# Interpolation process

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A process for obtaining a sequence of interpolation functions for an indefinitely-growing number of interpolation conditions. If the interpolation functions are represented by the partial sums of some series of functions, the series is sometimes called an interpolation series. The aim of an interpolation process often is, at least in the simplest basic problems of interpolating, the approximation (in some sense) by means of interpolation functions of an initial function about which one only has either incomplete information or whose form is too complicated to deal with directly.

A sufficiently general situation related to constructing interpolation processes is described in what follows. Let , , be an infinite triangular table of arbitrary but fixed complex numbers:

 (1)

called interpolation nodes or interpolation knots. Suppose that next to (1) there is an analogous table , , also consisting of arbitrary fixed complex numbers.

If the -th row , , of (1) consists of different numbers, or, otherwise said, if this row consists of simple nodes, then, using e.g. the Lagrange interpolation formula, one constructs the (unique) algebraic interpolation polynomial of degree at most satisfying the simple interpolation condition

 (2)

If, on the other hand, the point is a multiple node of multiplicity in the -th row, i.e. if it is encountered times in the -th row: , then the corresponding multiple interpolation condition at has the form

 (3)

In the general case in the presence of multiple nodes the (unique) algebraic interpolation polynomial of degree at most is constructed using, e.g., the Hermite interpolation formula. As an example, the system (1) may consist of the systems of , equally-spaced nodes on the unit circle. This situation is so-called interpolation at roots of unity (cf. [5]).

As a result of the interpolation process described one obtains a sequence of interpolation polynomials defined by the tables and . The main questions that arise here are: to determine the set of points of convergence of the sequence , at which exists, in dependence on and ; to determine the character of the limit function ; to determine the set of uniform convergence ; etc.

In the theory of functions of a complex variable the case where the table is constructed from the values of a regular analytic function and its derivatives at the interpolation nodes, such that (applied to a node of multiplicity ; cf. (3))

has been well-studied. In this case the interpolation polynomial can be written, by Hermite's formula, as a contour integral over a contour encircling the nodes , , on and inside which is regular:

 (4)

where

Formula (4) easily implies an integral representation for the remainder term of interpolation . Generally speaking, the sequence constructed from may diverge. If, however, it converges, then the limit function need not coincide with . The fundamental question is the study of the convergence of to , and the determination of those systems of nodes for which this convergence is optimal in a certain sense. Suppose, e.g., that is a regular function on a continuum containing at least two points and whose complement in the extended complex plane is a simply-connected domain containing the point at infinity. Let the nodes belong to . Then converges uniformly to on if and only if

where , and is the capacity of (cf. [4]).

The classical variant of an interpolation process is obtained if the , , form a sequence for which at the -th step the -th nodes are used to construct . For a regular function the polynomials are in this case the partial sums of the Newton interpolation series (cf. also Newton interpolation formula)

 (5)

In calculations an interpolation series of the form (5) has the advantage over the sequence that in the transition from the known polynomial to only one coefficient of the series has to be computed. Depending on the nodes and coefficients , the domain of convergence of (5) can be any simply-connected domain in with analytic boundary. In particular, if has only a limit point at infinity, if , and if (5) converges at at least one point , then (5) converges uniformly in any disc and, hence, its sum is an entire function. Stirling's interpolation series is a particular case of Newton's, for the sequence of nodes , , , . Other, similar, interpolation series have been investigated (cf. [3], [5]).

Interpolation processes with non-algebraic interpolation polynomials , constructed in systems of functions other than , e.g. in , are also an object of study (cf. [4], [6]).

The study of interpolation processes in the real domain has its own specifics, both in the formulation of problems as in the results (cf. [2], [4]). These specifics, first of all, are brought about by the natural (in the real domain) requirement of regularity of the function to be interpolated. It is known, e.g., that there is no system of nodes on that would guarantee the convergence of the interpolation processes for arbitrary continuous functions , . On the other hand, if a continuous function is given in advance, it is always possible to choose a system of nodes such that the interpolation process converges to .

Besides interpolation processes with polynomials , interpolation processes with rational functions , e.g. of the form , where and is a polynomial of degree at most , have drawn the attention of researchers. The interpolation conditions (1)–(3) remain in force, but conditions at the poles , which in the simple case are given by a triangular table , , similar to (1) must be given.

#### References

 [1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) [2] V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian) [3] A.O. [A.O. Gel'fond] Gelfond, "Differenzenrechnung" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) [4] V.I. Smirnov, A.N. Lebedev, "Functions of a complex variable" , Scripta Techn. (1968) (Translated from Russian) [5] J.L. Walsh, "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc. (1965) [6] A.F. Leont'ev, "Exponential series" , Moscow (1976) (In Russian)