Abel-Goncharov problem

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 serie d'Abel" Ann. Sci. Ecole Norm. Sup. (3) , 47 (1930) pp. 1–78
[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)