An interpolant in univariate interpolation is a function 
which agrees with a particular function 
at a set of
known points 
, 
,
, ..., 
and which is used to compute values for 
at points 
, 
, 1, 2, ..., 
.
Modulo a change of notation, the above definition translates verbatim to multivariate interpolation models as well.
Generally speaking, the properties required of the interpolant are the most fundamental designations between various interpolation models. For example, linear interpolation uses a piecewise linear function. This is a special case of spline interpolation, whose interpolants are piecewise polynomials with continuity conditions at their knots. A cubic spline with simple knots has continuous first and second derivatives, but need not be a C-infinity function.
