CC BY
61Probl. Anal. Issues Anal. Vol. 7 (25), Special Issue, 2018, pp. 113-123 113
DOI: 10.15393/j3.art.2018.5170
The paper is presented at the conference "Complex analysis and its applications" (COMAN 2018), Gelendzhik - Krasnodar, Russia, June 2-9, 2018.
UDC 517.537
K. G. Malyutin, A. L. GuSEV
THE INTERPOLATION PROBLEM IN THE SPACES OF ANALYTICAL 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 in terms of the canonical Nevanlinna product of nodes of interpolation are obtained. The solution of the interpolation problem is constructed in the form of the Jones interpolation series, which is a generalization of the Lagrange interpolation series.
Key words: half-plane, function of finite order, free interpolation, Nevanlinna product, interpolation series
2010 Mathematical Subject Classification: 30E05
1. Introduction and main result. Definitions and Notation.
In 1948, A. F. Leont'ev [4] first considered the interpolation problem in the space of entire functions [p,w] of finite order p > 0, which obtained subsequently the title of the free interpolation problem. In this paper we consider the problem of simple free interpolation in the spaces of analytical functions of finite order p > 1 in the half-plane.
Let C+ = {z : Im z > 0}. Denote by [p,ro] + the space of analytical functions of finite order p > 1 in C+ [2, Chapter I, §1], i.e., f G [p,<^]+ if
ln+ ln+ |f (reid )| limsup sup --- ^ p,
r^TO o<e<n ln r
where ln+ a =
ln+ a = ln a , a > 1, ln+ a = 0 , a < 1 .
@ Petrozavodsk State University, 2018
Let A = { a„}~=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. The sequence A is called an interpolation sequence in the space [p,to]+ if for any sequence of complex numbers {bn}%=1 satisfying the conditions:
ln+ ln+ |bn|
sup-:-:- < TO , (1)
„£n ln |a„| +2
!• ln+ ln+ |bn | . f0s
lim sup - :—:— < p, (2)
KK^ ln M
there exists a function F e [p,ro] + solving the interpolation problem
F (a„) = bn, n =1,2,.... (3)
The condition (1) and (2) are necessary restrictions on the sequence {bn}£=1. These restrictions are related to the fact that a function F(z), solving the interpolation problem (3), must belong to the space [p,<ro]+.
In 1975, B. Ya. Levin and N. Uen [6] considered the problem of simple interpolation in the space [p,to]+ for p > 1. Necessary conditions and sufficient conditions in terms of the canonical Nevanlinna product of interpolation nodes were obtained for the solvability of the corresponding interpolation problems. But between the two types of these conditions there was a gap that did not allow the interpolation nodes to "accumulate" at points of the real axis. The aim of this paper is to study the interpolation problem in the space [p,ro]+ for p > 1. We find necessary and sufficient conditions for the interpolation problem to be solvable. This conditions are formulated in terms of canonical product determined by the interpolation nodes. According to its content, the problem is a free interpolation problem, because only natural restrictions are imposed on the values of the function at the interpolation nodes, due to the requirement for the solution to be in the given space.
Denote by Bq(u,v) the Nevanlinna primary factor
-y(w — v)
q = 0,
v(u — v)
Bq(u,v) = 1 u ( 1 1 \\
Bo(u,v)ex^E — ^J l q e N .
Let A = {an = r„ei0nC C+, rn > ô0 > 0, be the sequence of distinct complex numbers such that all limit points of A are on the real axis and infinity, and for any e > 0
E
sin dn
rp+e
' n
< TO, p > 1 ,
(4)
then the function
E (z) = EA(z)
= : n ( T—T ) n Bq (z,an), q = [p]
v z am s
|a„|<1
|a„|^1
belongs to the space [p,w]+. We denote by [•] the integer part of a number. The function E(z) is called the canonical function of the sequence A. Our main result is the theorem stated below.
Theorem 1. The following two statements are equivalent.
1) The sequence A is an interpolation sequence in the space [p,ro] + .
2) Condition (4) is true and the canonical function E(z) of the sequence A satisfies the conditions:
1
sup—— neN in |an
I +2
1
limsup
|a„Kœ ln |an|
ln+ ln+
ln+ ln+
1
IE '(a
I Imc
1
|E'(an)| Im an
<00,
< p.
(5)
(6)
We now state the result of B. Ya. Levin and N. Uen from the paper [6]: In order for the sequence A to be an interpolation sequence in space [p,w] + , it is necessary that condition (4) is true and the canonical function E(z) of the sequence A satisfies condition (6); and sufficient that condition (4) is true and the canonical function E(z) of the sequence A satisfies the condition
lim sup
1
ln | an|
ln+ ln+
1
IE '(a
I Im2 an
< p.
(7)
Note that condition (7) implies the following restriction on the interpolation nodes: for all e > 0 the inequality Im an ^ exp(-|an|p+£) is true. This restriction allowed the authors to construct an interpolating function in the form of the Lagrange interpolation series. We also note that the
n
n
a
n
restriction (7) does not allow the interpolation nodes to have limit points on the real axis, and therefore condition (1) on the values of the interpolating function is unnecessary. In this formulation, the problem can not be regarded as a problem of free interpolation in space [p,to] +. These shortcomings are eliminated in this paper.
2. Preliminaries. Let
A / \ TT — ak) 771 / \ 771C \an(Z — an) A -1/ \
An(z) = II —(-r^T ,En(z)= -r A„ (z) .
0< X±/2 ak(z - «fc) an(z - «n) U<|a„-at | ^fn/2
We need the following statements. Lemma 1. If the sequence A satisfies (4) then
z
sup 1 supln+ ln+ |E„(z)| < TO, ee+ In |z| + 2 „eN
limsup-——j-supln+ ln+ |En(z)| ^ p. ln |z| neN
(8)
The lemma is proved by standard methods for estimating canonical products (see e.g. [2], [5]), and we omit the proof.
Lemma 2. If the sequence A satisfie (4), (5) and (6), then
^^ Im afc Im an k=i |an - ak |2(1+ r2)ep+L
Proof. We get from (5), (6) and (8) that
sup 1—;—1-77 ln+ ln+ ——*—-7 < TO,
new ln |rn| +2 |A n(an) |
limsup -—1—- ln+ ln+ *—— < p .
new ln |rn| |An(an)|
From these inequalities, the identity
-b
a — b
4Im a Im b
- |a - b|2
2
and the elementary inequality x ^ — ln(1 — x) (0 ^ x < 1) we get the following relations:
(9)
1 Im ak Im an sup —:—--ln > --_ < to ,
n£N ln |r" +2 ,„ |an - ak |2
1 Im afc Im a„
limsupln^ln E ~fa—^ p.
The condition (4) implies that for any e > 0 the series
TO T
EIm an
^ (10)
converges. From (9) and (10) we obtain the statement of the lemma. □
Definition 2. An absolutely continuous function p(r) on the real axis [0, + to) that satisfies the conditions
lim p(r) = p, lim p'(r)r lnr = 0 ,
r —+ TO r —+ TO
is called a proximate order.
We need the following lemma [5, Chapter I, Theorem 16].
Lemma 3. If f (r) is any continuous function that is positive for r > 0 and satisfies the condition
ln f (r) lim sup —--= p < to ,
r—^+to ln r
then the proximate order p(r) can be chosen so that for all positive values of r
f (r) < rp(r)
and for some sequence of values rn (n = 1, 2,...) tending to infinity
f (rn) = r"(rn) .
3. The proof of implication 1) ^ 2) of the Theorem 1. We
prove the implication 1) ^ 2) by contradiction. Let the sequence A be an interpolation sequence in the space [p,to] + . There exists a function
F € [p,to]+ such that F(a1) = 1 and F(an) =0, n > 2. Therefore the sequence A\{a1} belongs to the set of zeros of F and satisfies condition (4) [2, Chapter I, §3].
We now prove (5). Assume the contrary, that there exists a sequence {cn} c A such that
lim —;—;-ln+ ln+ ———--= oo . (11)
"—TO ln |C"| + 2 |E'(C")| Im C"
Using the Carleson theorem [1] and passing if necessary to a subsequence, we can assume that Blaschke product H(z) of the sequence {cn} satisfies the condition
inf |H'(C")| Im C" > S> 0 . (12)
n
Suppose further that F € [p,to]+ is such that F(cn) = 1 and F(an) = 0,
F(z)H(z)
an = C". Then the function f (z) = — — belongs to [p,to]+. We
1 f(C )
have ———- = „," s . The last equality with (12) contradicts (11) and
E (c") h (c")
f (z) € [p,to] + .
The inequality (5) is proved. The inequality (6) is proved similarly. 4. The proof of implication 2) ^ 1) of the Theorem 1. Since the series (4) converges (after a renumbering of the points an if it is necessary), we can assume that
Im an+1 Im an
--"+1 < --", n € N . (13)
1 + r2+i 1 + r2
We define
1 + ak z Im ak
TO
an(z)=
, s 1 + ak z Im ak
(z) = > —.-r-HIT, n = 1,2,....
W ^i(a-k - z)(1 + r2)''
The series defining the functions an(z) converges uniformly in each domain Dr,a = {z : |z| < r, Imz > S > 0}, because
1 + ak z
i(ak - z)
Im ak < (1 + |z|)(1 + rk) Im ak
(1 i 2\lpl+3 ^ S /1 i 2\[p'+3
(1 + r2) 2 S (1 + r2) 2
if z € , and it follows from (4) that the series
w T
y^ Im ak
converges for any e > 0.
Let us estimate Re an(z). We have
Re an(z) =
Im ak + Im z + r2 Im z + | z |2 Im ak Im ak
k=n ^ (1+ r2) ^
Using Lemma 2 and inequality (13), we obtain
_ = ^ Im ak (Im ak (1 + rn) + Im an(1 + ))
' an(an i -
Re an (an) =
^K^m/ ^ _ [p+3
k=n |ak - an|2(1 + r2) 2
— / Im ak Im an \ Im ak(1 + r^)(1 + rn)
E Im ak +
1 + r2 +
^V1 + r2 1+ rV |ak - an|2(1 + r2)1223
<
^ O 1 + rn ^ Im an Im ak , ,, ^ _
< ia——^[Pi+1 < M< TO, (14)
1+ rn k=n |ak - an| (1+ r2 )-HT-for some M > 0, and
Re an(z) ^liiriSfc. (15)
We put further
(1 + zan\[p]+3 ( z \Sn (2ImanV ^n(z) = (j+T J la;) exp(an(an) - an(z)),
where the integers Sn will be chosen below. We note that
^n(an) = 1, n = 1,2, ....
In addition, using the elementary inequality 1 + x < \J2(1 + x2), we obtain for |z| > 1:
1 + zaan
1 + r2
n
< |z|(1+ rn) < V2|
1 + rn " v7!^
z
Further
b"(z)l <
[p]+3
Sn 4 Im2 an |z - a"|2
x exp{Re[an(a") - a"(z)]}, n = 1,2,....
(16)
Let
Pn(z)
bn^n(z)
(z - a")E'(an) '
The formal series
F(z) = E(z) £ P"(z)
(17)
n=i
solves the interpolation problem (3).
We now show that the function F(z) belongs to [p,to]+ for the suitable choice of the sequence {Sn} of positive integers. Let C(z,r) be the disk of radius r around a point z. It follows from (14), (15) and (16) for all z € C+, z € C(an, Iman), that
|Pn(z)| < Mi
Im a"E'(an)
TO
exp - E
[p]+3
Sn 4Im2 an |z - a"
2
(Im ak )2
(1+ r2)^ |«k - z|
n = 1,2,...
k="
(18)
where M1 > 0 is some constant. Let
A"(z)
TO
E
(Im ak )2
k=n (1+ r2) lP2+3 K - z|2
so that
A"(z) - A"+i(z) =
(Im an)2
(1 + r") lP+3 K - z|2
, n € N.
It is clear that An(z) I 0 as n ^ to, z € C+.
x
b
z
n
x
x
2
Returning to (18), we get for z G C+, z G C(an, Iman) |Pn(z)| < Ml|z|[p]+3
b„
Im a„E'(a„) x(A„(z) - A„+i(z))exp(-A„(z)), n = 1,2,....
Using the elementary inequality t < e4 — 1, t > 0, for t = An(z) — An+1(z) gives us
|Pn(z)| < Mi|z|
[p]+3
bn
Im a„E'(a„)
x (exp(—An+i(z)) — exp(—A„(z))), n =1,2,.... (19)
We now choose a sequence of numbers Sn such that the function F(z) defined by series (17) belongs to the space [p,œ] + .
From the conditions (1), (2), the inequalities (5), (6) and Lemma 3, it follows that there exist M2 > 0 and a proximate order p(r), lim p(r) = p, such that
bn
Im a„E'(a„) From this and (19), we obtain
|P„(z)| < M2 exp(r;
< M2 exp(rp(r"}), n = 1, 2,....
p(rn)) |z| [P]+3
x (exp(—A„+i(z)) — exp(—A„(z))), n = 1,2,.... Set Sn = [rP(rn)] + 1, n =1,2,.... Then, by (20),
(20)
|P„(z)| < M2|z|
[p]+4
rP(rn) ]
x (exp(—A„+i(z)) — exp(—A„(z))), n =1, 2,... .
(21)
We represent the sequence A as a sum of two sequences A = A1 U A2: if an € A1 then |z| > rn, and if an € A2 then |z| < rn. By (21), we have for the finite subset A2 c A2 that
£ |Pn(z)| < M2|z|[p]+4 exp(—An2(z)):
(22)
S
x
n
x
n
where n2 is the largest member number from A2.
For the finite subset Ai c Ai, we have in the same way
E |p"(z)| < M2|z|[p]+4 E |z|[rn<rn)](exp(-A"+i(z))-exp(-A"(z))) <
< M2|z|[p]+4|z||z|P<W) exp(-A"i(z)), (23)
where ni is the largest member number from Ai.
From (22) and (23), we have for the finite subset A = Ai U A2 c A
E |E(z)P"(z)| <
an eA
< M2|z|[p]+4|E(z)|(|z||z|P<|z|) exp(-A"i(z)) + exp(-A"(z))), (24) if z € C+, z € U C(an, Im an).
aneA
Applying the maximum modulus principle to the analytic function
E |E(z)Pn(z)|
ane A
inside the disk C(an,Iman), we obtain that the estimate (24) is true for all z € C+. This implies the convergence of the series (17) on compact sets in C+ and its belonging to the space [p, to]+. The Theorem 1 is proved.
Remark 1. In this paper, we consider the interpolation problem in the space [p, to] +, p > 1. There are various definitions of the order of functions analytic in the half-plane [2,3,8, 9]. These definitions coincide for p > 1 and differ for 0 < p < 1. In our opinion, each case requires an independent study.
Remark 2. In 1994, K. G. Malytin [7] considered the problem of multiple interpolation in the space [p(r), to)+ of functions of at most normal type for the proximate order p(r), p > 1, in the upper half-plane C+.
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 18-01-00236
References
[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. Continuity and asymptotic continuity of subharmonic functions. I. Mat. Fiz., Anal., Geom., 1994, vol. 1, no. 2, pp. 193-215. (in Russian)
[4] Leont'ev A. F. On interpolation in class of entire functions of finite order. Dokl. Akad. Nauk USSR, 1948, vol. 5, pp. 785-787. (in Russian)
[5] Levin B. Ya. Distribution of Zeros of Entire Functions. English revised edition Amer. Math. Soc.: Providence, RI. 1980.
[6] Levin B. Ya., Uen N. T. On the interpolation problem in the half-plane in the class of analytic functions of finite order. Teoriya funkts., funkts. analiz i ikh pril., 1975, vol. 22, pp. 77-85. (in Russian)
[7] 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.
[8] Malyutin K. G. Fourier series and S-subharmonic functions of finite 7-type in a half-plane. Russian Acad. Sci. Sb. Math., 2001, vol. 192, no. 6, pp. 843-861.
[9] Ronkin L. I. Regularity of growth and D'-asymptotic of holomorphic functions in C+. Soviet Mathematics (Izvestiya VUZ. Matematika), 1990, vol. 34, no. 2, pp. 16-29.
Received April 19, 2018.
In revised form, July 23, 2018.
Accepted August 05, 2018.
Published online August 18, 2018.
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]
