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''Goncharov problem''
 
''Goncharov problem''
  
A problem in the theory of functions of a complex variable, consisting of finding the set of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a0101201.png" /> from some class satisfying the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a0101202.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a0101203.png" />). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a0101204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a0101205.png" /> are sequences of complex numbers admissible for the given class. The problem was posed by V.L. Goncharov [[#References|[2]]].
+
A problem in the theory of functions of a complex variable, consisting of finding the set of all functions $  f(z) $
 +
from some class satisfying the relations $  f ^ { (n) } ( \lambda _ {n} ) = A _ {n} $(
 +
$  n = 0, 1 , . . . $).  
 +
Here $  \{ A _ {n} \} $
 +
and $  \{ \lambda _ {n} \} $
 +
are sequences of complex numbers admissible for the given class. The problem was posed by V.L. Goncharov [[#References|[2]]].
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a0101206.png" /> is put in correspondence with the series
+
A function $  f(z) $
 +
is put in correspondence with the series
  
<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/a/a010/a010120/a0101207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\sum _ {n = 0 } ^  \infty  f ^ { ( n ) } ( \lambda _ {n} )
 +
P _ {n} ( z ) ,
 +
$$
  
the Abel–Goncharov interpolation series, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a0101208.png" /> is the Goncharov polynomial defined by the equalities
+
the Abel–Goncharov interpolation series, where $  P _ {n} (z) $
 +
is the Goncharov polynomial defined by the equalities
  
<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/a/a010/a010120/a0101209.png" /></td> </tr></table>
+
$$
 +
P _ {n} ^ { ( k ) } ( \lambda _ {k} )  = 0 ,
 +
\  k = 0 \dots n - 1 ; \  P _ {n} ^ { ( n ) } ( z )  \equiv  1 .
 +
$$
  
N.H. Abel gave a formal treatment of the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012011.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012013.png" /> are real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012014.png" /> (cf. [[#References|[1]]]). Here
+
N.H. Abel gave a formal treatment of the case $  \lambda _ {n} = a + nh $,  
 +
$  n = 0, 1 \dots $
 +
where a $
 +
and $  h $
 +
are real numbers $  (h \neq 0 ) $(
 +
cf. [[#References|[1]]]). Here
  
<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/a/a010/a010120/a01012015.png" /></td> </tr></table>
+
$$
 +
P _ {n} ( z )  =
 +
\frac{1}{n!}
 +
( z - a ) ( z - a - nh ) ^ {n - 1 } .
 +
$$
  
The series (*) is used to study the zeros of the successive derivatives of regular functions. The set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012016.png" /> representable by a series (*) is called the convergence class of the Abel–Goncharov problem.
+
The series (*) is used to study the zeros of the successive derivatives of regular functions. The set of functions $  f(z) $
 +
representable by a series (*) is called the convergence class of the Abel–Goncharov problem.
  
In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012017.png" />, the Abel–Goncharov convergence class has been expressed in terms of bounds on the order and the type of entire functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012018.png" />, depending on the growth of the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012019.png" /> [[#References|[2]]].
+
In the case when $  \lim\limits _ {n \rightarrow \infty }  | \lambda _ {n} | = \infty $,  
 +
the Abel–Goncharov convergence class has been expressed in terms of bounds on the order and the type of entire functions $  f(z) $,  
 +
depending on the growth of the quantity $  \sum _ {k=0}  ^ {n-1} | \lambda _ {k+1} - \lambda _ {k} | $[[#References|[2]]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012021.png" /> is a slowly increasing function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012022.png" />, the Abel–Goncharov convergence class has, in a sense, been exactly determined [[#References|[6]]]. Other Abel–Goncharov convergence classes for entire functions of finite and infinite order have been identified in terms of various constraints on the indicators of the respective classes of functions. The Abel–Goncharov problem has also been studied for entire functions of several variables. Exact estimates of Goncharov polynomials were obtained for some classes of interpolation nodes.
+
If $  \lambda _ {n} = n ^ {1/ \rho } l (n) $,  
 +
where $  l (n) $
 +
is a slowly increasing function, $  0 < \rho < \infty $,  
 +
the Abel–Goncharov convergence class has, in a sense, been exactly determined [[#References|[6]]]. Other Abel–Goncharov convergence classes for entire functions of finite and infinite order have been identified in terms of various constraints on the indicators of the respective classes of functions. The Abel–Goncharov problem has also been studied for entire functions of several variables. Exact estimates of Goncharov polynomials were obtained for some classes of interpolation nodes.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012023.png" /> be the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012024.png" /> of the form
+
Let $  A _ {r}  ^  \alpha  $
 +
be the class of functions $  f(z) $
 +
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/a/a010/a010120/a01012025.png" /></td> </tr></table>
+
$$
 +
f ( z )  = \sum _ {n = 0 } ^  \infty 
 +
( n ! ) ^ {- \alpha } a _ {n} z  ^ {n} ,
 +
\  \overline{\lim\limits}\; _ {n \rightarrow \infty }  | a _ {n} | ^ {1 / n }  \leq  r ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012026.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012027.png" /> be the class of all possible sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012030.png" />. By definition, the bound of convergence for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012031.png" /> is the supremum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012032.png" /> of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012033.png" /> for which every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012034.png" /> can be represented by a series (*). The infimum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012035.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012036.png" />-values for which there exist a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012037.png" /> and a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012040.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012041.png" />, is called the bound of uniqueness. The quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012043.png" /> are known, respectively, as the Whittaker constant and the Goncharov constant. It has been shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012044.png" /> (cf. [[#References|[6]]]); the more general statements
+
$  0 < r < \infty , 0 \leq  \alpha < \infty $,  
 +
and let $  \Lambda _  \alpha  $
 +
be the class of all possible sequences $  \{ \lambda _ {n} \} $
 +
such that $  | \lambda _ {n} | \leq  {(n + 1) } ^ {\alpha - 1 } $,
 +
$  n = 0, 1 ,\dots $.  
 +
By definition, the bound of convergence for the class $  \Lambda _  \alpha  $
 +
is the supremum $  S _  \alpha  $
 +
of the values of $  r $
 +
for which every function $  f(z) \in A _ {r}  ^  \alpha  $
 +
can be represented by a series (*). The infimum $  W _  \alpha  $
 +
of the $  r $-
 +
values for which there exist a function $  f(z) \in A _ {r}  ^  \alpha  $
 +
and a sequence $  \{ \lambda _ {n} \} \in \Lambda _  \alpha  $
 +
such that $  f ^ { (n) } ( \lambda _ {n} ) = 0 $,  
 +
$  n = 0, 1 \dots $
 +
$  f(z) \not\equiv 0 $,  
 +
is called the bound of uniqueness. The quantities $  W _ {1} $
 +
and $  W _ {0} $
 +
are known, respectively, as the Whittaker constant and the Goncharov constant. It has been shown that $  S _ {1} = W _ {1} $(
 +
cf. [[#References|[6]]]); the more general statements
  
<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/a/a010/a010120/a01012045.png" /></td> </tr></table>
+
$$
 +
S _  \alpha  = W _ {1} ,\  W _  \alpha  = W _ {1} ,
 +
0 \leq  \alpha < \infty ,
 +
$$
  
 
have also been proved [[#References|[5]]], [[#References|[10]]].
 
have also been proved [[#References|[5]]], [[#References|[10]]].
  
Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012046.png" />, the Abel–Goncharov problem is reduced to finding the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012047.png" />. Its precise numerical value is not known, but one has obtained the bounds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012048.png" /> [[#References|[9]]].
+
Thus, if $  \{ \lambda _ {n} \} \in \Lambda _  \alpha  $,  
 +
the Abel–Goncharov problem is reduced to finding the constant $  W _ {1} $.  
 +
Its precise numerical value is not known, but one has obtained the bounds $  0.7259 < W _ {1} < 0.7378 $[[#References|[9]]].
  
Relative to the Abel–Goncharov problem for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012049.png" /> of functions which are regular in the region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012050.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012051.png" />, the following has been shown: For any set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012053.png" />, which satisfy the condition
+
Relative to the Abel–Goncharov problem for the class $  A _ {1}  ^ {*} $
 +
of functions which are regular in the region $  | z | \geq  1 $
 +
and such that $  f ( \infty ) = 0 $,  
 +
the following has been shown: For any set of numbers $  \{ \lambda _ {k} \} $,  
 +
$  | \lambda _ {k} | \geq  1 $,  
 +
which satisfy the condition
  
<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/a/a010/a010120/a01012054.png" /></td> </tr></table>
+
$$
 +
\lim\limits _  \overline { {k \rightarrow \infty }}\;
 +
\frac{ n _ {k}  }{| \lambda _ {k} | }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012055.png" /> is an increasing sequence of natural numbers, the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012057.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012058.png" />. Moreover, for any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012059.png" /> there exist a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012060.png" />
+
= 0 ,
 +
$$
  
<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/a/a010/a010120/a01012061.png" /></td> </tr></table>
+
where  $  \{ n _ {k} \} $
 +
is an increasing sequence of natural numbers, the equations  $  f ^ { ( n _ {k} ) } ( \lambda _ {k} ) = 0 $,
 +
$  k = 0, 1 \dots $
 +
imply  $  f(z) \equiv 0 $.
 +
Moreover, for any number  $  b > 0 $
 +
there exist a sequence  $  \{ \lambda _ {n} \} $
  
and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012062.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012064.png" /> [[#References|[7]]].
+
$$
 +
\lim\limits _ {n \rightarrow \infty } 
 +
\frac{n}{| \lambda _ {n} | }
 +
  = b
 +
$$
  
The Abel–Goncharov problem includes the so-called two-point problem posed by J.M. Whittaker [[#References|[12]]]. Let the sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012066.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012068.png" />. The problem is to determine conditions under which there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012069.png" />, regular on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012070.png" />, satisfying the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012072.png" />. This problem was solved for various subclasses of the class of functions regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012073.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012074.png" />). The resulting conditions, which are in a sense precise, are expressed in terms of various bounds imposed on the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012075.png" /> of the expansions
+
and a function $  f(z) \not\equiv 0, f(z) \in A _ {1}  ^ {*} $,  
 +
for which  $  f ^ { (n) } ( \lambda _ {n} ) = 0 $,  
 +
$  n = 0, 1 , . . $[[#References|[7]]].
  
<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/a/a010/a010120/a01012076.png" /></td> </tr></table>
+
The Abel–Goncharov problem includes the so-called two-point problem posed by J.M. Whittaker [[#References|[12]]]. Let the sequences  $  \{ \nu _ {k} \} $
 +
and  $  \{ \mu _ {n} \} $
 +
be such that  $  \{ \nu _ {k} \} \cup \{ \mu _ {n} \} = \{ n \} $
 +
and  $  \{ \nu _ {k} \} \cap \{ \mu _ {n} \} = \emptyset $.
 +
The problem is to determine conditions under which there exists a function  $  f(z) \not\equiv 0 $,
 +
regular on the segment  $  [0, 1] $,
 +
satisfying the conditions  $  f ^ { ( \nu _ {k} ) } (1) = 0 $,
 +
$  f ^ { ( \mu _ {n} ) } (0) = 0 $.  
 +
This problem was solved for various subclasses of the class of functions regular in the disc  $  | z | < R $(
 +
$  R > 1 $).  
 +
The resulting conditions, which are in a sense precise, are expressed in terms of various bounds imposed on the coefficients  $  a _ {\nu _ {k}  } $
 +
of the expansions
  
depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012077.png" /> [[#References|[3]]]. This problem was generalized and solved using methods of the theory of infinite systems of linear equations [[#References|[4]]]. In the particular case where the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a01012078.png" /> forms an arithmetic progression and one is dealing with entire functions of exponential type, the two-point problem has, in a certain sense, been solved completely [[#References|[8]]].
+
$$
 +
f ( z )  = \sum _ { k=0 } ^  \infty  a _ {\nu _ {k}  } z ^ {\nu _ {k} }
 +
$$
 +
 
 +
depending on  $  \{ \nu _ {k} \} $[[#References|[3]]]. This problem was generalized and solved using methods of the theory of infinite systems of linear equations [[#References|[4]]]. In the particular case where the sequence $  \{ \nu _ {k} \} $
 +
forms an arithmetic progression and one is dealing with entire functions of [[Function of exponential type|exponential type]], the two-point problem has, in a certain sense, been solved completely [[#References|[8]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.H. Abel,  "Sur les fonctions génératrices et leurs déterminants" , ''Oeuvres complètes, nouvelle éd.'' , '''2''' , Grondahl &amp; Son , Christiania  (1839)  pp. 77–88  (Edition de Holmboe)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.L. [V.L. Goncharov] Gontcharoff,  "Recherches sur les dérivées successives des fonctions analytiques. Géneralisation de la serie d'Abel"  ''Ann. Sci. Ecole Norm. Sup. (3)'' , '''47'''  (1930)  pp. 1–78</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.O. Gel'fond,  I.I. Ibragimov,  "On functions, the derivatives of which vanish in two points"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''11'''  (1947)  pp. 547–560  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Dzhrbashyan,  "Uniqueness and representability theorems for analytic functions"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''16'''  (1952)  pp. 225–252  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.M. Dragilev,  O.P. Chukhlova,  "On convergence of certain interpolation series"  ''Sibirsk. Mat. Zh.'' , '''4''' :  2  (1963)  pp. 287–294  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.A. Evgrafov,  "The Abel–Goncharov interpolation problem" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  Yu.A. Kaz'min,  "On zeros of sequences of derivatives of analytic functions"  ''Vestnik Moskov. Univ. Ser. I Math. Mekh.'' :  1  (1963)  pp. 26–34  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Yu.A. Kaz'min,  "On a problem of Gel'fond–Ibragimov"  ''Vestnik Moskov. Univ. Ser. I Math. Mekh.'' , '''19''' :  6  (1965)  pp. 37–44  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.S. Macintyre,  "On the zeros of successive derivatives of integral functions"  ''Trans. Amer. Math. Soc.'' , '''67'''  (1949)  pp. 241–251</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  Yu.K. Suetin,  "Coincidence of constants of uniqueness and convergence of certain interpolation problems"  ''Vestnik Moskov. Univ. Ser. I Math. Mekh.'' , '''21''' :  5  (1966)  pp. 16–25  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  V.A. Oskolkov,  "On estimates for Gončarov polynomials"  ''Math. USSR-Sb.'' , '''21''' :  1  (1973)  pp. 57–62  ''Mat. Sb.'' , '''92''' :  1  (1973)  pp. 55–59</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  J.M. Whittaker,  "Interpolatory function theory" , Cambridge Univ. Press  (1935)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N.H. Abel,  "Sur les fonctions génératrices et leurs déterminants" , ''Oeuvres complètes, nouvelle éd.'' , '''2''' , Grondahl &amp; Son , Christiania  (1839)  pp. 77–88  (Edition de Holmboe)</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top">  W.L. [V.L. Goncharov] Gontcharoff,  "Recherches sur les dérivées successives des fonctions analytiques. Géneralisation de la série d'Abel"  ''Ann. Sci. Ecole Norm. Sup. (3)'' , '''47'''  (1930)  pp. 1–78 {{ZBL|56.0260.01}}</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  A.O. Gel'fond,  I.I. Ibragimov,  "On functions, the derivatives of which vanish in two points"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''11'''  (1947)  pp. 547–560  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Dzhrbashyan,  "Uniqueness and representability theorems for analytic functions"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''16'''  (1952)  pp. 225–252  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.M. Dragilev,  O.P. Chukhlova,  "On convergence of certain interpolation series"  ''Sibirsk. Mat. Zh.'' , '''4''' :  2  (1963)  pp. 287–294  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.A. Evgrafov,  "The Abel–Goncharov interpolation problem" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  Yu.A. Kaz'min,  "On zeros of sequences of derivatives of analytic functions"  ''Vestnik Moskov. Univ. Ser. I Math. Mekh.'' :  1  (1963)  pp. 26–34  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Yu.A. Kaz'min,  "On a problem of Gel'fond–Ibragimov"  ''Vestnik Moskov. Univ. Ser. I Math. Mekh.'' , '''19''' :  6  (1965)  pp. 37–44  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.S. Macintyre,  "On the zeros of successive derivatives of integral functions"  ''Trans. Amer. Math. Soc.'' , '''67'''  (1949)  pp. 241–251</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  Yu.K. Suetin,  "Coincidence of constants of uniqueness and convergence of certain interpolation problems"  ''Vestnik Moskov. Univ. Ser. I Math. Mekh.'' , '''21''' :  5  (1966)  pp. 16–25  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  V.A. Oskolkov,  "On estimates for Gončarov polynomials"  ''Math. USSR-Sb.'' , '''21''' :  1  (1973)  pp. 57–62  ''Mat. Sb.'' , '''92''' :  1  (1973)  pp. 55–59</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  J.M. Whittaker,  "Interpolatory function theory" , Cambridge Univ. Press  (1935)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====

Latest revision as of 11:49, 10 April 2023


Goncharov problem

A problem in the theory of functions of a complex variable, consisting of finding the set of all functions $ f(z) $ from some class satisfying the relations $ f ^ { (n) } ( \lambda _ {n} ) = A _ {n} $( $ n = 0, 1 , . . . $). Here $ \{ A _ {n} \} $ and $ \{ \lambda _ {n} \} $ are sequences of complex numbers admissible for the given class. The problem was posed by V.L. Goncharov [2].

A function $ f(z) $ is put in correspondence with the series

$$ \tag{* } \sum _ {n = 0 } ^ \infty f ^ { ( n ) } ( \lambda _ {n} ) P _ {n} ( z ) , $$

the Abel–Goncharov interpolation series, where $ P _ {n} (z) $ is the Goncharov polynomial defined by the equalities

$$ P _ {n} ^ { ( k ) } ( \lambda _ {k} ) = 0 , \ k = 0 \dots n - 1 ; \ P _ {n} ^ { ( n ) } ( z ) \equiv 1 . $$

N.H. Abel gave a formal treatment of the case $ \lambda _ {n} = a + nh $, $ n = 0, 1 \dots $ where $ a $ and $ h $ are real numbers $ (h \neq 0 ) $( cf. [1]). Here

$$ P _ {n} ( z ) = \frac{1}{n!} ( z - a ) ( z - a - nh ) ^ {n - 1 } . $$

The series (*) is used to study the zeros of the successive derivatives of regular functions. The set of functions $ f(z) $ representable by a series (*) is called the convergence class of the Abel–Goncharov problem.

In the case when $ \lim\limits _ {n \rightarrow \infty } | \lambda _ {n} | = \infty $, the Abel–Goncharov convergence class has been expressed in terms of bounds on the order and the type of entire functions $ f(z) $, depending on the growth of the quantity $ \sum _ {k=0} ^ {n-1} | \lambda _ {k+1} - \lambda _ {k} | $[2].

If $ \lambda _ {n} = n ^ {1/ \rho } l (n) $, where $ l (n) $ is a slowly increasing function, $ 0 < \rho < \infty $, the Abel–Goncharov convergence class has, in a sense, been exactly determined [6]. Other Abel–Goncharov convergence classes for entire functions of finite and infinite order have been identified in terms of various constraints on the indicators of the respective classes of functions. The Abel–Goncharov problem has also been studied for entire functions of several variables. Exact estimates of Goncharov polynomials were obtained for some classes of interpolation nodes.

Let $ A _ {r} ^ \alpha $ be the class of functions $ f(z) $ of the form

$$ f ( z ) = \sum _ {n = 0 } ^ \infty ( n ! ) ^ {- \alpha } a _ {n} z ^ {n} , \ \overline{\lim\limits}\; _ {n \rightarrow \infty } | a _ {n} | ^ {1 / n } \leq r , $$

$ 0 < r < \infty , 0 \leq \alpha < \infty $, and let $ \Lambda _ \alpha $ be the class of all possible sequences $ \{ \lambda _ {n} \} $ such that $ | \lambda _ {n} | \leq {(n + 1) } ^ {\alpha - 1 } $, $ n = 0, 1 ,\dots $. By definition, the bound of convergence for the class $ \Lambda _ \alpha $ is the supremum $ S _ \alpha $ of the values of $ r $ for which every function $ f(z) \in A _ {r} ^ \alpha $ can be represented by a series (*). The infimum $ W _ \alpha $ of the $ r $- values for which there exist a function $ f(z) \in A _ {r} ^ \alpha $ and a sequence $ \{ \lambda _ {n} \} \in \Lambda _ \alpha $ such that $ f ^ { (n) } ( \lambda _ {n} ) = 0 $, $ n = 0, 1 \dots $ $ f(z) \not\equiv 0 $, is called the bound of uniqueness. The quantities $ W _ {1} $ and $ W _ {0} $ are known, respectively, as the Whittaker constant and the Goncharov constant. It has been shown that $ S _ {1} = W _ {1} $( cf. [6]); the more general statements

$$ S _ \alpha = W _ {1} ,\ W _ \alpha = W _ {1} , \ 0 \leq \alpha < \infty , $$

have also been proved [5], [10].

Thus, if $ \{ \lambda _ {n} \} \in \Lambda _ \alpha $, the Abel–Goncharov problem is reduced to finding the constant $ W _ {1} $. Its precise numerical value is not known, but one has obtained the bounds $ 0.7259 < W _ {1} < 0.7378 $[9].

Relative to the Abel–Goncharov problem for the class $ A _ {1} ^ {*} $ of functions which are regular in the region $ | z | \geq 1 $ and such that $ f ( \infty ) = 0 $, the following has been shown: For any set of numbers $ \{ \lambda _ {k} \} $, $ | \lambda _ {k} | \geq 1 $, which satisfy the condition

$$ \lim\limits _ \overline { {k \rightarrow \infty }}\; \frac{ n _ {k} }{| \lambda _ {k} | } = 0 , $$

where $ \{ n _ {k} \} $ is an increasing sequence of natural numbers, the equations $ f ^ { ( n _ {k} ) } ( \lambda _ {k} ) = 0 $, $ k = 0, 1 \dots $ imply $ f(z) \equiv 0 $. Moreover, for any number $ b > 0 $ there exist a sequence $ \{ \lambda _ {n} \} $

$$ \lim\limits _ {n \rightarrow \infty } \frac{n}{| \lambda _ {n} | } = b $$

and a function $ f(z) \not\equiv 0, f(z) \in A _ {1} ^ {*} $, for which $ f ^ { (n) } ( \lambda _ {n} ) = 0 $, $ n = 0, 1 , . . . $[7].

The Abel–Goncharov problem includes the so-called two-point problem posed by J.M. Whittaker [12]. Let the sequences $ \{ \nu _ {k} \} $ and $ \{ \mu _ {n} \} $ be such that $ \{ \nu _ {k} \} \cup \{ \mu _ {n} \} = \{ n \} $ and $ \{ \nu _ {k} \} \cap \{ \mu _ {n} \} = \emptyset $. The problem is to determine conditions under which there exists a function $ f(z) \not\equiv 0 $, regular on the segment $ [0, 1] $, satisfying the conditions $ f ^ { ( \nu _ {k} ) } (1) = 0 $, $ f ^ { ( \mu _ {n} ) } (0) = 0 $. This problem was solved for various subclasses of the class of functions regular in the disc $ | z | < R $( $ R > 1 $). The resulting conditions, which are in a sense precise, are expressed in terms of various bounds imposed on the coefficients $ a _ {\nu _ {k} } $ of the expansions

$$ f ( z ) = \sum _ { k=0 } ^ \infty a _ {\nu _ {k} } z ^ {\nu _ {k} } $$

depending on $ \{ \nu _ {k} \} $[3]. This problem was generalized and solved using methods of the theory of infinite systems of linear equations [4]. In the particular case where the sequence $ \{ \nu _ {k} \} $ forms an arithmetic progression and one is dealing with entire functions of exponential type, the two-point problem has, in a certain sense, been solved completely [8].

References

[1] N.H. Abel, "Sur les fonctions génératrices et leurs déterminants" , Oeuvres complètes, nouvelle éd. , 2 , Grondahl & Son , Christiania (1839) pp. 77–88 (Edition de Holmboe)
[2] W.L. [V.L. Goncharov] Gontcharoff, "Recherches sur les dérivées successives des fonctions analytiques. Géneralisation de la série d'Abel" Ann. Sci. Ecole Norm. Sup. (3) , 47 (1930) pp. 1–78 Zbl 56.0260.01
[3] A.O. Gel'fond, I.I. Ibragimov, "On functions, the derivatives of which vanish in two points" Izv. Akad. Nauk SSSR Ser. Mat. , 11 (1947) pp. 547–560 (In Russian)
[4] M.M. Dzhrbashyan, "Uniqueness and representability theorems for analytic functions" Izv. Akad. Nauk SSSR Ser. Mat. , 16 (1952) pp. 225–252 (In Russian)
[5] M.M. Dragilev, O.P. Chukhlova, "On convergence of certain interpolation series" Sibirsk. Mat. Zh. , 4 : 2 (1963) pp. 287–294 (In Russian)
[6] M.A. Evgrafov, "The Abel–Goncharov interpolation problem" , Moscow (1954) (In Russian)
[7] Yu.A. Kaz'min, "On zeros of sequences of derivatives of analytic functions" Vestnik Moskov. Univ. Ser. I Math. Mekh. : 1 (1963) pp. 26–34 (In Russian)
[8] Yu.A. Kaz'min, "On a problem of Gel'fond–Ibragimov" Vestnik Moskov. Univ. Ser. I Math. Mekh. , 19 : 6 (1965) pp. 37–44 (In Russian)
[9] S.S. Macintyre, "On the zeros of successive derivatives of integral functions" Trans. Amer. Math. Soc. , 67 (1949) pp. 241–251
[10] Yu.K. Suetin, "Coincidence of constants of uniqueness and convergence of certain interpolation problems" Vestnik Moskov. Univ. Ser. I Math. Mekh. , 21 : 5 (1966) pp. 16–25 (In Russian)
[11] V.A. Oskolkov, "On estimates for Gončarov polynomials" Math. USSR-Sb. , 21 : 1 (1973) pp. 57–62 Mat. Sb. , 92 : 1 (1973) pp. 55–59
[12] J.M. Whittaker, "Interpolatory function theory" , Cambridge Univ. Press (1935)

Comments

For an introduction to the general area, cf. [a1].

References

[a1] R.P. Boas, "Entire functions" , Acad. Press (1954)
How to Cite This Entry:
Abel-Goncharov problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel-Goncharov_problem&oldid=18670
This article was adapted from an original article by V.A. Oskolkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article