ISSN 0001-4346, Mathematical Notes, 2011, Vol. 89, No. 4, pp. 598–601. © Pleiades Publishing, Ltd., 2011.
Original Russian Text © S. A. Imomkulov, Z. Sh. Ibragimov, 2011, published in Matematicheskie Zametki, 2011, Vol. 89, No. 4, pp. 637–640.
SHORT
COMMUNICATIONS
Pluripolarity of the Graphs
of Gonchar Quasi-Analytic Functions
S. A. Imomkulov
1*
and Z. Sh. Ibragimov
2**
1
Navoi State Pedagogical Institute, Uzbekistan
2
Urgench State University, Uzbekistan
Received November 10, 2009; in final form, June 11, 2010
DOI: 10.1134/S0001434611030321
Keywords: pluripolar set, plurisubharmonic function, rational function, quasi-analytic func-
tion, logarithmic capacity, Green function.
Let f be a function defined and continuous on a closed interval Δ=[a, b] of the real line R, and let
ρ
n
(f ) be the minimal deviation of f on Δ from rational functions of degree ≤ n,
ρ
n
(f ) = inf
{rn}
‖f − r
n
‖
Δ
,
where ‖·‖
Δ
is the uniform norm and the infimum is taken in the class of all rational functions of the form
r
n
(x)=
a
0
x
n
+ a
1
x
n−1
+ ··· + a
n
b
0
x
n
+ b
1
x
n−1
+ ··· + b
n
.
As usual, let e
n
(f ) be the minimal deviation of f on Δ from polynomials of degree ≤ n. Obviously,
ρ
n
(f ) ≤ e
n
(f ) for each n =0, 1, 2,... .
Gonchar [1], [2] proved that the function class
R(Δ) =
f ∈ C (Δ) : lim
n→∞
n
ρ
n
(f ) < 1
possesses one of the most important properties of the class of analytic functions. Namely, if
lim
n→∞
n
ρ
n
(f ) < 1
and f (x)=0 on a set E ⊂ Δ of positive logarithmic capacity, then f (x) ≡ 0, x ∈ Δ.
By analogy with the class
B(Δ) =
f ∈ C (Δ): lim
n→∞
n
e
n
(f ) < 1
of Bernstein quasi-analytic functions (see [3]), the class R(Δ) will be called the class of Gonchar
quasi-analytic functions.
It is well known that the analytic functions on Δ are characterized by the condition
lim
n→∞
n
e
n
(f ) < 1
(Bernstein’s theorem). It follows that the class A(Δ) of analytic functions on Δ is a subclass of the
classes B(Δ) ⊂ R(Δ); i.e., A(Δ) ⊂ B(Δ) ⊂ R(Δ). Clearly, A(Δ) = B(Δ). Gonchar [1] showed that
*
E-mail: sevdi@rambler.ru
**
E-mail: z.ibragim@gmail.com
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