On interpolation in the class of analytic functions in the unit disk with the Nevanlinna characteristic from Lp-spaces Текст научной статьи по специальности «Математика»

УДК 517.53

On Interpolation in the Class of Analytic Functions in the Unit Disk with the Nevanlinna Characteristic from Lp-spaces

Eugenia G. Rodikova*

*

Bryansk State University Bezhitskaya, 14, Bryansk, 241036

Russia

Received 02.11.2015, received in revised form 08.12.2015, accepted 12.01.2016

In this paper we solve the interpolation problem for the class of analytic functions in the unit disk with the Nevanlinna characteristic from Lp-spaces under the condition that interpolation nodes are contained in a finite union of Stolz angles.

Keywords: interpolation, holomorphic functions, the Nevanlinna characteristic, Stolz angles. DOI: 10.17516/1997-1397-2016-9-1-69-78.

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 all 0 < p < a > —1 we define the class Sp as (see [14]):

where T(r, f) = — ln+ \f (rel<p)\d^ is the Nevanlinna characteristic of the function f, 2n J-n

ln+ \a\ = max(0, ln \a\), a G C (see [8]).

Note that Sp-classes are a natural generalization of the Nevanlinna-Djrbashian classes. In this paper we investigate the questions of interpolation in Sp-spaces. In solving the problem of free interpolation, that is, with minimal restrictions imposed on the interpolated function, it is important to find a natural class to which the restriction of on the interpolation set to belong. We denote it ip.

In [9] it was set that if f G Sp, then

where M(r,f) = max \f (z)\.

\z\=r

It is clear that if f G Sp and {akis a sequence of points from the unit disk, then the operator R(f) = (f (ai),..., f (ak), ■■■) maps the class Sp into the class of sequences

1

(1)

* [email protected] © Siberian Federal University. All rights reserved

In this article we answer the question under what conditions on the sequence {a.k}+=°l the operator R(f) maps the class SS onto the class IS.

Definition 1. A sequence {ak}+=1 is called interpolating for Sa if R(SS) = lS•

Let us note that the solution of interpolation problems in the various classes of analytic functions has been widely discussed by Russian and foreign scientists: A. G Naftalevic [7], H. Shapiro and A. Shields [11], S.A.Vinogradov, V.P.Havin [16], M.Djrbashian [4], N.A. Shirokov and A. M. Kotochigov [6], K. Seip [12], A.Hartmann [5], V. A. Bednazh and F.A. Shamoyan [1] and etc. The fundamental result in this area belongs to the L. Carleson [2]. This work continues the research started in [10] in solving the interpolation problem in the classes of analytic functions in the unit disk with power growth of the Nevanlinna characteristic.

The paper is organized as follows: in the first section we present the formulation of main result of the article and give 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 ¡3 > — 1 we denote n"(z, ak) as M. M. Djrbashian's infinite product with zeros at points of the sequence {ak}+=1 (see [3]):

( z A

ß(z, ak) = JJ ( 1--j exp(-Uß(z, ak)),

¡.—1 V ak /

k=i where

TT ( ) 2(3 + 1) f1 f (1 — P2)"ln I1 — € IM d (2)

U?(z,ak) = ^^ j0 ddpdp- (2)

We denote np,n(z, ak) as infinite product (z, ak) without n-th factor. As stated in [3], the infinite product (z,ak) is absolutely and uniformly convergent in the unit disk D if and only if the series converges:

£(1 — \ak\f+2 < k=1

Definition 2. The angle of the nS, 0 < 5 < 1, contained in D, with vertex at the point e%e and with bisector re10, 0 ^ r < 1 is said to be the Stolz angle (0)•

For any 0 < p < to we denote by BS p the O. Besov space on T of 0 < s < 2 order. The main result of this article is the proof of the following theorem:

Theorem 1.1. Let a > —1, 0 < p < {ak}+=1 be the arbitrary sequence of complex numbers from D, which is contained in a finite union of Stolz angles, Le^

{ak} C (j rs(9s),

s=l

p

with certain 0 < 5 < - . .

2(a + p +1)

The following statements are equivalent:

i) {ak}+=1 is an interpolating sequence in SS;

ii) the series converges:

p

E 2k( n+p+i) < (3)

k=i

nk = card|zk : \zk\ < 1 — —k|, k = 1, 2,..., and there exists a sequence {ek}+=i, e(k) ^ 0, k ^ such that

\n'p(ak)\ > exp --—£(k)a+±+ -, , (4)

for all ß >

(1 -Ki)^

a + 1

p

The proof of the theorem is based on the following statements.

a + 1

Theorem 1.2 (see [14, 15]). Let a > —1, 3 > -. The following assertions are equivalent:

p

1) f G Sp;

2) function f allows the following representation in D

rf ^ A / ^ i 1 } ^(eie)d0 I f (z) = c\z (z, zk) exp I — J (1 — e-iezy+i I , z G D,

where {zk}+=i is an arbitrary sequence from D, satisfying the condition

Enk

2k(p+a +1) <

2k(p+a +i) k=1

a +1

$ G Bs = 3--—, A G Z, cA G C.

p

Here and in the sequel, unless otherwise noted, we denote by c, ci,..., cn(a, ¡3,...) some arbitrary positive constants depending on a, 3,..., whose specific values are immaterial.

For the further exposition of the results we introduce metrics in spaces Sp and lpa as follows:

p(f, g) = Jo (1 — r)a(y In (1 + \f (reie) — g(reie)\) d6^ dr for 0 <p < 1,

P(f,g) = ( f (1 — r)a(f In (1 + \f (reie) — g(reie)\) d^' dr^" for p> 1. for all f,g G Sp.

:(a, b) = sup|(1 - iaki)S++1 ln(1 + iak - bki)j •

k>i ^ '

for all a = {ak}, b = {bk} G ip. It is easy to check that these spaces are complete metric spaces with respect to these metrics.

The following statement is valid:

Lemma 1.3. If the operator R(f) = (f (ai),...,f (an),...) maps space Sp onto space ip, then

gn(z)}+=

there exists the sequence of the functions {gn(z)}+^l=y'i G Sp such that

sup pspa (gn, 0) < C, C> 0

n>1

and

( ) ( ) I 0, for all k = n,

gn K) = lu , where Ik = \ exp-, for k = n,

I (i—|«fc|) p +1

where k,n = 1, 2,..., S(k) = o(l), k ^ +to.

The proof of Lemma 1.3 repeats the reasoning conducted in the proof of the corresponding assertions from [10] with a+l + 1 index.

2. Proof of main result

Let prove the implication i) ^ ii).

We assume that {ak}+=1 G D is an interpolation sequence in the class Sa, (a > -1, 0 < p < to), i.e. for any {jk} G ¡a there exists a function f G Sa such that f (ak) = jk, k = 1, 2,....

Let consider the sequence {jk}+=1: Y1 = 1, 72 = 73 = ... = 0. Evidently, {jk}+=1 G ¡a. Since {ak}+=° is zero-sequence for the function f G Sa, then we have an estimate (3) as follows from Theorem 1.2.

In order to show (4) we fix n G N and take the sequence Ykn) = 0, k = n, Ykn) = S(n)

exp-+—, k = n, where S(n) —> 0, n —> +oo. By Lemma 1.3 there exists a func-

(1 -\an\) ^+1 n

tion gn G Sa such that pSp (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) = Ynn). According to Theorem 1.2, any function gn G Sa can be represented as

gn(z) = cXnzXnn^n(z,ak)exp{^n(z)}, z G D,

h ut^ 1 f ^ ) q ^ a + 1

where hn{z) = -J ^ (I—iT+T^ ß > ~' So,

,(n)|____p S(n)

\gn(an)\ = Iy^I = exp-5+r-T = \cxn \\an\Xn\nß,n(an,ak )| | exp{h„(a„)}|,

(1 - \an\) p +

where S(n) ^ 0, n ^ +to.

Since exp{^n(z)} G Sa, then taking into account the estimate (1) we have:

i / M S(n) V (n) \gn(an)\ = exp-a+m ^ C1\nPn(an,ak)\exp-a±X77,

(1 - \an\) p + (1 - \an\) p +

where the sequence v(n) ^ 0, n ^ +to, is chosen so that v(n) > S(n). From the last inequality we obtain:

\nP,n(an)\ > exp-^L + 1 , (5)

(1 - \an\) p +

where e(n) = (v(n) — S(n)) is positive infinitesimal sequence.

The same manner as in [10], it can be shown that (5)^(4). Thus, 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 (3), (4)

are valid. Let us show that there exists a function ^ e Sa such that ^(ak) = jk, k = 1,2,... for each [lk 1+=°! e ll, where [lk 1+=°! e laa, i.e.

6(k)

Yk =exp-, (6)

(1 "K\)~

S(k) ^ 0, k ^

We denote by {qk} the sequence with general term qk = Sk + ek, k = 1,2,..., where {ek} is an infinitesimal sequence from the estimate (4), {¿k} is an infinitesimal sequence from the estimate (6).

We construct a function ^(z) as follows:

hOO

^(z)=Y" k (z,aj) 1 f1 \ak h f (z) (7)

^ (z - ak) nß (ak ,aj)\1 - akz J f (ak)'

a+1 a+1 where ß > -, m >--+ 1,

f (z) = expV y qm-(1 pmT +1 , 2 e D,

W V ^ k (1_ zo e-ies f+ S+1+ 1

s=1 m=1 (1 zpme )

- a +1

where 0 < 3 <--+ 1, qk ^ 0, k ^ Here the sequence {qk} is depending on the

P

sequence of nodes {ak}, therefore the function f depends on k, that is f = fk.

3 a + 1

For brevity ¡' = 3 +---+ 1. It is obvious, that $(a„) = y„, n = 1, 2,....

P

Without lost of generality we assume that interpolating nodes are contained in the Stolz angle r (0). Then we have

+ (1 - P 2 )<® f (z) = fk(z) = exp £ qm (1 " , 2 e D.

1 - zpme-ie)ß'

m=1 v '

Let us show that ^ e Sa. First we estimate f (ak) in the angle of r(0).

+ (1 - P 2)/3

\f (ak)\ = exp ^ qm^-^-Pm -_.e , =

^ (1 " ak Pm e lU )P

m=1 v '

+ ~ qm(1_ P 2y§ ^(1 " 0~kPme-ie f

exp£ qm(1 - Pm2)^ 7 ß .

11 - akPme \2ß

But

K(1 - akPme-ie)ß' = 1 - rkPme-i(ipk-e)) =

1 - Pm rk + Pmrk (1 - e-i(^-e))) ß = 1 - Pmrk + f^k (1 - e-i^k-e)) )

= (PmrkP)ß' 1 - Pmrk + e-iAß , PmrkP

n5

where ak = rkeilfk, (1 - e-i(vk-e)) = Pe-iv, \p\ < y < -ß■ So we have K(1 - äkPme-ie)ß > c1 (PmrkP)ß', C1 > 0, by Lemma, established in [13].

From the other hand,

therefore

\1 - e-i(tpk-9)\ß' = 2ß' sinß' ^= 2ß' sinß' |,

Cl(pm rk )ß' 2ß' sinß' 2ß' sinß' f

5R--^77 > -----7T7 >

•(1 - akpme-ey ^ -pmrk)2 + 4sin^e-^ p^' (1 - pmrk)2ß' • ( 1 + (—T^) Since {ak} c r(-), we have

sin(

(1 - rk)

Hence,

< C.

sß_1_ > C(ß)

(1 - akpme-ie)ß' " (1 - pmrk)ß' .

Thus we have the following estimate for f (ak) in the angle of r(0):

(1 - n 2)3

\f (ak)\ * expc(p') £ qm (1 _ Pm )p,.

m_1 (1 rkPm)

1

We will continue this estimate. For this purpose we split the interior sum into two parts:

s = Eqm(S1_ Pm2w = E (...)+ E (...) = si(*) + w

= 1 (1 rkPm)ß (1-pm)^(1-rfc) (1-pm)>(1-rfc)

We estimate each of them separately. Let estimate the sum S2.

■ (1 — Pm")3 (1 — rk Pm)3

Q /] \ V"^ m (1 pm )ß .

S2(k) = qm (1 _ rin ß >

(1-pm )>(1-rk) -

> V qm-1-- >

Hk (1 _ 2 )ß'-ß ^ pm<Tk "mj

> 1 E qmv, ,+1

2 pm<Tk (1 - Pm) p ^

Without lost of generality we assume that 1 — pm = qm, 0 < q < 1, and we find that the sum

above converges to S2(k) = —^-3--= o(1), k ^ +to.

q( 3'—¡3) — qk Now we get lower estimate for the sum S1 .

S1 (k) = (1 £1 ) c (—pm^ *

(1 — PmX,(1— rk)

(1 Pm 2 ) ß

Pm^Tk " (1 - rkPm) ~ + 1(1 - rkPm)ß

V^ m_(1 - pm )_ .

> qk ~ 1 1 ^ß >

> 1 ^ m (1 - Pm2)ß

> (1 - ri ) ^ + 1 pf^Tk qk (1 - rk Pm)ß .

Since

(1 - Pm2)*

E qm tP^ <E q)m,

pm^rk (1 rk Pm) m=1

then Si(k) > -—, where ¿i(k) « —, Si(k) ^ 0, k ^

(1 _ r2k )p +1 1 - qk

From the estimation for Si, Sk we finally set:

S ^-+ Sk(k),

(1 _ rk)S+L+1 2U'

whence we conclude that

So(k)

\f (ak)| > exp--"2 a+1 +i, (8)

(1 - rk)

qk

where S0(k) = sup(k), j = 1, 2. It is obvious, that S0(k) .

j 1 _ qk

From condition (3) we conclude (see [15])

]T(1 _|«k|)m < (9)

k=1

a + 1

for all m >--+ 1. Taking into account the convergence of the series (9), we obtain that the

P

infinite product (z,aj) and the series (7) are absolutely and uniformly convergent in D. Now we need to prove that function ^(z) is analytic in D and ^ € Sa. Now we get an upper estimate for the function |^(z)|. Since {jk}+=1 € la and condition (4) is valid, we have:

Mß (z,aj)| 1 ( 1 - \ak n m \f (z) ^^RO^OTi liT^i) -

k= Iz ~ |"ßV<Jk, M1 - ak z\) \f (ak )\

V exp S(k) \nß (z, aj )\ exp £(k) ( 1 — \ak\ \ m f)

k=1 (1 — \ak\) +1 \z - ak \ (1 — \ak\) O^1 V\1 - W \f(ak

<

We estimate the factor ^J3(z, aj)|. Using the well-known estimate for the Djrbashian product

lz _ ak |

(see [13]):

1 _ ^-k^ | |1 _ akz^

,»+kß k (z, aj )K , £( ¿-M)'

k=1

we get:

\nß(z,aj)\ 1 \ak - z\ _ \nß,k(z,<

:\nß,k(z,aj-—L\exp(-^ß(z,ak))\ < Cß-!^'

\z - ak \ \z - ak \ ' ' \ak \ ' \1 - akz

\nß,k(z,aj) \ ^ _Cß_ex^c ß ( 1 - \ a"i

( f 1 - \ aw \ N ß+2\

r n=1t ^^)

for all ß >

\ 1 - ak z \ \ 1 - akz \ y ^=1 V \ 1 - anz\/

a +1

j

P

Therefore

S(k)+ e(k) 1 (1 -\ak \

/ , 1 _ I I X ß+2\

\*(z) \ < exp Oß^l -\a" X\f (z) \xcß E exp-

V n=1 ^ \ anz\' ) k=l (

,\1 — an z\) ^ k^r" (1 — \ak +1 \f (ak )\ \1 — akkz\m+1'

Now we consider the last factor in the product:

qk 1 (1 — \ak\r

exp-

k_1

where qk = S(k) + e(k).

^eXP (1 - \ak\)+1 ' \f (ak)\ \1 - Okz\m+^

We split the sum into n parts:

^ qk 1 (1 — \ak\)

exp-

S_1 aker5(0s)

^ ^ eXP(1 -\ak\)^+1' \f (ak)\\1 - a~kz\m+1-

n p

Since {ak} C M r(0S) for certain 0 < S < —--, we can apply for each part the

s=i 2(a + p + 1)

equation (8). Thus we have:

qk - So(k) (1 -K\)m

exp

S=1 ak er (0s)

(1 -\ak\) +1 \1 - äkz\m+1-

Since qk — S0(k) = qk — — = —qk < 0, we have the following estimate: 1 — qk

exp qk — So k < 1,

p(1 — \ak\) ^+1 < ,

for all k = 1, 2, ... . Thus we have:

i*(z)K exp (,§( (z)ix c,.g

Taking into account the convergence of the series (9), we have:

(1 — \ ak \)m < c |a hm < C1

\ 1 — akz\m+^ (1 — \ z \)m+1 " (1 — \ z \)m+1

a +1

for all m >--+ 1.

p

The estimate of function \^(z) \ takes form:

1 — an c2

( / 1-1 I \ ß+2\ \ *(z) K exp(OßE( ^^ ) Jx\ fk(z) \x

\ 1 — a.nz \J J '•"twl (1 — \ z \ )m+^

From the works of F.A.Shamoyan (see [14], also [15, p. 132]) and of the author (see [9]) it follows that a majorizing function in the last inequality belongs to Sa space, and hence ^ G Sa for all a > —1, 0 < p < +to.

This shows that ii) ^ i). □

The author thanks Professor F.A. Shamoyan for carefully reading of the manuscript and helpful comments.

This work was supported by the Ministry of Education and Science of the Russian Federation (grant 1.1704.2014K) and by the Russian Foundation for Basic Research (grant 13-01-97508).

m

References

[1] V.A.Bednazh, Description of traces, characteristic of the principal parts in the Laurent expansion of classes of meromorphic functions with restrictions on growth characteristics Nevanlinna, Thesis Abstract, St. Peter., 2007 (in Russian).

[2] L.Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math., 80(1958), 921-930.

[3] M.M.Djrbashian, On the representation problem of analytic functions, Soob. Inst. Math. i Mekh. AN ArmSSR, 2(1948), 3--40 (in Russian).

[4] M.M.Djrbashian, Basicity of some biorthogonal systems and the solution of the multiple interpolation problem in the classes Hp in the halfplane, Izv. Acad. Nauk SSSR. Ser. Matem., 42(1978), no. 6, 1322-1384 (in Russian).

[5] A.Hartmann, X.Massaneda, A.Nicolau, P.Thomas, Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants, J. Funct. Anal., 217(2004), 1-37.

[6] A.M.Kotochigov, Free interpolation in Jordan domains: Thesis Abstract — The Russian Academy of Sciences, St. Petersburg Department of Steklov Institute of Mathematics, St. Peter., 2001 (in Russian).

[7] 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).

[8] R.Nevanlinna, Eindeutige analytische Funktionen, 2nd ed., Springer-Verlag, 1953.

[9] E.G.Rodikova, On estimates of the expansion coefficients of certain classes of analytic functions in the disk, Abstracts of Petrozavodsk international conference "Complex analysis and its applications", Petrozavodsk St. Univ., 2012, 64-69 (in Russian).

[10] F.A.Shamoyan, E.G.Rodikova, On interpolation in the class of analytic functions in the unit disk with power growth of the Nevanlinna characteristic, Journal of Sberian Federal University. Mathematics & Physics, 2014, 7(2), 235-243.

[11] H.Shapiro, A.Shields, On some interpolation problems for analytic functions, Amer. J. Math., 83(1961), 513-532.

[12] K.Seip, Interpolation and sampling in spaces of analytic functions, University Lecture Series, Amer. Math. Soc., Providence, 33(2004).

[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, Parametric representation and description of the root sets of weighted classes of holomorphic functions in a disk, Sib. Mat. Zhurn., 40(1999), no. 6, 1422-1440 (in Russian).

[15] F.A.Shamoyan, Weighted spaces of analytic functions with mixed norm, Bryansk, Bryansk Gos. Univ., 2014 (in Russian).

[16] S.A.Vinogradov, V.P.Havin, Free interpolation in Hand in some other classes of functions, J. of Math. Sci., 47(1974), 15-54.

Об интерполяции в классах аналитических в круге функций с характеристикой Р. Неванлинны из Lp-весовых пространств

Евгения Г. Родикова

В статье получено решение интерполяционной задачи в классе аналитических функций в единичном круге, характеристика Р. Неванлинны которых принадлежит Lp-весовым пространствам, при условии, что узлы интерполяции принадлежат конечному числу углов Штольца.

Ключевые слова: интерполяция, аналитические функции, характеристика Р. Неванлинны, углы Штольца.