CC BY
7
УДК 517.53
On Interpolation in the Class of Analytic Functions in the Unit Disk with Power Growth of the Nevanlinna Characteristic
Faizo A. Shamoyan* Eugenia G. Rodikova^
Bryansk State University, Bezhitskaya, 14, Bryansk, 241036 Russia
Received 25.12.2013, received in revised form 23.02.2014, accepted 18.03.2014 In this paper we solve the interpolation problem for the class of analytic functions in the unit disk with power growth of the Nevanlinna characteristic under the condition that interpolation nodes are contained in a finite union of Stolz angles.
Keywords: interpolation, holomorphic functions, the Nevanlinna characteristic, Stolz angles.
Introduction
Let C be the complex plane, D be the unit disk on C, H(D) be the set of all functions, holomorphic in D. For any a > 0 we define the class S^0 as:
Sa H f e H (D): T (r,f ) < Cf
(1 - r)a
where Cf > 0 is a positive constant, depending on the function f, r e [0,1), T(r, f) =
i r ■
— ln+ |f (relv)|df is the Nevanlinna characteristic of the function f, ln+ |a| = max(0, ln |a|), J-n
a e C (see [1]).
It is well known that if f e S0°, then
M (r, f) = ^ |f (Z)| < ex^ (Tf+T } (1)
for all a > 0, cf > 0 (see [1]).
It is clear that if f e S^0 and {ak}+=J is a sequence of points from the unit disk, then the operator R(f) = (f (aT), ...,f (ak),...) maps the class S^0 into the class of weighted sequences
la = {Y = {Yk}+=°i : |Yk| < exp(1 _ ^|)a+1, A > o} .
In this article we answer the question under what conditions on the sequence {ak}+=1 the operator R(f ) maps the class S™ onto the class la.
Definition 1. A sequence {ak}+=1 is called interpolating for S™, if R(S™) = la.
* shamoyanfa@yandex. ru tevheny@yandex.ru © Siberian Federal University. All rights reserved
Let us note that interpolation theory has become intensively developed since the fundamental work of L. Carleson (see [2]) about interpolation in the class of bounded analytic functions. The term of "free interpolation" was first introduced in [3]. The interpolation problem in subclasses of the bounded type functions N was investigated there. This problem in the Nevanlinna and Smirnov classes was solved in [4,5]. These questions in the Hardy and Bergman spaces was studied in works [6,7].
The paper is organized as follows: in the first section we present the formulation of main result of the article and prove some auxiliary results, in the second section we present the proof of main result.
1. Formulation of main result and proof of auxiliary results
To formulate and proof the results of the work we introduce some more notation and definitions:
For any fl > -1 we denote np(z,ak) as M. M. Djrbashian's infinite product with zeros at points of the sequence {ak }+=1 (see [8]):
( Z \
np(z, ak) = JJ ( 1--j exp(-Up(z, ak)), (2)
k=i ^ ak '
where
U ( ) 2(fi +1) f1 fn (1 - P2)P In I1 - CI d e (3)
UP(z,ak) = Jo J_n (1 - zpe-iO)P+2 pdpdd- (3)
We denote na,n (z,ak) as infinite product np (z,ak) without n-th factor. As stated in [8], the infinite product np(z,ak) is absolutely and uniformly convergent in the unit disk D if and only if the series converges:
]T(1 -K|)p+2 < (4)
k=i
Let us remark that the product np (z,ak) appear naturally in the integral representations of the holomorphic functions by the kernel
K(Z z) = a + 1 (1 -IZI2)" z z G D Ka(z,z)= n (1 - Zz)a+2 ' Z,z G D'
as there is the Blaschke product in the integral representation of the bounded type functions by the Poisson-Jensen formula (see [1,9]).
If fl = p G Z+ then product (2) takes a form (see [8]):
, ^ ak (ak - z) ^ 1/1 - |ak 3
np(z,akHII^—=—exp^- i—•
k=i 1 - akz 3=1 ^ 1 - akz j
Definition 2. The part of the angle with vertex at the point e%e, less then n and contained in D, whose bisector coincides with radius connecting the center of the disk and the point e%e is said to be the Stolz angle with vertex at the point e%e, i.e.
rs(e) := {z G D : | arg(eie - z) - 0| < ^}
where 0 ^ 6 < 1.
The main result of this article is the proof of the following theorem:
Theorem 1.1. Let {ak }+=1 be the arbitrary sequence of complex numbers from D, which is contained in a finite union of Stolz angles, i.e.
n
{ak }cU rs (es), (5)
S = 1
with certain 0 < 6 <
a +1
The following statements are equivalent:
i) {ak}+= is an interpolating sequence in S™, a > 0,
ii)
c
n(r) = {cardak : |afc| <r < 1} < — a+1, (6)
for some c > 0;
, -M
|na(an,ak)| > ex^--¡-—j, (7)
for some M > 0 and all ft > a — 1.
For presentation of auxiliary result we need also the O. Besov class Bf™ on the unite circle (see [10, p. 151]). Let 0 < s < 2; function ^ integrable on the unite circle belongs to the class Bf™ if and only if
sup i —---^-"--dd) < +cx.
0<t<1 I J-n |t|S J
The proof of the theorem is based on the following statements.
Theorem A.(see [11]). The class S^ coincides with the class of analytic in D functions represented as
( i rn é(ée) 1
f (z) = cxz^p(z,ak )expj — j ^ (l _ ze_¿e^+2 dd j , z G D
(z,ak Î (1 +2 dd}, z G D, (8)
for all fi > a — 1, where ^>(el6) is real-valued function from the O. Besov class 0+ 1, X G Z+, c\ G C, and the sequence {ak j+0 satisfies the condition
n(rf) < (l-^
for some cf > 0.
Here and in the sequel, unless otherwise noted, we denote by c, cr,cn(a, fl,...) some arbitrary positive constants depending on a, fl,whose specific values are immaterial. It is clear that the space la coincides with the space of sequences {jk}+=1 such that
sup{(l -K|)a+1 ln(l + |YkI)} <
k>1
where {ak }+=°1 C D.
k=i
For the further exposition of the results we introduce metrics in spaces S™ and la as follows:
Vf, g g S
a
n
a.
PS- (f, g)= sup \ (i _ r)a I ln(l + If (rei0 ) _ g(rel ) |)d^ ,
0<r<1
Va = |afc}, b = {bk} e la
Pia (a, b) = sup {(1 - K |)a+1 ln(1 + |ak - bfc|)} . k>1
It is easy to check that SO0 and la are complete metric spaces with respect to these metrics, and the space SO° is invariant regarding the shift.
Lemma 1.2. If the operator R(f) = (f (a1),...,f (an),...) maps space SO° onto space la, then
gn(z)}+=i e SO°
there exists the sequence of the functions {gn(z)}+=1 e SO° such that
suppS—(gn, 0) ^ C, C > 0
and
(n) (n) I 0, for all k = n,
gn(«k) = Y( , where Y(n) = < _1
exp(1 -|ak|)a+1, for k = n
where k, n = 1, 2,...
Proof. Let SL, L > 0, be the set of the sequences c = {ck }+= from the space lO such that ck = F(ak), k =1,2,... for certain function F e satisfying condition pS—(F, 0) < L, that is
sup <J (1 - r)a / ln(1 + |F(reie)|)d^ < L.
0<r<1 [ J-n J
By hypothesis, R(S^°) = la, equivalently to lO = U+=1SL. Let prove that the sets SL are closed for all L in lO, in this terms if c(m) = {ckm)}+=°1 e SL and pl (c(m), c(0)) ^ 0 as m ^ then c(0) e SL.
By assumption, ckm) = Fm(ak), k = 1, 2,..., Fm e SO° with
PS- (Fm, 0) < L (9)
and pla (c(m), c(0)) ^ 0 as m ^
In this notation, we need to prove that there exists function F0 e such that F0(ak) = c k = 1, 2,... and pS- (F0,0) < L.
From (9) it follows that for any m = 1, 2,...
ln+ |Fm(reie)| < C(L)
(0)
(1 - r)
O+1 '
where C(L) is independent on m.
By Montel's theorem we can choose the subsequence of functions {Fmfc(z)}, uniformly convergent to function F0 on compact subsets of the unit disk. Let check that F0 e SO°. Using inequality (9), we get:
sup ((1 - r)a / ln(1 + |Fmfc(reie)|)d»l < L.
;r^R<1 I J-n J
0<r^R<1
Taking in this inequality the limit as k ^ we obtain:
sup |(1 - r)a / ln(1 + |F0(reie )|)dtfl < L.
0<r^R<1 [ J-n J
Whence guiding R ^ 1 — 0, we have:
(Fo, 0) < L. (10)
Thus, Fo e
Since Fmfc(z) ^ F0(z) as k ^ for all z e D, we have c"mfc^ = Fmfc(an) ^ F0(an) = c"0) as k ^ for all |an| < 1, n = 1, 2,.... Taking into account the estimate (10), we conclude: c(0) = |ci0)}+=1 e SL. Thus, we prove that the set SL is closed in la and la = U+=1SL.
By Baire's theorem there exists a number L0 such that the set SLo includes the ball with the center at one of its interior points, for example,
B(F0(a„),d) := {c = |cfc} : (c, F0(an)) = sup(1 — |afc|)a+1 ln(1 + |cfc — F0K)|) < d},
fc>i
where the sequence {F0(an)} = {c"0)} is the center of the ball, d > 0 is its radius.
It means that for any sequence c = {ck} e B there exists a function F e S^ such that F (afc ) = cfc, k =1, 2,....
So, if (7,0) < d for some 7 = {7^}, then there exists a function g e S0° such that (g, 0) < 2L0 and g(ak) = for all k. It is sufficient to put Yk = ck — F0(ak), k = 1, 2,..., g(z) = F(z) — F0(z), where F is the function with the above properties.
Let Y(n) = {7kn)}, n e N, where 7(n) = 4"° — F0K) k = 1,2,..., 4"° e B. Then
Pia(Y(n), 0) < d. We put 4 = Fo(afc) for k = n, ck"} = Fo(afc) + exp --.—p—r for
(1 - |«fc|)"-1
ckn) = F0(afc) for k = n, 4" " ' ' 1
k = n. Arguing as above, we see that there exists a function gn(z) = Fn(z) — F0(z) such that (gn, 0) < 2L0 and g„(afc) = 7k" =0 for k = n, g„(afc) = = exp -jakjj^+I for k = n. The proof of Lemma 1.2 is complete with the constant C = 2L0.
Remark 1.1. The idea of proving Lemma 1.2 is adopted from P. Koosis (see [12, p. 200]), who first applied this for solving the interpolation problem in the class of the bounded analytic functions.
Lemma 1.3. (see [13]) If members of the sequence {ak}+= are contained in a finite union of
n 1
Stolz angles, i.e. {ak} C |l r(0s) for certain 0 < S < --, then for any function
s=i a +1
nC
g(z) = = exp (1--¿e )a+1' z e D, a > —1 the following estimate is valid
C
> c0 exP (1 — |a |)a+1, s = 1, 2,...,n, (11)
where c0, C are some positive constants.
2. Proof of main result
Let prove the implication i) ^ ii).
We assume that {ak}+=I e D is an interpolation sequence in the class a > 0, i.e. for any {Yk} e la there exists a function f e S^0 such that f (ak) = , k = 1, 2,....
Let consider the sequence {7^}+=I: Y1 = 1,72 = Y3 = ... = 0. Evidently, {7^}+=I e la. Since {ak }+=2 is zero-sequence for the function f e a > 0, we have
n(r) ^
(1 — r)a+1
by Theorem A. The estimate (6) is established.
In order to show (7) we fix n e N and take the sequence {Ykn)}+=i as follows: y kn) =0, k = n,
Yin) = exp —-:--—-T, k = n. By Lemma 1.2 there exists a function gn e S?° such that
Ik y _ |a k |)a+;L' J rna
Ps~ (gn, 0) < C and gn(ak) = Y(n) for all k =1, 2,..., where the constant C > 0 is independent on n. In particular, gn(an) = Yr?^. According to Theorem A, any function gn e S^0, a > 0 can be represented as
gn(z) = cXnzXnnß,n(z,ak)exp{hn(z)}, z e D,
1 ^ (eie)
where hn(z) = — J-n (1 _ nze_ie )ß+2 d0, ß>a _ L
So,
|gn(an)| = |Ynn)| = exP (1 - | 1 |)a+1 = Ic\IIanlXlInp,n(an,ak)|1 exP{h(an)}|. (1 |an|)
Since exp{hn(z)} G then taking into account the estimate (1) we have:
1 c2 |gn(«n)| = exp —-:-r—+i < ci|nß,n(«n, ak)| exp-
(1 _ |«n|)°+^ ^ (i _ |£
)a+1 -
where c2 is independent on n according to Lemma 1.2. From the last inequality we obtain:
- M1
|np,n(an, ak)| > exp —--j, Mi > 0. (12)
(1 |an|)
We show, that (12)^(7). Really, differentiating the function np, we get:
hTO
nß(z,ak) = _-1 exp(_Uß(z,aj)) _ ^ 1 _ exp(_Uß(z,aj))Uß(z,ajx nßj(z,ak),
where Uß(z,aj) defined by (3).
/ a \
Since nßj(an,ak)= n 1--- exp(_Uß(an,ak))=0 for all j = 1, 2,..., j = n, we
k=i,k=j V ak J
have:
1
|np (an,ak )| = |-11 exp(-Up (an, an))| Inp,n (an,ak )|. (13)
|an |
Using now estimate (12), from (3) and (13) we obtain:
. -M
|np^ ak)| > exP ty-—^rj, M >
(1 |an|)
Thus, (12)^(7). The implication i) ^ ii) is established.
Now we prove that ii) ^ i). Suppose that {ak }+=1 be the arbitrary sequence of complex numbers from D, which is contained in a finite union of Stolz angles, and the estimates (6), (7) are valid. Let us show that there exists a function ^ G S0° such that ^(ak) = Yk, k = 1, 2,... for each {Yk }+=°1 G la.
We construct a function ^(z) as follows:
V^ Y nß (z,aj) 1 1 _Kh f(z) (14)
^(z) = Yk-,-T —f--^ 77—^ (14)
k=1 (z _ ak) nß(ak, a.j)\1 _ akzj f (ak)
with ß > a — 1, m > a + 1, and
n
I = TT exp
(1 — )«+i
' C
f (z)^exp(i — ^ )a+i , z g D (15)
s = 1
It is obvious, that $(a„) = y„, n =1, 2,...
Now we need to prove that function ^(z) is analytic in D and ^ e First, from the estimate (6) we conclude:
]T(1 -Kl)m < (16)
k=1
for all m > a +1. Taking into account the convergence of the series (16) and Lemma 1.3, we obtain that the infinite product ng (z, aj) and the series (14) are absolutely and uniformly convergent in D.
Now we get an upper estimate for the function |^(z)|. Since {Yk}+=1 G 1a and condition (7) is valid, we have:
(z,aj)| 1 f 1 -|afc| \m |f(z)i
j^(z)l < ¿1l7k| |z —'afc | |^(afc, a j )| U — akzj |f(afc)| <
< c V cxp A ^(z,aj)l cxp M ( 1 — aI NT JfM.
" P(1 — |afc|)«+1 |z — afc | P(1 — |afc |1 — a*z|J |f(afc )|"
We estimate the factor (z, aj)|. Using the well-known estimate for the Djrbashian product
|z — afc |
(see [13]):
+ ^ ( 1 — | | N ,8+2
ln+ |n,,fc(z,aj)| < jT-fkZ| , (17)
fc=1 \| k \/
we get:
(z, aj)| 1 |afc — z| , TT / \w ^ ~ l^ß.fc(z, aj) r|nß,fc (z,aj -—exp( —Uß(z,afc))| < Cß-!^'' JJ
|z — afc | |z — afc | ' ' |afc | ' |1 — afc z|
|nß,k(z,aj)| Cß_ exp fCß ^ ( 1 — |an| X
|1 — afcz| |1 — afcz| y n^ V |1 — anz|
for all ß > a — 1. Therefore
,T, M / f 1 — |an| A^+2\ ,,, „ A + M 1 (1 — |afc|)r
i™ <exp (ra) J ^a—^« .
Now we consider the last factor in the product:we obtain the following estimate
A + M 1 (1 — K |)m
Cxp(1 — |afc|)a+1 ' |f(afc)||1 — akz|m+1.
We split the sum into n parts:
" A + M 1 (1 — |afc |)m
exp
s=i afc£r,(,.) (1 — |ak|)a+1 |f (afc)| |1 — akz|m+1'
n 1
Since {ak} c |l rs (es) for certain 0 < 6 < -, we can apply Lemma 1.3 for each part of the
s=1 a +1
sum. Thus we have: n
^^ ^^ A + M - C (1 - I I )m
exP ■
s=1 afceri(0s)
Choosing the positive constant C such that A + M - C < 0, we obtain the following estimate:
V exP A + M - c (! -Kl)r
^ P(1 -|afc|)a+! ^ |1 - akzlm+1 ■
for all k = 1,2, ■■■. Thus we have:
A + M - C
eXP(1 -K l)«+ ^ 1
)l ^ ( 1 -Kb^ |f( )| ~ ^ (1 -|akl)r ^(z)| < exM x lf (z)| x cß •
^ n=1 J1 - anzD J MV 71 p f=1 |1 - akz|m+1'
Taking into account the convergence of the series (16), we have:
(1 - |ak|)m , c V^(1 -| |)m ^ c1
Z^ |1 - a-zIm+1 ^ (1 -|zI)m+^Z^(1 |ak|) ^
kT] |1 _ äkz|m+^ (1 _ |z|)m+] ' ^ " (1 _ |z|)m+]
for all m > a + 1.
The estimate of function |^(z)| takes form:
( / 1 _ | | \ ß+^
^(z)| < ex^cß £ (j x |f(z)| x ^^. (18)
1 !'n C
Now we show that ^(z) e S£°, in this notation T(r, = 2~ J ln+ |^(reie)|d0 < (1-^,
where a > 0, C > 0. Using (18), we get
+ ^ fn / 1 _ | | \ ß+2 /.n
T f. *) < cß • d0 + L ln+f (re" >"» +2n ln
We estimate each of summands in this sum separately. As established in [11] (see also [14]),
+ TO „ n / I IN ß+2
E/ in1 _|an'J d0 < —, (19)
|1 _ änrei0 y ^ (1 _ r)a v ;
for all ß > a _ 1.
Further, from (15) we have:
/n n fn
ln+ |f (reie)|d0 /
C
|1 — rei(0-0s)|a+1 ' Applying elementary estimate, we obtain:
/n
n ln+ |f (re")d < ^^. (20)
Combining (19), (20), we obtain that function ^(z) belongs to the class SO. This shows that ii) - i).
The proof of Theorem 1.1 is complete. □
Remark 2.1. Note that if a = 0, then condition (5) is necessary, as established in [4].
The work is supported by the Russian Foundation for Basic Research, project 13-01-97508
References
[1] R.Nevanlinna, Eindeutige analytische Funktionen, 2nd ed., Springer-Verlag, 1953.
[2] L.Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math., 80(1958), 921-930.
[3] S.A.Vinogradov, V.P.Havin, Free interpolation in and in some other classes of functions. I, Zap. Nauchn. Sem. LOMI, 47(1974), 15-54 (in Russian).
[4] A.G.Naftalevic, On interpolation by functions of bounded characteristic, Vilniaus Valst. Univ. Mokslu Darbai. Mat. Fiz. Chem. Mokslu Ser., 5(1956), 5-27 (in Russian).
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[10] E.M.Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 1970.
[11] F.A.Shamoyan, E.N.Shubabko, Parametrical representations of some classes of holomorphic functions in the disk, Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, 113(2000), 331-338.
[12] P.Koosis, Introduction to Spaces, Cambridge University Press, 1998.
[13] F.A.Shamoyan, M. M. Dzhrbashyan's factorization theorem and characterization of zeros of functions analytic in the disk with a majorant of bounded growth, Izv. Akad. Nauk ArmSSR, Matematika, 13(1978), no. 5-6, 405-422 (in Russian).
[14] F.A.Shamoyan, E.N.Shubabko, Introduction to the theory of weighted Lp-classes of mero-morphic functions, Bryanskiy Gosudarstven. Universitet, Bryansk, 2009 (in Russian).
Об интерполяции в классах аналитических в круге функций со степенным ростом характеристики Р. Неванлинны
Файзо А. Шамоян Евгения Г. Родикова
В статье получено 'решение интерполяционной задачи в классе аналитических функций в единичном круге, характеристика Р. Неванлинны которых имеет степенной рост при приближении к единичной окружности, при условии, что узлы интерполяции принадлежат конечному числу углов Штольца.
Ключевые слова: интерполяция, аналитические функции, характеристика Р. Неванлинны, углы Штольца.
