Bernstein-type characterization of entire functions
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irk-123456789-1929852023-07-28T15:26:58Z Bernstein-type characterization of entire functions Dovgoshey, O.A. Prestin, J. Shevchuk, I.O. Математика 2023 Article Bernstein-type characterization of entire functions / O.A. Dovgoshey, J. Prestin, I.O. Shevchuk // Доповіді Національної академії наук України. — 2023. — № 1. — С. 10-15. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2023.01.010 http://dspace.nbuv.gov.ua/handle/123456789/192985 517 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Bernstein-type characterization of entire functions |
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Bernstein-type characterization of entire functions / O.A. Dovgoshey, J. Prestin, I.O. Shevchuk // Доповіді Національної академії наук України. — 2023. — № 1. — С. 10-15. — Бібліогр.: 15 назв. — англ. |
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10 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 1: 10—15
C i t a t i o n: Dovgoshey O.A., Prestin J., Shevchuk I.O. Bernstein-type characterization of entire functions.
Dopov. Nac. akad. nauk Ukr. 2023. No 1. P. 10—15. https://doi.org/10.15407/dopovidi2023.01.010
© Publisher PH «Akademperiodyka» of the NAS of Ukraine, 2023. This is an open acsess article under the
CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
МАТЕМАТИКА
MATHEMATICS
https://doi.org/10.15407/dopovidi2023.01.010
UDC 517
O.A. Dovgoshey1, 2
, https://orcid.org/0000-0002-6496-2466
J. Prestin2
, https://orcid.org/0000-0001-5985-7939
I.O. Shevchuk3
, https://orcid.org/0000-0003-1140-373X
1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yansk
2 Institute of Mathematics, University of Lubeck, Germany
3 Taras Shevchenko National University of Kyiv
E-mail: oleksiy.dovgoshey@gmail.com, prestin@math.uni-luebeck.de, shevchuk@univ.kiev.ua
Bernstein-type characterization of entire functions
Presented by Corresponding Member of the NAS of Ukraine I.O. Shevchuk
Let be the set of all entire functions on the complex plane . Let us consider the class EX of all complex Banach
spaces X such that X ⊇ . For ( , )X ⋅ ∈ EX and g X∈ we write , ( ) inf { : },n X nE g g p p= − ∈Π where nΠ is the set
of all polynomials with degree at most n. We describe all X ∈ EX for which the relation 1/
,lim ( ) 0( ) n
n X
n
E g
→∞
= holds
if and only if g ∈ .
Keywords: Bernstein theorem, entire function, polynomial approximation, Shauder basis, transfinite diameter.
1. Introduction. The initial Bernstein theorem. Let f be a real-valued continuous function on
[–1,1] and let , [ 1, 1] ( )nE f− be the minimum error in the Chebyshev approximation of f on [–1,1]
by polynomials of degree at most n .
Theorem 1. (Bernstein theorem). The equality 1/
, [–1, 1]lim ( ) 0n
nn
E f
→∞
= holds if and only if f is the
restriction of an entire function to [–1,1] .
This theorem was published in the classical book [1].
The Introduction briefly describes the early development of Bernstein theorem 1. In Sec-
tion 2 we formulate two new theorems and two conjectures describing the structure of Banach
spaces for which “the Bernstein theorem” remains valid.
The Walsh theorem. In 1926 J. L. Walsh [2] published the following result.
Theorem 2 (Walsh theorem). Let { }= ∪ ∞ be the one-point compactification of the complex
plane, K be a compact subset of and let K be a simply connected regular for Dirichlet prob-
lem domain. Then the following statements are equivalent for every continuous function :f K → :
(i) f is the restriction to K of an entire function;
11ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 1
Bernstein-type characterization of entire functions
(ii) the equality 1/
,lim ( ) 0n
n K
n
E f
→∞
= holds, where , ( )n KE f is the minimum error in uniform approxi-
mation of f on K by polynomials with degree at most n .
Recall that the domain K is regular for Dirichlet problem if and only if it possesses the
classical Green function with pole at infinity.
Considering the over-convergence of polynomials of the best uniform approximation J. Walsh
and H. Russell obtained (see [3]) a result which implies that the equivalence (i) (ii)⇔ in Walsh
theorem 2 remains valid if K is an arbitrary regular for Dirichlet problem domain.
The extension of Bernstein theorem by R. S. Varga. For more than thirty years, the Ber-
nstein-Walsh-Russell theorems do not actually attract the attention of mathematicians till the
paper of R. S. Varga [4] who characterized the order and type of an entire function f by minimum
error sequence , [–1, 1]( ( ))n nE f ∈ .
It should be noted here that these remarkable characteristics and the results associated with
them are not the subject of present paper, and we limit ourselves to studying the equivalence
1/lim ( ) 0 iff is entire.n
n
n
E f f
→∞
=
Reformulation of Bernstein-Walsh-Russell theorems. For further it is convenient to give
some suitable reformulations of the Bernstein-Walsh-Russell theorems.
Let us denote by the set of all entire functions :f → and write nΠ for the set of all
polynomials of degree at most n . Now we define the class EX as follows.
Definition 1. By EX we denote the class of all complex Banach linear spaces ( , )X ⋅ such
that ( , )X ⋅ belongs to EX if and only if X⊆ .
For ( , )X ⋅ ∈ EX , we define the set X as 1/
,: { : lim ( ( )) 0},n
X n X
n
f X E f
→∞
= ∈ = where, for
every n∈ , , ( ) inf { : }n X nE f f p p= − ∈Π . We will also denote by KC the set of all continuous
complex-valued functions on the compact K ⊆ and write : sup ( )
z K
f f z∞
∈
= for Kf C∈ .
Now the classical results of Bernstein, Walsh, and Walsh-Russell can be formulated as follows.
Theorem 3. (Bernstein theorem). Let [–1, 1]( , ) ( , )X C ∞⋅ = ⋅ . Then the equality
X X= (1)
holds.
Theorem 4. (Walsh theorem). Let ( , ) ( , )KX C ∞⋅ = ⋅ . Equality (1) holds if K is a
simply connected regular for Dirichlet problem domain.
Theorem 5. (Walsh-Russell theorem). Let ( , ) ( , )KX C ∞⋅ = ⋅ . Equality (1) holds if K
is a regular for Dirichlet problem domain.
2. The main results. In this section we formulate new Theorems 6, 7, and Conjectures 1, 2.
Theorem 6. L et X ∈ EX . Then equality (1) holds iff
→∞
< τ 1/
,0 inf (l m )i n
n X
n
and
→∞
< ∞1/
,(lim sup ) n
n X
n
m hold.
Theorem 7. Let a complex Banach space Y have a Shauder basis. Then Y is linearly isometric
to a space X ∈ EX .
By Mazur’s theorem, every infinite-dimensional vector normed space contains an infinite-
dimensional subspace that has a Shauder basis (see, for example, Theorem 6.3.3 in [5]). Hence,
Theorem 7 implies the following.
12 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 1
O.A. Dovgoshey, J. Prestin, I.O. Shevchuk
Corollary 1. Every infinite-dimensional complex Banach space contains a subspace Y which is
linearly isometric to some X ∈ EX .
Let us consider now some corollaries of Theorem 6 for the case of uniform approximation.
It is clear that
Walsh�Russell theorem Walsh theorem Bernstein theorem⇒ ⇒
In what follows we will use the concept of transfinite diameter.
For K ⊆ and 1u , … , nu K∈ , we write
1
,
( , , ) : ( )n k l
k l
k l
V u u u u
<
… = −∏
and
1( ) : sup{ ( , , ) : ,1 }.n n n jV V K V u u u K j n= = … ∈
In accordance with M. Fekete, the transfinite diameter of K is the number
2
( 1)( ) lim .n n
n
n
d K V −
→∞
=
Let ⋅ ∈( , )X EX . If n nf ∈Π is the monomial ( ) n
nf z z= , we write
,n X nm f= and
1
, inf .
n
n X
p
f p
−∈Π
τ = −
Fekete [6] proved that 1/
,lim ( ) n
n X
n→∞
τ (the Chebyshev constant) exists for ∞⋅ = ⋅( , ) ( , )KX C .
In this case he also showed in [7] that
1/
,lim ( ) ( ).n
n X
n
d K
→∞
τ =
(2
The existence of Green function for the domain K implies that the Robin constant
( )Kγ is strictly positive,
( ) 0.Kγ > (3
Now, from the equality
( ) ( ),d K K= γ (4
we have
mT Whe lore hm 6 R& (3) s & (4) a s � us ell theore⇒
Remark 1. Inequality (3) and equality (4) follow, respectively, from Theorem 1 and Theorem
2 of Goluzin’s book [8, p. 311].
Using Faber’s polynomials A. V. Batyrev [9] proved the following.
13ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 1
Bernstein-type characterization of entire functions
Theorem 8 (Batyrev theorem). A function f , holomorphic on a compact set K ⊆ with the
positive transitive diameter ( )d K , and with the simply connected K , can be extended to an
entire function if and only if 1/
,lim ( ) 0,n
n K
n
E f
→∞
= where , ( ) inf { : }n K nE f f p p= − ∈Π .
Batyrev theorem was extended by T. Winiarski [10] for the case when K is not necessar-
ily simply connected. Using our notation we can formulate this result as follows.
Theorem 9 (Winiarski theorem). If K is a compact subset of with ( ) 0d K > , then (1)
holds for ∞⋅ = ⋅( , ) ( , )KX C .
Thus, we obtain
Theorem 6 & (2) Winiarski theorem Batyrev theorem
Walsh�Russell theorem
⇒ ⇒
⇓
Theorem 6 and (2) also imply the following result which shows that the converse to Winiar-
ski theorem is valid.
Corollary 2 [11]. Let K be a compact set in with | |K = ∞ . Then, for ∞⋅ = ⋅( , ) ( , )KX C ,
equality (1) holds if and only if ( ) 0d K > .
Corollary 2 can be strengthen as follows.
Theorem 10 [12]. Let K be a compact subset of . Then the following statements are equiva-
lent for the space ∞⋅ = ⋅( , ) ( , )KX C :
(i) the equality { | : is holomorphic on }X Kf f K= holds;
(ii) the transfinite diameter of K equals zero.
The original formulation of Theorem 10 contains the condition: “The logarithmic capacity of
K is zero” instead of statement (ii); but it was shown by P.J. Myrberg [13] that the logarithmic
capacity coincides with the transfinite diameter for every compact K ⊆ .
We conclude this brief survey of “uniform” generalizations of the Bernstein theorem by fol-
lowing.
Theorem 11 [10]. Let K ⊆ be a compact set with K = ∞ and let ∞∈ ⋅( , )Kf C . The func-
tion f can be extended to an entire function if and only if
1/
1
,
2
( )
lim ( ) 0.
( )
n
n
n K
n n
V K
E f
V K
+
→∞ +
⎡ ⎤
=⎢ ⎥
⎣ ⎦
The last theorem is valid even if ( ) 0d K = . This result and the equality
1/
1 ( )
( ) lim
( )
n
n
n n
V K
d K
V K
+
→∞
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
imply Winiarski theorem.
Theorem 11 can be derived also from the results A. G. Naftalevich, whose paper [14], appar-
ently, is the first attempt to consider the polynomial approximation of entire functions on com-
pact sets of zero transfinite diameter.
Let us turn to the weighted polynomial approximation.
14 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 1
O.A. Dovgoshey, J. Prestin, I.O. Shevchuk
Let K be a bounded subset of and let : [0, )w K → ∞ be a weight on K . We denote by
wX X= the set of all functions :f K → such that
∞
∈
= < ∞
,
sup ( ) ( ) .
w
z K
f f z w z
Then ∞⋅ → ∞
,
: [0, )
w
X is a seminorm on .X Furthermore, the space ∞⋅( , ) belongs to
EX if and only if the set –1 (0)K w has an infinite cardinality.
Conjecture 1. Let K be a bounded subset of and let : [0, )w K → ∞ be a weight on K such
that –1 (0)K w = ∞ . Then the following statements are equivalent:
(i) equality (1) holds for ∞⋅ ,
( , )
w
X ;
(ii) there is a constant (0, )c∈ ∞ such that 2/( ( 1))l ,i ( ) 0i nfm n n
n n
n
V K−
→∞
> where
{ : ( ) }n
nK z K w z c= ∈ .
We conclude the paper by the following conjecture that can be considered as a “weighted
generalization” of the Walsh-Russell theorem.
Conjecture 2. The following statements are equivalent for every compact K ⊆ with
| |K = ∞ and connected K .
(i) Equality (1) holds for every ∞⋅ ,
( , )
w
X with continuous ( ) 0w z ≡/ .
(ii) The domain K is regular for Dirichlet problem.
We conclude the paper by the following.
Problem 1. Does every X ∈ EX have a Shauder basis?
Remark 2. The first example of separable Banach space which does not have any Shauder basis
was constructed by P. Enflo [15]. So, if the above formulated problem has a positive solution, then
using Theorem 6 we can characterize the complex Banach spaces with a basis as spaces linearly
isometric to EX -spaces.
O. Dovgoshey’s research is partially financed by Volkswagen Stiftung Project “From Modeling
and Analysis to Approximation”.
REFERENCES
1. Bernstein, S. N. (1926). Leçons sur les propriétés extrémales et la meilleure approximation des fonctions
analytiques d’une variable réelle. Paris: Gauthier-Villars.
2. Walsh, J. L. (1926). Über den Grad der Approximation einer analyti schen Funktion. In Sitzungsberichte der
Mathematisch-Naturwissenschaftlichen Abteilung der Bayerischen Akademie der Wissenschaften zu Mün-
chen (heft 2) (pp. 223-229). München: Oldenbourg Wissenschaftsverlag.
https://doi.org/10.1515/9783486751932-002
3. Walsh, J. L. & Russell, H. G. (1934). On the convergence and overcon vergence of sequences of polynomials
of best simultaneous approximation to several functions analytic in distinct regions. Trans. Amer. Math. Soc.,
36, pp. 13-28. https://doi.org/10.2307/1989705
4. Varga, R. S. (1968). On an extension of a result of S. N. Bernstein. J. Approx. Theory, 1, pp. 176-179.
https://doi.org/10.1016/0021-9045(68)90020-8
5. Kadets, M. I. & Kadets, V. M. (1997). Series in Banach spaces: condit ional and unconditional convergence.
Basel, Boston, Berlin: Birkhäuser.
6. Fekete, M. (1923). Über die Verteilung der Wurzeln bei gewissen algebr aischen Gleichungen mit ganzzahligen
Koeffizienten. Math. Z., 17, pp. 228-249. https://doi.org/10.1007/BF01504345
15ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 1
Bernstein-type characterization of entire functions
7. Fekete, M. (1930). Über den transfiniten Durchmesser ebener Punktmengen , II. Math. Z., 32, pp. 215-221.
https://doi.org/10.1007/BF01194630
8. Goluzin, G. M. (1969). Geometric theory of functions of a complex variab le. Providence: American Ma the-
matical Society.
9. Batyrev, A. V. (1951). On the problem of best approximation of an analyt ic function by polynomials. Dokl.
Akad. Nauk SSSR, 26, pp. 173-175 (in Russian).
10. Winiarski, T. (1970). Approximation and interpolation of entire function s. Ann. Pol. Math., 23, pp. 259-273.
11. Dovgoshey, A. A. (1995). Uniform polynomial approximation of entire func tions on arbitrary compact sets in
the complex plane. Math. Notes, 58, No. 3, pp. 921-927. https://doi.org/10.1007/BF02304768
12. Walsh, J. L. (1946). Taylor’s series and approximation to analytic funct ions. Bull. Amer. Math. Soc., 52,
pp. 572—579.
13. Myrberg, P. J. (1933). Über die Existenz der Greenschen Funktionen auf e nier Gegebenen Riemannschen
Fläche. Acta Math., 61, pp. 39-79. https://doi.org/10.1007/BF02547786
14. Naftalevich, A. G. (1969). On the approximation of analytic functions by a lgebraic polynomials. Litovsk.
Matem. Sb., 9, No. 3, pp. 577-588 (in Russian).
15. Enflo, P. (1973). A counterexample to the approximation problem in Banach spaces. Acta Math., 130, No. 1,
pp. 309-317. https://doi.org/10.1007/BF02392270
Received 27.09.2022
О.А. Довгоший1, 2, https://orcid.org/0000-0002-6496-2466
Ю. Престін2, https://orcid.org/0000-0001-5985-7939
І.О. Шевчук3, https://orcid.org/0000-0003-1140-373X
1 Інститут прикладної математики і механіки НАН України, Слов’янськ
2 Інститут математики університету Любека, Німеччина
3 Київський національний університет ім. Тараса Шевченка
E-mail: oleksiy.dovgoshey@gmail.com, prestin@math.uni-luebeck.de, shevchuk@univ.kiev.ua
ХАРАКТЕРИЗАЦІЯ ЦІЛИХ ФУНКЦІЙ НЕРІВНОСТЯМИ ТИПУ БЕРНШТЕЙНА
Нехай — це множина усіх цілих функцій, що задані на комплексній площині . Розглянемо клас EX
усіх Банахових комплексних просторів X таких, що X ⊇ . Для X ∈ EX і g X∈ позначено
, ( ) inf { : },n X nE g g p p= − ∈Π де nΠ — це множина всіх многочленів степеня не вище n. Описано усі X ∈ EX ,
для яких співвідношення 1/
,lim ( ) 0( ) n
n X
n
E g
→∞
= виконується тоді і тільки тоді, коли g ∈ .
Ключові слова: теорема Бернштейна, ціла функція, наближення многочленами, базис Шаудера, трансфі-
нітний діаметр.
|