Namespaces
Variants
Actions

Difference between revisions of "Interpolation process"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A process for obtaining a sequence of interpolation functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i0519701.png" /> for an indefinitely-growing number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i0519702.png" /> of [[Interpolation|interpolation]] conditions. If the interpolation functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i0519703.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i0519704.png" /> of an initial function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i0519705.png" /> about which one only has either incomplete information or whose form is too complicated to deal with directly.
+
<!--
 +
i0519701.png
 +
$#A+1 = 127 n = 0
 +
$#C+1 = 127 : ~/encyclopedia/old_files/data/I051/I.0501970 Interpolation process
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A sufficiently general situation related to constructing interpolation processes is described in what follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i0519706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i0519707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i0519708.png" /> be an infinite triangular table of arbitrary but fixed complex numbers:
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/i/i051/i051970/i0519709.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A process for obtaining a sequence of interpolation functions  $  \{ f _ {n} ( z) \} $
 +
for an indefinitely-growing number  $  n $
 +
of [[Interpolation|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.
  
called interpolation nodes or interpolation knots. Suppose that next to (1) there is an analogous table <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197012.png" /> also consisting of arbitrary fixed complex numbers.
+
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:
  
If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197013.png" />-th row <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197015.png" />, of (1) consists of different numbers, or, otherwise said, if this row consists of simple nodes, then, using e.g. the [[Lagrange interpolation formula|Lagrange interpolation formula]], one constructs the (unique) algebraic interpolation polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197016.png" /> of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197017.png" /> satisfying the simple interpolation condition
+
$$ \tag{1 }
  
<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/i/i051/i051970/i05197018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\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}
  
If, on the other hand, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197019.png" /> is a multiple node of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197020.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197021.png" />-th row, i.e. if it is encountered <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197022.png" /> times in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197023.png" />-th row: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197024.png" />, then the corresponding multiple interpolation condition at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197025.png" /> has the form
+
$$
  
<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/i/i051/i051970/i05197026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
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.
  
<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/i/i051/i051970/i05197027.png" /></td> </tr></table>
+
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|Lagrange interpolation formula]], one constructs the (unique) algebraic interpolation polynomial  $  p _ {n} ( z) $
 +
of degree at most  $  n $
 +
satisfying the simple interpolation condition
  
In the general case in the presence of multiple nodes the (unique) algebraic interpolation polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197028.png" /> of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197029.png" /> is constructed using, e.g., the [[Hermite interpolation formula|Hermite interpolation formula]]. As an example, the system (1) may consist of the systems of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197031.png" /> equally-spaced nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197032.png" /> on the unit circle. This situation is so-called interpolation at roots of unity (cf. [[#References|[5]]]).
+
$$ \tag{2 }
 +
p _ {n} ( a _ {nk} ) = \
 +
w _ {nk} ,\  k = 0 \dots n .
 +
$$
  
As a result of the interpolation process described one obtains a sequence of interpolation polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197033.png" /> defined by the tables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197035.png" />. The main questions that arise here are: to determine the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197036.png" /> of points of convergence of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197037.png" />, at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197038.png" /> exists, in dependence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197040.png" />; to determine the character of the limit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197041.png" />; to determine the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197042.png" /> of uniform convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197043.png" />; etc.
+
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
  
In the theory of functions of a complex variable the case where the table <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197044.png" /> is constructed from the values of a regular analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197045.png" /> and its derivatives at the interpolation nodes, such that (applied to a node <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197046.png" /> of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197047.png" />; cf. (3))
+
$$ \tag{3 }
 +
p _ {n} ( a _ {n0} )  = \
 +
p _ {n}  ^ {(} 0)
 +
( a _ {n0} )  = w _ {n0} ,
 +
$$
  
<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/i/i051/i051970/i05197048.png" /></td> </tr></table>
+
$$
 +
p _ {n}  ^  \prime  ( a _ {n0} )  = w _ {nk _ {1}  } \dots p _ {n} ^ {( \nu _ {0} - 1 ) } ( a _ {n0} )  = w _ {nk _ {\nu _ {0}  - 1 } } .
 +
$$
  
<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/i/i051/i051970/i05197049.png" /></td> </tr></table>
+
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|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. [[#References|[5]]]).
  
has been well-studied. In this case the interpolation polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197050.png" /> can be written, by Hermite's formula, as a contour integral over a contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197051.png" /> encircling the nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197053.png" />, on and inside which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197054.png" /> is regular:
+
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.
  
<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/i/i051/i051970/i05197055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
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
 
where
  
<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/i/i051/i051970/i05197056.png" /></td> </tr></table>
+
$$
 +
\omega _ {n} ( z)  = ( z - a _ {n0} ) \dots ( z - a _ {nn} ) .
 +
$$
  
Formula (4) easily implies an integral representation for the remainder term of interpolation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197057.png" />. Generally speaking, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197058.png" /> constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197059.png" /> may diverge. If, however, it converges, then the limit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197060.png" /> need not coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197061.png" />. The fundamental question is the study of the convergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197062.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197063.png" />, and the determination of those systems of nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197064.png" /> for which this convergence is optimal in a certain sense. Suppose, e.g., that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197065.png" /> is a regular function on a continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197066.png" /> containing at least two points and whose complement in the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197067.png" /> is a simply-connected domain containing the point at infinity. Let the nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197068.png" /> belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197069.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197070.png" /> converges uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197071.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197072.png" /> if and only if
+
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
  
<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/i/i051/i051970/i05197073.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty } \
 +
M _ {n} ^ {1 / ( n+ 1) }  = c ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197074.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197075.png" /> is the [[Capacity|capacity]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197076.png" /> (cf. [[#References|[4]]]).
+
where $  M _ {n} = \sup \{ {\omega _ {n} ( z) } : {z \in \partial  K } \} $,  
 +
and $  c $
 +
is the [[Capacity|capacity]] of $  K $(
 +
cf. [[#References|[4]]]).
  
The classical variant of an interpolation process is obtained if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197079.png" /> form a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197080.png" /> for which at the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197081.png" />-th step the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197082.png" />-th nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197083.png" /> are used to construct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197084.png" />. For a regular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197085.png" /> the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197086.png" /> are in this case the partial sums of the Newton interpolation series (cf. also [[Newton interpolation formula|Newton interpolation formula]])
+
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|Newton interpolation formula]])
  
<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/i/i051/i051970/i05197087.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
q ( z)  = \sum _ { n= } 0 ^  \infty  c _ {n} \omega _ {n} ( z) ,
 +
$$
  
<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/i/i051/i051970/i05197088.png" /></td> </tr></table>
+
$$
 +
\omega _ {n} ( z)  = ( z - a _ {0} ) \dots ( z - a _ {n} ) ,
 +
$$
  
<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/i/i051/i051970/i05197089.png" /></td> </tr></table>
+
$$
 +
c _ {n}  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma 
 +
\frac{f ( t)  d t }{\omega _ {n} ( t) }
 +
.
 +
$$
  
In calculations an interpolation series of the form (5) has the advantage over the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197090.png" /> that in the transition from the known polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197091.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197092.png" /> only one coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197093.png" /> of the series has to be computed. Depending on the nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197094.png" /> and coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197095.png" />, the domain of convergence of (5) can be any simply-connected domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197096.png" /> with analytic boundary. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197097.png" /> has only a limit point at infinity, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197098.png" />, and if (5) converges at at least one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i05197099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970100.png" /> then (5) converges uniformly in any disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970101.png" /> and, hence, its sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970102.png" /> is an entire function. Stirling's interpolation series is a particular case of Newton's, for the sequence of nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970106.png" />. Other, similar, interpolation series have been investigated (cf. [[#References|[3]]], [[#References|[5]]]).
+
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. [[#References|[3]]], [[#References|[5]]]).
  
Interpolation processes with non-algebraic interpolation polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970107.png" />, constructed in systems of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970108.png" /> other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970109.png" />, e.g. in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970110.png" />, are also an object of study (cf. [[#References|[4]]], [[#References|[6]]]).
+
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. [[#References|[4]]], [[#References|[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. [[#References|[2]]], [[#References|[4]]]). These specifics, first of all, are brought about by the natural (in the real domain) requirement of regularity of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970111.png" /> to be interpolated. It is known, e.g., that there is no system of nodes on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970112.png" /> that would guarantee the convergence of the interpolation processes for arbitrary continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970114.png" />. On the other hand, if a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970115.png" /> is given in advance, it is always possible to choose a system of nodes such that the interpolation process converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970116.png" />.
+
The study of interpolation processes in the real domain has its own specifics, both in the formulation of problems as in the results (cf. [[#References|[2]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970117.png" />, interpolation processes with rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970118.png" />, e.g. of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970119.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970121.png" /> is a polynomial of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970122.png" />, have drawn the attention of researchers. The interpolation conditions (1)–(3) remain in force, but conditions at the poles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970124.png" /> which in the simple case are given by a triangular table <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051970/i051970127.png" /> similar to (1) must be given.
+
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|Abel–Goncharov problem]]; [[Bernstein interpolation method|Bernstein interpolation method]].
 
See also [[Abel–Goncharov problem|Abel–Goncharov problem]]; [[Bernstein interpolation method|Bernstein interpolation method]].
Line 61: Line 238:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.L. Goncharov,  "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.O. [A.O. Gel'fond] Gelfond,  "Differenzenrechnung" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Smirnov,  A.N. Lebedev,  "Functions of a complex variable" , Scripta Techn.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.L. Walsh,  "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc.  (1965)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.F. Leont'ev,  "Exponential series" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.L. Goncharov,  "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.O. [A.O. Gel'fond] Gelfond,  "Differenzenrechnung" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Smirnov,  A.N. Lebedev,  "Functions of a complex variable" , Scripta Techn.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.L. Walsh,  "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc.  (1965)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.F. Leont'ev,  "Exponential series" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 22:13, 5 June 2020


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)
How to Cite This Entry:
Interpolation process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interpolation_process&oldid=12630
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article