈ (−π, π), and hence

f (x) =

c j e 2πixj/L , x ∈ − ,

f (x)e −2πixω dx, ω = , (4.6.33)

L then by (4.6.32) we have c j = (1/L)g L (ω) , and hence

−L/2

−L f (x) L

g L (ω)e 2πixω

Now by passing to the limit L → ∞, one avoids making an artificial periodic continuation outside a finite interval. The sum in (4.6.34) is a “sum of rectangles” similar to the sum

which appears in the definition of a definite integral. But here the argument varies from

4.6. Fourier Methods 495 −∞ to +∞, and the function g L (t ) depends on L. By a somewhat dubious passage to the

limit, then, the pair of formulas (4.6.33) and (4.6.34) becomes the pair

g(ω) =

f (x)e −2πixω dx ⇐⇒ f (x) = g(ω)e 2πixω dω. (4.6.35)

One can, in fact, after a rather complicated analysis, show that the above result is correct; see, e.g., Courant–Hilbert [83]. The proof requires, besides the previously mentioned “local” regularity conditions on f , the “global” assumption that

|f (x)| dx

is convergent. The beautiful, almost symmetric relation of (4.6.35) is called the Fourier integral theorem . This theorem, and other versions of it, with varying assumptions under which they are valid, is one of the most important aids in both pure and applied mathematics. The function g is called the Fourier transform 160 of f . The Fourier transform is one of the most important tools of applied analysis. It plays a fundamental role in problems relating to input–output relations, e.g., in electrical networks.

Clearly the Fourier transform is a linear operator. Another elementary property that can easily be verified is

f (ax) ⇐⇒

|a|

Whether the function f (x) has even or odd symmetry and is real or purely imaginary leads to relations between g(ω) and g(−ω) that can be used to increase computational

efficiency. Some of these properties are summarized in Table 4.6.1.

Table 4.6.1. Useful symmetry properties of the continuous Fourier transform.

Function

Fourier transform

f (x) real

g( −ω) = g(ω)

f (x) imaginary

g( −ω) = −g(ω)

f (x) even

g( −ω) = g(ω)

f (x) odd

g( −ω) = −g(ω)

f (x) real even

g(ω) real even

f (x) imaginary odd g(ω) real odd

Example 4.6.3.

The function f (x) = e −|x| has Fourier transform

g(ω) =

e −|x| e −2πixω dx =

e −(1+2πiω)x +e −(1−2πiω)x dx

The terminology in the literature varies somewhat as to the placement of the factor 2π; it can be taken out of the exponent by a simple change of variable.

496Chapter 4. Interpolation and Approximation Here f (x) is real and an even function. In agreement with Table 4.6.1, the Fourier transform

is also real and even. From (4.6.35) it follows that

It is not so easy to prove this formula directly. Many applications of the Fourier transform involve the use of convolutions.

Definition 4.6.5.

The convolution of f 1 and f 2 is the function

h(ξ ) = conv (f 1 ,f 2 ) =

f 1 (x)f 2 (ξ − x) dx. (4.6.36)

It is not difficult to verify that conv (f 1 ,f 2 ) = conv (f 2 ,f 1 ) . The following theorem states that the convolution of f 1 and f 2 can be computed as the inverse Fourier transform of the product g 1 (ω)g 2 (ω) . This fact is of great importance in the application of Fourier analysis to differential equations and probability theory.

Theorem 4.6.6.

Let f 1 and f 2 have Fourier transforms g 1 and g 2 , respectively. Then the Fourier transform g of the convolution of f 1 and f 2 is the product g(ω) = g 1 (ω)g 2 (ω).

Proof. By definition the Fourier transform of the convolution is

g(ω) =

e −2πiξω

f 1 (x)f 2 (ξ − x) dx dξ

e −2πi(x+ξ−x)ω f 1 (x)f 2 (ξ − x) dx dξ

e −2πixω f 1 (x) dx

e −2πi(ξ−x)ω f 2 (ξ − x) dξ

e −2πixω f 1 (ξ ) dx

e −2πixω f 2 (x) dx =g 1 (ω)g 2 (ω).

The legitimacy of changing the order of integration is here taken for granted. In many physical applications, the following relation, analogous to Parseval’s identity

(corollary to Theorem 4.5.14), is of great importance. If g is the Fourier transform of f , then

2 |g(ω)| 2 dω = |f (ξ)| dξ.

In signal processing this can be interpreted to mean that the total power in a signal is the same whether computed in the time domain or the frequency domain.

4.6. Fourier Methods 497