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''of a stationary stochastic process or of a homogeneous random field in  $  n $-
dimensional space''

The [[Fourier transform|Fourier transform]] of the covariance function of a stochastic process which is stationary in the wide sense (cf. [[Stationary stochastic process|Stationary stochastic process]]; [[Random field, homogeneous|Random field, homogeneous]]). Stationary stochastic processes and homogeneous random fields for which the Fourier transform of the covariance function exists are called processes with a spectral density.

Let

$$ 
X ( t) = \{ X _ {k} ( t) \} _ {k=} 1  ^ {n}
$$

be an  $  n $-
dimensional stationary stochastic process, and let

$$ 
X ( t)  =  \int\limits e ^ {i t \lambda } \Phi ( d \lambda ) ,\  \Phi  = \ 
\{ \Phi _ {k} \} _ {k=} 1  ^ {n}
$$

be its spectral representation ( $  \Phi _ {k} $
is the [[Spectral measure|spectral measure]] corresponding to the  $  k $-
th component  $  X _ {k} ( t) $
of the multi-dimensional stochastic process  $  X ( t) $). 
The range of integration is  $  - \pi \leq  \lambda \leq  \pi $
in the case of discrete time  $  t $, 
and  $  - \infty < \lambda < + \infty $
in the case of continuous time  $  t $. 
The process  $  X ( t) $
has a spectral density

$$ 
f ( \lambda )  =  \{ f _ {k,l} ( \lambda ) \} _ {k,l=} 1  ^ {n} ,
$$

if all the elements

$$ 
F _ {k,l} ( \Delta )  =  {\mathsf E}
\Phi _ {k} ( \Delta ) \overline{ {\Phi _ {l} ( \Delta ) }}\; ,\ \ 
k , l = {1 \dots n } ,
$$

of the spectral measure  $  F = \{ F _ {k,l} \} _ {k,l=} 1  ^ {n} $
are absolutely continuous and if

$$ 
f _ {k,l} ( \lambda )  = \ 

\frac{F _ {k,l} ( d \lambda ) }{d \lambda }
 .
$$

In particular, if the relation

$$ 
\sum _ {l = - \infty } ^  \infty  
| B _ {k,l} ( t) |  <  \infty ,\ \ 
k , l = {1 \dots n } ,
$$

holds for  $  X ( t) $, 
$  t = 0 , \pm  1 \dots $
where

$$ 
B ( t)  =  \{ B _ {k,l} ( t) \} _ {k,l=} 1  ^ {n}  = \ 
\{ {\mathsf E} X _ {k} ( t + s )
\overline{ {X _ {l} ( s) }}\; \} _ {k,l=} 1  ^ {n}
$$

is the covariance function of  $  X ( t) $, 
then  $  X ( t) $
has a spectral density and

$$ 
f _ {k,l} ( \lambda )  =  ( 2 \pi )  ^ {-} 1
\sum _ {t = - \infty } ^  \infty  
B _ {k,l} ( t)  \mathop{\rm exp} \{ - i \lambda t \} ,
$$

$$ 
- \infty  <  \lambda  <  \infty ,\  k , l  =  {1 \dots n } .
$$

The situation is similar in the case of processes  $  X ( t) $
in continuous time  $  t $. 
The spectral density  $  f ( \lambda ) $
is sometimes called the second-order spectral density, in contrast to higher spectral densities (see [[Spectral semi-invariant|Spectral semi-invariant]]).

A homogeneous  $  n $-
dimensional random field  $  X ( t _ {1} \dots t _ {n} ) $
has a spectral density  $  f ( \lambda _ {1} \dots \lambda _ {n} ) $
if its [[Spectral resolution|spectral resolution]]  $  F ( \lambda _ {1} \dots \lambda _ {n} ) $
possesses the property that its mixed derivative  $  \partial   ^ {n} F / \partial  \lambda _ {1} \dots \partial  \lambda _ {n} $
exists almost-everywhere, and then

$$ 
f ( \lambda _ {1} \dots \lambda _ {n} )  = \ 

\frac{\partial   ^ {n} F }{\partial  \lambda _ {1} \dots \partial  \lambda _ {n} }

$$

and

$$ 
F ( \lambda _ {1} \dots \lambda _ {n} )  = \ 
\int\limits _ { \lambda _ {01} } ^ { {\lambda _ 1 } } \dots
\int\limits _ { \lambda _ {0n} } ^ { {\lambda _ n } }
f ( \mu _ {1} \dots \mu _ {n} ) \ 
d \mu _ {1} \dots d \mu _ {n} + \textrm{ const } .
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,   Yu.A. Rozanov,   "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.A. Rozanov,   "Stationary random processes" , Holden-Day  (1967)  (Translated from Russian)</TD></TR></table>