View source for Abel-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 function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010120/a0101206.png" /> 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>

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

<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>

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

<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>

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.

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]]].

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.

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

<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>

<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

<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>

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]]].

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

<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>

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" />

<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>

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]]].

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

<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>

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]]].

====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>



====Comments====
For an introduction to the general area, cf. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P. Boas,   "Entire functions" , Acad. Press  (1954)</TD></TR></table>