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The problem of estimating the values of a [[Stochastic process|stochastic process]]  $  X ( t) $
on some interval  $  a < t < b $
using its observed values outside this interval. Usually one has in mind the interpolation estimator  $  \widehat{X}  ( t) $
for which the mean-square error of interpolation is minimal compared to all other estimators:

$$ 
{\mathsf E} | \widehat{X}  ( t) - X ( t) |  ^ {2}  =  \min ;
$$

the interpolation is called linear if one restricts attention to linear estimators. One of the first problems posed and solved was that of linear interpolation of the value  $  X ( 0) $
of a stationary sequence. This problem is analogous to the following one: In the space  $  L _ {2} $
of square-integrable functions on the interval  $  - \pi < \lambda \leq  \pi $, 
one must find the projection of  $  \phi ( \lambda ) \in L _ {2} $
onto the subspace generated by the functions  $  e ^ {i \lambda k } \phi ( \lambda ) $, 
$  k = \pm  1 , \pm  2 ,\dots $. 
This problem has been greatly generalized in the theory of stationary stochastic processes (cf. [[Stationary stochastic process|Stationary stochastic process]]; [[#References|[1]]], [[#References|[2]]]). One application is the problem of interpolation of the stochastic process arising from the system

$$ 
L X ( t)  =  Y ( t) ,\  t > t _ {0} ,
$$

where  $  L $
is a linear differential operator of order  $  l $, 
and  $  Y ( t) $, 
$  t > t _ {0} $, 
is a [[White noise|white noise]] process. For given initial values  $  X  ^ {(} k) ( t _ {0} ) $, 
$  k = 1 \dots l - 1 $, 
independent of the white noise, the optimal interpolation estimator  $  X ( t) $, 
$  a < t < b $, 
is the solution of the corresponding boundary value problem

$$ 
L  ^ {*} L \widehat{X}  ( t)  =  0 ,\ \ 
a < t < b ,
$$

where  $  L  ^ {*} $
denotes the formal adjoint operator,

$$ 
\widehat{X}  {}  ^ {(} k) ( s)  =  X  ^ {(} k) ( s) ,\ \ 
k = 0 \dots l ,
$$

with boundary conditions at the boundary points  $  s = a , b $. 
For systems of stochastic differential equations the problem of interpolation of some components given the values of other observed components reduces to similar interpolation equations. (See [[#References|[3]]].)

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   "Stationary sequences in Hilbert space"  ''Byull. Moskov. Gos. Univ. Sekt. A'' , '''2''' :  6  (1941)  pp. 1–40  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.A. Rozanov,   "Stationary stochastic processes" , Holden-Day  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.S. Liptser,   A.N. Shiryaev,   "Statistics of stochastic processes" , '''1–2''' , Springer  (1977–1978)  (Translated from Russian)</TD></TR></table>

====Comments====
The interpolation problem is usually defined as the estimation of an unobserved stochastic process on some time interval given a related stochastic process that is observed outside this time interval. One distinguishes two special cases: 1) linear least-squares interpolation, in which the estimator is constrained to be linear and minimizes a least-squares criterion, see [[#References|[a1]]], [[#References|[a3]]]; and 2) interpolation in which the conditional distribution of the estimator given the observations is determined, see [[#References|[a2]]].

For the Western literature on interpolation see [[#References|[a5]]], Sect. 5.3 and [[#References|[a1]]], Sect. 4.13. Additional Russian references that have been translated are [[#References|[a6]]]; [[#References|[a7]]], Sect. 37. For recent developments using stochastic realization theory see [[#References|[a3]]], [[#References|[a4]]]. Results for the interpolation problem may also be deduced from those for the smoothing problem [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Dym,   H.P. McKean,   "Gaussian processes, function theory, and the inverse spectral problem" , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Pardoux,   "Equations du filtrage nonlinéaire, de la prédiction et du lissage"  ''Stochastics'' , '''6'''  (1982)  pp. 193–231</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Pavon,   "New results on the interpolation problem for continuous-time stationary increment processes"  ''SIAM J. Control Optim.'' , '''22'''  (1984)  pp. 133–142</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Pavon,   "Optimal interpolation for linear stochastic systems"  ''SIAM J. Control Optim.'' , '''22'''  (1984)  pp. 618–629</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Wiener,   "Extrapolation, interpolation, and smoothing of stationary time series: with engineering applications" , M.I.T.  (1949)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.M. Yaglom,   "Extrapolation, interpolation and filtration of stationary random processes with rational spectral density"  ''Amer. Math. Soc. Sel. Transl. Math. Statist.'' , '''4'''  (1963)  pp. 345–387  ''Tr. Moskov. Mat. Obshch.'' , '''4'''  (1955)  pp. 333–374</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.M. Yaglom,   "An introduction to the theory of stationary random functions" , Prentice-Hall  (1962)  (Translated from Russian)</TD></TR></table>