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The problem of estimating the values of a [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s0902701.png" /> on some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s0902702.png" /> using its observed values outside this interval. Usually one has in mind the interpolation estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s0902703.png" /> for which the mean-square error of interpolation is minimal compared to all other estimators:

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s0902705.png" /> of a stationary sequence. This problem is analogous to the following one: In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s0902706.png" /> of square-integrable functions on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s0902707.png" />, one must find the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s0902708.png" /> onto the subspace generated by the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s0902709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027010.png" />. 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027012.png" /> is a linear differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027015.png" />, is a [[White noise|white noise]] process. For given initial values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027017.png" />, independent of the white noise, the optimal interpolation estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027019.png" />, is the solution of the corresponding boundary value problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027020.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027021.png" /> denotes the formal adjoint operator,

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with boundary conditions at the boundary points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090270/s09027023.png" />. 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>