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The problem of extending a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c0203201.png" /> to a power series representing a regular function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c0203202.png" /> which realizes the least value of the supremum of the modulus on the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c0203203.png" /> in the class of all regular functions in the unit disc having the given polynomial as initial segment of the MacLaurin series. The solution to this problem is given by the following theorem.
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The problem of extending a polynomial in  $  z $
 +
to a power series representing a regular function in the disc $  | z | < 1 $
 +
which realizes the least value of the supremum of the modulus on the disc $  | z | < 1 $
 +
in the class of all regular functions in the unit disc having the given polynomial as initial segment of the MacLaurin series. The solution to this problem is given by the following theorem.
  
 
Carathéodory–Fejér theorem [[#References|[1]]]. Let
 
Carathéodory–Fejér theorem [[#References|[1]]]. Let
  
<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/c/c020/c020320/c0203204.png" /></td> </tr></table>
+
$$
 +
P (z)  = c _ {0} +
 +
c _ {1} z + \dots + c _ {n-1} z  ^ {n-1}
 +
$$
 +
 
 +
be a given polynomial,  $  P (z) \not\equiv 0 $.
 +
There exists a unique rational function  $  R (z) = R ( z , c _ {0} \dots c _ {n-1} ) $
 +
of the form
  
be a given polynomial, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c0203205.png" />. There exists a unique rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c0203206.png" /> of the form
+
$$
 +
R (z)  = \lambda
  
<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/c/c020/c020320/c0203207.png" /></td> </tr></table>
+
\frac{\overline \alpha \; _ {n-1} +
 +
\overline \alpha \; _ {n-2} z + \dots + \overline \alpha \; _ {0} z  ^ {n-1} }{\alpha _ {0} + \alpha _ {1} z + \dots + \alpha _ {n-1} z  ^ {n-1} }
 +
,\ \
 +
\lambda > 0 ,
 +
$$
  
regular in the unit disc and having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c0203208.png" /> as the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c0203209.png" /> coefficients of its MacLaurin expansion. This function, and only this, realizes the minimum value of
+
regular in the unit disc and having $  c _ {0} \dots c _ {n-1} $
 +
as the first $  n $
 +
coefficients of its MacLaurin expansion. This function, and only this, realizes the minimum value of
  
<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/c/c020/c020320/c02032010.png" /></td> </tr></table>
+
$$
 +
M _ {f}  = \
 +
\sup _
 +
{| z | < 1 } \
 +
| f (z) |
 +
$$
  
in the class of all regular functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c02032011.png" /> in the unit disc of the form
+
in the class of all regular functions $  f (z) $
 +
in the unit disc of 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/c/c020/c020320/c02032012.png" /></td> </tr></table>
+
$$
 +
f (z)  = P (z) + a _ {n} z  ^ {n} + \dots ,
 +
$$
  
and this minimum value is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c02032013.png" />.
+
and this minimum value is $  \lambda = \lambda ( c _ {0} \dots c _ {n-1} ) $.
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c02032014.png" /> is equal to the largest positive root of the following equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c02032015.png" />:
+
The number $  \lambda ( c _ {0} \dots c _ {n-1} ) $
 +
is equal to the largest positive root of the following equation of degree $  2 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/c/c020/c020320/c02032016.png" /></td> </tr></table>
+
$$
 +
\left |
 +
\begin{array}{cccccccc}
 +
- \lambda  & 0  &\dots  & 0  &c _ {0}  &c _ {1}  &\dots  &c _ {n-1}  \\
 +
0  &- \lambda  &\dots  & 0  & 0  &c _ {0}  &\dots  &c _ {n-2}  \\
 +
\cdot  &\cdot  &\dots  &\cdot  &\cdot  &\cdot  &\dots  &\cdot  \\
 +
0  & 0  &\dots  &- \lambda  & 0  & 0  &\dots  &c _ {0}  \\
 +
\overline{c}\; _ {0}  & 0  &\dots  & 0  &- \lambda  & 0  &\dots  & 0  \\
 +
\overline{c}\; _ {1}  &\overline{c}\; _ {0}  &\dots  & 0  & 0  &- \lambda  &\dots  & 0  \\
 +
\cdot  &\cdot  &\dots  &\cdot  &\cdot  &\cdot  &\dots  &\cdot  \\
 +
\overline{c}\; _ {n-1}  &\overline{c}\; _ {n-2}  &\dots  &c
 +
bar _ {0}  & 0  & 0  &\dots  &- \lambda  \\
 +
\end{array}
 +
\right |  = 0 .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c02032017.png" /> are real, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c02032018.png" /> is the largest of the absolute values of the roots of the following equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020320/c02032019.png" />:
+
If $  c _ {0} \dots c _ {n-1} $
 +
are real, then $  \lambda ( c _ {0} \dots c _ {n-1} ) $
 +
is the largest of the absolute values of the roots of the following equation of degree $  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/c/c020/c020320/c02032020.png" /></td> </tr></table>
+
$$
 +
\left |
 +
\begin{array}{ccccc}
 +
- \lambda  & 0  &\dots  & 0  &c _ {0}  \\
 +
0  &- \lambda  &\dots  &c _ {0}  &c _ {1}  \\
 +
\cdot  &\cdot  &\dots  &\cdot  &\cdot  \\
 +
\cdot  &\cdot  &\dots  &\cdot  &\cdot  \\
 +
c _ {0}  &c _ {1}  &\dots  &c _ {n-1}  &- \lambda  \\
 +
\end{array}
 +
\right |  = 0 .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéodory,  L. Fejér,  "Ueber den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und den Picard–Landau'schen Satz"  ''Rend. Circ. Mat. Palermo'' , '''32'''  (1911)  pp. 218–239</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéodory,  L. Fejér,  "Ueber den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und den Picard–Landau'schen Satz"  ''Rend. Circ. Mat. Palermo'' , '''32'''  (1911)  pp. 218–239</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>

Latest revision as of 06:30, 30 May 2020


The problem of extending a polynomial in $ z $ to a power series representing a regular function in the disc $ | z | < 1 $ which realizes the least value of the supremum of the modulus on the disc $ | z | < 1 $ in the class of all regular functions in the unit disc having the given polynomial as initial segment of the MacLaurin series. The solution to this problem is given by the following theorem.

Carathéodory–Fejér theorem [1]. Let

$$ P (z) = c _ {0} + c _ {1} z + \dots + c _ {n-1} z ^ {n-1} $$

be a given polynomial, $ P (z) \not\equiv 0 $. There exists a unique rational function $ R (z) = R ( z , c _ {0} \dots c _ {n-1} ) $ of the form

$$ R (z) = \lambda \frac{\overline \alpha \; _ {n-1} + \overline \alpha \; _ {n-2} z + \dots + \overline \alpha \; _ {0} z ^ {n-1} }{\alpha _ {0} + \alpha _ {1} z + \dots + \alpha _ {n-1} z ^ {n-1} } ,\ \ \lambda > 0 , $$

regular in the unit disc and having $ c _ {0} \dots c _ {n-1} $ as the first $ n $ coefficients of its MacLaurin expansion. This function, and only this, realizes the minimum value of

$$ M _ {f} = \ \sup _ {| z | < 1 } \ | f (z) | $$

in the class of all regular functions $ f (z) $ in the unit disc of the form

$$ f (z) = P (z) + a _ {n} z ^ {n} + \dots , $$

and this minimum value is $ \lambda = \lambda ( c _ {0} \dots c _ {n-1} ) $.

The number $ \lambda ( c _ {0} \dots c _ {n-1} ) $ is equal to the largest positive root of the following equation of degree $ 2 n $:

$$ \left | \begin{array}{cccccccc} - \lambda & 0 &\dots & 0 &c _ {0} &c _ {1} &\dots &c _ {n-1} \\ 0 &- \lambda &\dots & 0 & 0 &c _ {0} &\dots &c _ {n-2} \\ \cdot &\cdot &\dots &\cdot &\cdot &\cdot &\dots &\cdot \\ 0 & 0 &\dots &- \lambda & 0 & 0 &\dots &c _ {0} \\ \overline{c}\; _ {0} & 0 &\dots & 0 &- \lambda & 0 &\dots & 0 \\ \overline{c}\; _ {1} &\overline{c}\; _ {0} &\dots & 0 & 0 &- \lambda &\dots & 0 \\ \cdot &\cdot &\dots &\cdot &\cdot &\cdot &\dots &\cdot \\ \overline{c}\; _ {n-1} &\overline{c}\; _ {n-2} &\dots &c bar _ {0} & 0 & 0 &\dots &- \lambda \\ \end{array} \right | = 0 . $$

If $ c _ {0} \dots c _ {n-1} $ are real, then $ \lambda ( c _ {0} \dots c _ {n-1} ) $ is the largest of the absolute values of the roots of the following equation of degree $ n $:

$$ \left | \begin{array}{ccccc} - \lambda & 0 &\dots & 0 &c _ {0} \\ 0 &- \lambda &\dots &c _ {0} &c _ {1} \\ \cdot &\cdot &\dots &\cdot &\cdot \\ \cdot &\cdot &\dots &\cdot &\cdot \\ c _ {0} &c _ {1} &\dots &c _ {n-1} &- \lambda \\ \end{array} \right | = 0 . $$

References

[1] C. Carathéodory, L. Fejér, "Ueber den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und den Picard–Landau'schen Satz" Rend. Circ. Mat. Palermo , 32 (1911) pp. 218–239
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
How to Cite This Entry:
Carathéodory-Fejér problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carath%C3%A9odory-Fej%C3%A9r_problem&oldid=12120
This article was adapted from an original article by G.V. Kuz'mina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article