Extrapolation

of a function

An extension of the function beyond the boundary of its domain of definition, in which the extended (as a rule, analytic) function belongs to a given class. The extrapolation of functions is usually performed by means of formulas using information about the behaviour of the function at a finite collection of points (interpolation nodes) in its domain of definition.

The concept of interpolation of functions is used as the opposite concept to that of extrapolation (in a restricted sense of this term), and consists in constructively (possibly, approximately) re-establishing the values of functions in their domains of definition.

Example. If the values of a function $f : [a,b] \to \mathbf{R}$ at nodes $x_k \in [a,b]$, $k=0,\ldots,n$, are given, then the Lagrange interpolation polynomial $L_n(x)$ (see Lagrange interpolation formula), which is defined on the whole real axis $\mathbf{R}$, is, in particular, the extrapolation of $f$ outside $[a,b]$ in the class of polynomials of degree not exceeding $n$.

Sometimes extrapolation of a function does not use its entire domain of definition, but only a part of it, that is, the extrapolation constructed is, in fact, that of the values of the restriction of the given function to the part in question. In this case the extrapolation formulas yield, in particular, the values (generally speaking, approximate) of the function at the corresponding points in the domain of definition. This is often the method used in the solution of practical problems when the information required to compute the values of a function outside a certain part of its domain of definition is not sufficient.