Interpolation process

A process for obtaining a sequence of interpolation functions $ \{ f _ {n} ( z) \} $ for an indefinitely-growing number $ n $ of interpolation conditions. If the interpolation functions $ f _ {n} ( z) $ 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 $ f _ {n} ( z) $ of an initial function $ f ( z) $ 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 $ ( a _ {jk} ) $, $ 0 \leq k \leq j $, $ j = 0 , 1 \dots $ be an infinite triangular table of arbitrary but fixed complex numbers:

$$ \tag{1 } \begin{array}{llll} a _ {00} &{} &{} &{} \\ a _ {10} &a _ {11} &{} &{} \\ \dots &\dots &\dots &{} \\ a _ {n0} &a _ {n1} &\dots &a _ {nn} \\ \dots &\dots &\dots &\dots , \\ \end{array} $$

called interpolation nodes or interpolation knots. Suppose that next to (1) there is an analogous table $ ( w _ {jk} ) $, $ 0 \leq k \leq j $, $ j= 0 , 1 \dots $ also consisting of arbitrary fixed complex numbers.

If the $ n $- th row $ a _ {nk} $, $ k = 0 \dots n $, 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 $ p _ {n} ( z) $ of degree at most $ n $ satisfying the simple interpolation condition

$$ \tag{2 } p _ {n} ( a _ {nk} ) = \ w _ {nk} ,\ k = 0 \dots n . $$

If, on the other hand, the point $ a _ {n0} $ is a multiple node of multiplicity $ \nu _ {0} > 1 $ in the $ n $- th row, i.e. if it is encountered $ \nu _ {0} $ times in the $ n $- th row: $ a _ {n0} = a _ {nk _ {1} } = {} \dots = a _ {nk _ {\nu _ {0} - 1 } } $, then the corresponding multiple interpolation condition at $ a _ {n0} $ has the form

$$ \tag{3 } p _ {n} ( a _ {n0} ) = \ p _ {n} ^ {(} 0) ( a _ {n0} ) = w _ {n0} , $$

$$ p _ {n} ^ \prime ( a _ {n0} ) = w _ {nk _ {1} } \dots p _ {n} ^ {( \nu _ {0} - 1 ) } ( a _ {n0} ) = w _ {nk _ {\nu _ {0} - 1 } } . $$

In the general case in the presence of multiple nodes the (unique) algebraic interpolation polynomial $ p _ {n} ( z) $ of degree at most $ n $ is constructed using, e.g., the Hermite interpolation formula. As an example, the system (1) may consist of the systems of $ j + 1 $, $ j = 0 , 1 \dots $ equally-spaced nodes $ a _ {jk} = e ^ {2 \pi i k / ( j+ 1 ) } $ 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 $ \{ p _ {n} ( z) \} $ defined by the tables $ ( a _ {jk} ) $ and $ ( w _ {jk} ) $. The main questions that arise here are: to determine the set $ E \subset \mathbf C $ of points of convergence of the sequence $ \{ p _ {n} ( z) \} $, at which $ \lim\limits _ {n \rightarrow \infty } p _ {n} ( z) = g ( z) $ exists, in dependence on $ ( a _ {jk} ) $ and $ ( w _ {jk} ) $; to determine the character of the limit function $ g ( z) $; to determine the set $ F \subset E $ of uniform convergence $ p _ {n} ( x) \rightarrow g ( z) $; etc.

In the theory of functions of a complex variable the case where the table $ ( w _ {jk} ) $ is constructed from the values of a regular analytic function $ f ( z) $ and its derivatives at the interpolation nodes, such that (applied to a node $ a _ {n0} $ of multiplicity $ \nu _ {0} \geq 1 $; cf. (3))

$$ p _ {n} ( a _ {n0} ) = p _ {n} ^ {(} 0) ( a _ {n0} ) = \ f ( a _ {n0} ) , $$

$$ p _ {n} ^ \prime ( a _ {n0} ) = f ^ { \prime } ( a _ {n0} ) \dots p _ {n} ^ {( \nu _ {0} - 1 ) } ( a _ {n0} ) = \ f ^ { ( \nu _ {0} - 1 ) } ( a _ {n0} ) , $$

has been well-studied. In this case the interpolation polynomial $ p _ {n} ( z) $ can be written, by Hermite's formula, as a contour integral over a contour $ \Gamma $ encircling the nodes $ a _ {nk} $, $ k = 0 \dots n $, on and inside which $ f ( z) $ is regular:

$$ \tag{4 } p _ {n} ( z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\omega _ {n} ( t) - \omega _ {n} ( z) }{\omega _ {n} ( t) ( t - z ) } f ( t) d t , $$

where

$$ \omega _ {n} ( z) = ( z - a _ {n0} ) \dots ( z - a _ {nn} ) . $$

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

$$ \lim\limits _ {n \rightarrow \infty } \ M _ {n} ^ {1 / ( n+ 1) } = c , $$

where $ M _ {n} = \sup \{ {\omega _ {n} ( z) } : {z \in \partial K } \} $, and $ c $ is the capacity of $ K $( cf. [4]).

The classical variant of an interpolation process is obtained if the $ a _ {jk} = a _ {k} $, $ 0 \leq k \leq j $, $ j= 0 , 1 \dots $ form a sequence $ \{ a _ {jk} \} $ for which at the $ n $- th step the $ n $- th nodes $ a _ {0} \dots a _ {n} $ are used to construct $ p _ {n} ( z) $. For a regular function $ f ( z) $ the polynomials $ p _ {n} ( z) $ are in this case the partial sums of the Newton interpolation series (cf. also Newton interpolation formula)

$$ \tag{5 } q ( z) = \sum_{n=0}^\infty c _ {n} \omega _ {n} ( z) , $$

$$ \omega _ {n} ( z) = ( z - a _ {0} ) \dots ( z - a _ {n} ) , $$

$$ c _ {n} = \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{f ( t) d t }{\omega _ {n} ( t) } . $$

In calculations an interpolation series of the form (5) has the advantage over the sequence $ \{ p _ {n} ( z) \} $ that in the transition from the known polynomial $ p _ {n} ( z) $ to $ p _ {n+} 1 ( z) $ only one coefficient $ c _ {n+} 1 $ of the series has to be computed. Depending on the nodes $ a _ {k} $ and coefficients $ c _ {k} $, the domain of convergence of (5) can be any simply-connected domain in $ \mathbf C $ with analytic boundary. In particular, if $ \{ a _ {k} \} $ has only a limit point at infinity, if $ \sum _ {k=m} ^ \infty 1 / | a _ {k} | < \infty $, and if (5) converges at at least one point $ z _ {0} \neq a _ {k} $, $ k = 0 , 1 \dots $ then (5) converges uniformly in any disc $ | z | \leq R $ and, hence, its sum $ q ( z) $ is an entire function. Stirling's interpolation series is a particular case of Newton's, for the sequence of nodes $ a _ {0} = 0 $, $ a _ {1} = - 1 $, $ a _ {2} = 1 \dots a _ {2k-1} = - k $, $ a _ {2k} = k ,\dots $. Other, similar, interpolation series have been investigated (cf. [3], [5]).

Interpolation processes with non-algebraic interpolation polynomials $ p _ {n} ( z) = \sum _ {\nu = 0 } ^ {n} b _ \nu \phi _ \nu ( z) $, constructed in systems of functions $ \{ \phi _ \nu ( z) \} $ other than $ \{ z ^ \nu \} $, e.g. in $ \{ e ^ \alpha \nu ^ {z} \} $, 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 $ f ( z) $ to be interpolated. It is known, e.g., that there is no system of nodes on $ [ a , b ] $ that would guarantee the convergence of the interpolation processes for arbitrary continuous functions $ f ( x) $, $ x \in [ a , b ] $. On the other hand, if a continuous function $ f ( x) $ is given in advance, it is always possible to choose a system of nodes such that the interpolation process converges to $ f ( x) $.

Besides interpolation processes with polynomials $ p _ {n} ( z) $, interpolation processes with rational functions $ r _ {n} ( z) $, e.g. of the form $ r _ {n} ( z) = q _ {n} ( z) / \omega _ {n-1} ( z) $, where $ \omega _ {n-1} ( z) = ( z - b _ {0} ) \dots ( z - b _ {n-1} ) $ and $ q _ {n} ( z) $ is a polynomial of degree at most $ n $, have drawn the attention of researchers. The interpolation conditions (1)–(3) remain in force, but conditions at the poles $ b _ {k} $, $ k = 0 , 1 \dots $ which in the simple case are given by a triangular table $ ( b _ {jk} ) $, $ 0 \leq k \leq j $, $ j = 0 , 1 \dots $ similar to (1) must be given.

See also Abel–Goncharov problem; Bernstein interpolation method.

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)

Comments

A very general interpolation scheme is Birkhoff interpolation, cf. Hermite interpolation formula; Interpolation formula, and [a5]. See also the various articles on approximation of functions.

For interpolation with rational functions see also Padé approximation.

Good references for interpolation (and approximation) in the complex domain are [a3], [a4]. See also Interpolation.

References

[a1]	J.F. Steffenson, "Interpolation" , Chelsea, reprint (1950)
[a2]	P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126
[a3]	D. Gaier, "Lectures on complex approximation" , Birkhäuser (1987) (Translated from German)
[a4]	J.-B. Garnett, "Bounded analytic functions" , Acad. Press (1981)
[a5]	G.G. Lorentz, K. Jetter, S.D. Riemenschneider, "Birkhoff interpolation" , Wiley (1983)