In this chapter we review the de¬nitions and some properties of Fourier

transforms. We ¬rst treat one-dimensional non-periodic functions f (x) with

Fourier transform F (k), the domain of both coordinates x and k being the

set of real numbers, while the function values may be complex. The funct-

∞

ions f and F are piecewise continuous and ’∞ |f (x)| dx exists. The domain

of x is usually called the real space while the domain of k is called reciprocal

space. Such transforms are applicable to wave functions in quantum me-

chanics. In Section 12.6 we consider Fourier transforms for one-dimensional

periodic functions, leading to discrete transforms, i.e., Fourier series instead

of integrals. If the values in real space are also discrete, the computationally

e¬cient fast Fourier transform (FFT) results (Section 12.7). In Section 12.9

we consider the multidimensional periodic case, with special attention to tri-

clinic periodic 3D unit cells in real space, for which Fourier transforms are

useful when long-range forces are evaluated.

12.1 De¬nitions and properties

The relations between f (x) and its Fourier transform (FT) F (k) are

∞

1

f (x) = √ F (k) exp(ikx) dk, (12.1)

2π ’∞

∞

1

F (k) = √ f (x) exp(’ikx) dx. (12.2)

2π ’∞

√

The factors 1/ 2π are introduced for convenience in order to make the

transforms symmetric; one could use any arbitrary factors with product 2π.

The choice of sign in the exponentials is arbitrary and a matter of convention.

Note that the second equation follows from the ¬rst by using the de¬nition

315

316 Fourier transforms

of the δ-function:

∞

exp[±ikx] dk = 2πδ(x), (12.3)

’∞

and realizing that

∞

δ(x ’ x) f (x ) dx .

f (x) = (12.4)

’∞

The following relations are valid:

(i) if f (x) is real then F (’k) = F — (k)

(ii) if F (k) is real then f (’x) = f — (x);

(iii) if f (x) is real and f (’x) = f (x) then F (k) is real and F (’k) = F (k)

(cosine transform);

(iv) if f (x) is real and f (’x) = ’f (x) then F (k) is imaginary and

F (’k) = ’F (k) (sine transform);

def

(v) the FT of g(x) = f (x + x0 ) is G(k) = F (k) exp(ikx0 );

def

(vi) the FT of g(x) = f (x) exp(ik0 x) is G(k) = F (k ’ k0 );

def

(vii) the FT of g(x) = df (x)/dx is G(k) = ikF (k);

def

(viii) the FT of g(x) = xf (x) is G(k) = ’i dF (k)/dk.

12.2 Convolution and autocorrelation

The convolution h(x) of two functions f (x) and g(x) is de¬ned as

∞

f — (ξ ’ x)g(ξ) dξ

def

h(x) = (12.5)

’∞

∞

f — (ξ)g(x + ξ) dξ,

= (12.6)

’∞

with short notation h = f — g. Its Fourier transform is

√

H(k) = 2πF — (k)G(k), (12.7)

and hence

∞

F — (k)G(k) exp(ikx) dk.

h(x) = (12.8)

’∞

√

∞

def

If h(x) = ’∞ f (ξ ’ x)g(ξ) dξ then H(k) = 2πF (k)G(k).

A special case is

∞ ∞

—

F — (k)G(k) dk.

h(0) = f (x)g(x) dx = (12.9)

’∞ ’∞

12.3 Operators 317

The autocorrelation function is a self-convolution:

∞

f — (ξ ’ x)f (ξ) dξ,

def

h(x) = (12.10)

’∞

√

2πF — (k)F (k),

H(k) = (12.11)

∞

F — (k)F (k) exp(ikx) dk,

h(x) = (12.12)

’∞

∞ ∞

—

F — (k)F (k) dk.

h(0) = f (x)f (x) dx = (12.13)

’∞ ’∞

Equation (12.13)is known as Parseval™s theorem. It implies that, if the

∞

function f (x) is normalized in the sense that ’∞ f — f dx = 1, then its Fourier

transform is, in the same sense in k-space, also normalized.

We note that the de¬nitions given here for square-integrable functions

di¬er from the autocorrelation and spectral density functions for in¬nite

time series discussed in Section 12.8 (page 325).

12.3 Operators

When the function f — f (x) is interpreted as a probability density, the expec-

tation of some function of x (indicated by triangular brackets) is the average

of that function over the probability density:

∞

h(x)f — f (x) dx.

def

h(x) = (12.14)

’∞

Functions of k are similarly de¬ned by averages over the probability density

F — F (k) in k-space:

∞

h(k)F — F (k) dk.

def

h(k) = (12.15)

’∞

It can be shown that for polynomials of k the average can also be obtained

in x-space by

∞