Geometric meaning of the interpolation conditions in the class of functions of finite order in the half-plane Текст научной статьи по специальности «Математика»

96

Probl. Anal. Issues Anal. Vol. 8 (26), No3, 2019, pp. 96-104

DOI: 10.15393/j3.art.2019.6690

UDC 517.537

K. G. Malyutin, A. L. Gusev

GEOMETRIC MEANING OF THE INTERPOLATION CONDITIONS IN THE CLASS OF FUNCTIONS OF FINITE ORDER IN THE HALF-PLANE

Abstract. The aim of this paper is to study the interpolation problem in the spaces of analytical functions of finite order p > 1 in the half-plane. The necessary and sufficient conditions for its solvability are found in terms of the measure defined by the nodes of interpolation.

Key words: half-plane, function of finite order, free interpolation, Nevanlinna measure, interpolation sequence

2010 Mathematical Subject Classification: 30E05

1. Introduction, Definitions, and Notations. Interpolation problem is called simple free interpolation problem if multiplicities of the interpolation nodes are equal to unity and restrictions on the values of interpolation function F at these nodes are necessary restrictions related to the fact that the function F must to belong to the considered space. In [6], the problem of simple free interpolation in the spaces of analytical functions of finite order p > 1 in the half-plane C+ = {z : Im z > 0} was considered. In this article, we use the definitions and notation of [6]. Denote by [p, + the space of analytical functions of finite order p > 1 in C+ [2, Chapter I, §1]. Let A = {an}r=1 C C+ be a sequence of distinct complex numbers such that all limit points of A are on the real axis and infinity.

Definition 1. A sequence A is called an interpolation sequence in the space [p, if for any sequence of complex numbers {bn}r= 1 satisfying the conditions

ln+ ln+ |bn| < m

sup --:-:-— < TO , (1)

n€N ln |an| + 2

© Petrozavodsk State University, 2019

v ln+ ln+ |bn| . (2)

Iimsup—--:-:— ^ p, (2)

there exists a function F E [p, + that solves the interpolation problem

F (an) = bn, n = 1, 2,

(3)

Here ln+ b

0, b ^ 0, ln b, b > 0.

Denote by Bq(u,v) the Nevanlinna primary factor

q = 0,

Bq (u, v) = <

v(u — v) v(u — v)

Bo(u,v)exp ( V u- ( i — 1

i \v- v-

q G N.

Let A = {an = rne n C C+, rn > S0 > 0, be a sequence of distinct complex numbers such that all limit points of A are on the real axis and infinity, and for any e > 0

V

n=1

sin 9n

rP+£

rn

< œ, p > 1 ;

(4)

then the function E (z) =

n

|«n|<1 v n/ |«n

Y[ Bq (z,an^ q = [p]

belongs to the space [p, ro]+. Denote by [■] the integer part of a number. The function E(z) is called the canonical function of the sequence A.

The following theorem was proved in [6]. Theorem A. The following two statements are equivalent:

1) The sequence A is an interpolation sequence in the space [p, + .

2) Condition (4) is true and the canonical function E(z) of the sequence A satisfies the conditions

1

sup -—!—!--

n€N ln |an| + 2

lim sup --:—-

KK^ ln |an|

ln+ ln+

ln+ ln+

1

|E'(an)| Ima„ 1

|E'(an)| Im an

<,

(5)

p

The aim of this paper is to obtain necessary and sufficient conditions for solvability of this interpolation problem in terms of measure determined by the sequence A. We need the following definitions. By C(z,R) we denote the open disk of radius R with centre in a point z.

Definition 2. An absolutely continuous function p(r) on the half-axis (0, + to) that satisfies the conditions

lim p(r) = p, lim p'(r)r ln r = 0 ,

r—r—

is called a proximate order.

Here p'(r) is the maximum absolute value of the derivative numbers. Let us denote V(r) := rp(r) and Vi(r) := rPi(r), where p^(r) is a proximate order such that lim p»(r) = p. From the sequence A, we define

r —+

the Nevanlinna measure p(G) := pA(G) := sin0n and the families of

a„eG

functions

$ + (a) = p(C(z,a|z|) \ an), i+(a) = , a> 0 ,

where an is the point closest to z (if there are several such points, choose the one with the largest sin 0n). Set

T+f n • n i i+(a) da I+(z,S) = sin0 z y. , 0 = argz, J a(a + sin0)2 0

I+(z, S1,S2) := I +(z, ¿2) - I +(z, ¿i), ¿i ^ ¿2 .

Our main results are stated in the following two theorems.

Theorem 1. The following two statements are equivalent:

1) The sequence A is an interpolation sequence in the space [p, to] + .

2) Condition (4) is true, and for any 5 > 0

lim sup -— ln

r—^ ln r

5 ~

1 , r . „ f i+(a) da

sin 0

J a(a + sin0)2-

0

^ p, z = re%e

sup

zec+ ln(r + 2)

ln

sin Q

$+ (a)da a(a + sin 9)2.

< oo.

Theorem 2. The following two statements are equivalent:

1) The sequence A is an interpolation sequence in the space [p, + .

2) Condition (4) is true, and for any 5 > 0 there exists a proximate order p(r), lim p(r) = p > 1, such that

ln

an

an ak

$+(a) ^ a, $+(a) ^ sin 9 jln

^ V(rn), n = k sin 9

^ a ^ 5,

2

sin 9

0 ^ a ^

sin Q

:iq)

a 2

where Q = arg z.

2. Preliminaries. Denote by [p(r), the space of functions, analytic in the half-plane C+ and of at most normal type for p(r), i.e., such that ln |f (z)| ^ CfV(|z|), where Cf is a finite constant.

Definition 3. The sequence A is called an interpolation sequence in the space [p(r), if for any sequence of complex numbers satisfying

the condition

ln+ |6n

lim sup

< œ,

12)

V (|an|)

there exists a function F E [p(r), that solves the interpolation problem (3).

The following theorem is the corollary of [5, Theorem 1, Theorem 2]. Theorem B. The following three statements are equivalent:

1) The sequence A is an interpolation sequence in the space [p(r),

2) The canonical function E(z) of the sequence A satisfies the condition

11

sup

ln+

ne n v (|an|) |e'(an)|im a«

<,

1/2

• n C $+(a) da suP sin 9 \ —(—, ■ m2 zec+ J a(a + sin 9)2

<,

arg z.

13)

:i4)

à

i

r—fOO

9

We will need the following inequality [5, p. 264]:

$++(a) ^ M„$+ (a) (15)

for a ^ 1/2, where an is the point of A closest to z.

3. Proof of Theorem 1. Now, we prove Theorem 1.

Proof. Let the sequence A be an interpolation sequence in the space [p, Then conditions (4), (5) and (6) are true. It follows from (5) and (6), that there exists a proximate order p(r), lim p(r) = p > 1, such that

r—y^o

condition (13) is true. By Theorem B, we obtain condition (14) from (13). Obviously, condition (14) implies conditions (7) and (8).

Now, let conditions (7) and (8) be true. Then there exists a proximate order pi(r), lim pi(r) = p, such that condition (14) is true. Let a

r—rx

sequence |6n}^L1 satisfy conditions (1) and (2). There exists a proximate order p2(r), lim p2(r) = p, such that condition (12) is true. Let p(r),

lim p(r)= p, be a proximate order such that

r—rx

p(r) ^ max{p1(r),p2(r) : r> 0}.

Conditions (12) and (14) are true by p(r). By Theorem B, the sequence A is the interpolation sequence in the space [p(r), Therefore, there

exists a function F E [p(r), solving the interpolation problem (3). Since [p(r), C [p, for each proximate order p(r), such that lim p(r) = p, then F E [p, □

4. Proof of Theorem 2. Now let us prove Theorem 2. Proof. Let conditions (9), (10) and (11) be true. Then

T+/ t-N T+ ( sin T+ ( sin ^ f

/+(z, 5) = /+ i z, — j + / + ( z, — ,5

Using (10), we estimate the second term:

/ +(z, ^^ ^ sin e / da

(sin 0)/2 5

(a + sin e)2

sin e

a + sin e

(sin 0)/2

2

< 3. (i6)

Let us estimate the first term. To do this, we will prove

sin 9

sup 1 ( z,

zec+ \ 2

/+( z,—)< œ . (17)

From inequalities (9) and

an ak

an ak

^ |an — ak1

^ Im an

we obtain |an — ak | ^ Im an exp(-M\V(rn)), n = k, where M1 > 0 is some constant. From this it follows that the disks C(an, Im an exp(—M2V(rn)) do not intersect for some M2 > 0. Then

I+ (a>n,= I + ^an, sin9n exp(—M2V(rn)),^n9^ ^

sin 8n/2 _ 1

< f f\ sin 9n \- sin2 9n da

a J a(a + sin 9n)2

sin d„ exp(-M2V(rn))

exp(M2V(rn)) exp(M2V(rn))

i a da ^ i da ^

J (a + 1)2 ln a^ J a ln a^ 22

^ ln M2 + ln V(rn) = ln M2 + p(rn) ln rn . Let pi(r), lim pi(r) = p, be a proximate order, such that for r > 0

r—y^o

p1(r) ^ p(r) ln r. For this proximate order

+ ( sin 9n\ ^

sup I+ I an, 2 1 < w.

To conclude the proof of (17), it remains to see that $+(a) ^ M0$+n(a) (see (15)) for a ^ 1/2, where an is the point of A closest to z. Condition (14) follows from (16) and (17).

Conversely, let conditions (7) and (8) be true. Then, for some proximate order p(r), lim p(r) = p, condition (14) is true. Let ^ E [(sin 9)/2,$].

Then, for some M3 > 0

2,3

M3 > 1+(z,8, 28) > sin9$+(z,8) i >

J a(a + sin9)2

,

23 23

> $+(z,8) / sin9da _ $+(z,8)sin9 |23

3

28 J a(a + sin 9)2 28(a + sin(

3

Since ——'-- > - for 8 > sin9/2, we obtain (10).

8 + sin 9 3 ^ ' ' v y

Similarly, for 8 E [0, sin 9/2]

M3 > 1+ (z,8, 28) > 1 +(z,8, sin 9) >

sin 0

> $+(z,8w da _ ^+(z,8) ln sin 9

^ 4 sin 9 J a sin 9 8

3

From this, we obtain (11). □

5. Interpolation in the space H^. Let H^ be the space of bounded functions in C+.

Definition 4. The sequence A is called an interpolation sequence in the space H^, if for any bounded sequence of complex numbers there

exists a function F E H^ that solves the interpolation problem (3).

The following theorem is the famous Carleson theorem [1]. Carleson Theorem. The following three statements are equivalent. -) The sequence A is an interpolation sequence in the space H^. 2) The Blaschke product B(z) satisfies the condition

inf{Im an|B'(an)|} > 0 . (18)

n

an ak

inf

n=k

an ak

>0

and the measure p(z) = Iman#(z — an) is Carleson's measure, i. e., for all x E R and all h > 0

p((x,x + h) x (0, h)) ^ Kh, (19)

where K > 0 is some constant independent of x and h, 5(z) is the Dirac delta function.

Our result for the space H^ is stated in the following proposition.

Proposition. The following two statements are equivalent.

1) The sequence A is an interpolation sequence in the space H^.

2) The following inequality is true

This proposition is a corollary of Theorem B for the case p(r) = 0, r > 0.

Remark. Condition (13) is an analogue of condition (18) in the space [p(r), œ). Condition (14) is an analogue of condition (19) in the space [p(r), œ). In contrast to condition (19), which gives the boundary density of the distribution of the imaginary parts of the interpolation nodes, condition (20) gives the interior density of the distribution of the arguments of the interpolation nodes.

Acknowledgment. The authors are thankful to the referee for valuable suggestions towards the improvement of the paper.

The reported study was funded by RFBR according to the research project No 18-01-00236.

[1] Carleson L. An interpolation problem for bounded analytic functions. Amer. J. Math., 1958, vol. 80, pp. 921-930.

[2] Govorov N. V. Riemann's boundary problem with infinite index. Basel; Boston; Berlin: Birkhauser. 1994.

[3] Grishin A. F. On regularity of the growth of subharmonic functions. Teor. Funktsii Funktsional. Anal. i Prilozhen., 1968, vol. 7, pp. 59-84 (in Russian).

[4] Levin B. Ya. Distribution of Zeros of Entire Functions. English revised edition Amer. Math. Soc.: Providence, RI. 1980.

[5] Malyutin K. G. The problem of multiple interpolation in the half-plane in the class of analytic functions of finite order and normal type. Russian Acad. Sci. Sb. Math., 1994, vol. 78, no. 1, pp. 253-266.

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References

[6] Malyutin K. G., Gusev A. L. The interpolation problem in the spaces of analytical functions of finite order in the half-plane. Probl. Anal. Issues Anal., Special Issue, 2018, vol. 7 (25), pp. 113-123. DOI: https://doi .org/10.15393/j3.art. 2018.5170

Received July 11, 2019. In revised form,, November 3, 2019. Accepted November 3, 2019. Published online November 9, 2019.

K. G. Malyutin Kursk State University 33 Radischeva str., Kursk 305000, Russia E-mail: [email protected]

A. L. Gusev

Kursk State University

33 Radischeva str., Kursk 305000, Russia

E-mail: [email protected]