Journal of Siberian Federal University. Mathematics & Physics 2014, 7(2),

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1 Journal of Siberian Federal University. Mathematics & Physics 204, 72, УДК On Interpolation in the Class of Analytic Functions in the Unit Dis with Power Growth of the Nevanlinna Characteristic Faizo A. Shamoyan Eugenia G. Rodiova Bryans State University, Bezhitsaya, 4, Bryans, Russia Received , received in revised form , accepted In this paper we solve the interpolation problem for the class of analytic functions in the unit dis 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 dis on C, HD be the set of all functions, holomorphic in D. For any > 0 we define the class S as: { S := f HD : Tr,f C f r, where C f > 0 is a positive constant, depending on the function f, r [0,, Tr,f = π ln + fre iϕ dϕ is the Nevanlinna characteristic of the function f, ln + a = max0,ln a, 2π a C see []. It is well nown that if f S, then { c f Mr,f = max fz z r r for all > 0, c f > 0 see []. It is clear that if f S and { + is a sequence of points from the unit dis, then the operator Rf = f,...,f,... maps the class S into the class of weighted sequences { l = γ = {γ + : γ λ, λ > 0. In this article we answer the question under what conditions on the sequence { + the operator Rf maps the class S onto the class l. Definition. A sequence { + is called interpolating for S, if RS = l. shamoyanfa@yandex.ru evheny@yandex.ru c Siberian Federal University. All rights reserved 235

2 Faizo A.Shamoyan, Eugenia G.Rodiova On Interpolation in the Class of Analytic Functions... Let us note that interpolation theory has become intensively developed since the fundamental wor 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 wors [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.. Formulation of main result and proof of auxiliary results To formulate and proof the results of the wor we introduce some more notation and definitions: For any β > we denote π β z, as M.M.Djrbashian s infinite product with zeros at points of the sequence { + see [8]: + π β z, = z U β z,, 2 where U β z, = 2β + π 0 ρ 2 β ln ρeiθ zρe iθ β+2 ρdρdθ. 3 We denote π,n z, as infinite product π β z, without n-th factor. As stated in [8], the infinite product π β z, is absolutely and uniformly convergent in the unit dis D if and only if the series converges: β+2 < +. 4 Let us remar that the product π β z, appear naturally in the integral representations of the holomorphic functions by the ernel K ζ,z = + π ζ 2,ζ,z D, ζz +2 as there is the Blasche product in the integral representation of the bounded type functions by the Poisson-Jensen formula see [,9]. If β = p Z + then product 2 taes a form see [8]: π p z, = + z z p+ j j= 2 j. z Definition 2. The part of the angle with vertex at the point e iθ, less then π and contained in D, whose bisector coincides with radius connecting the center of the dis and the point e iθ is said to be the Stolz angle with vertex at the point e iθ, i.e. where 0 δ <. Γ δ θ := {z D : arge iθ z θ πδ 2, 236

3 Faizo A.Shamoyan, Eugenia G.Rodiova On Interpolation in the Class of Analytic Functions... The main result of this article is the proof of the following theorem: Theorem.. Let { + be the arbitrary sequence of complex numbers from D, which is contained in a finite union of Stolz angles, i.e. { n Γ δ θ s, 5 with certain 0 < δ < +. The following statements are equivalent: i { + is an interpolating sequence in S, > 0, ii for some c > 0; for some M > 0 and all β >. nr = {card : < r < π β n, c, 6 r M, 7 n For presentation of auxiliary result we need also the O.Besov class B s, on the unite circle see [0, p.5]. Let 0 < s < 2; function ψ integrable on the unite circle belongs to the class B s, if and only if { sup 0<t< ψe iθ+t 2ψe iθ + ψe iθ t t s dθ < +. The proof of the theorem is based on the following statements. Theorem A.see []. The class S coincides with the class of analytic in D functions represented as { π fz = c λ z λ ψe iθ π β z, 2π ze iθ dθ, z D, 8 β+2 for all β >, where ψe iθ is real-valued function from the O. Besov class B β,, λ Z +, c λ C, and the sequence { + satisfies the condition nr,f c f r for some c f > 0. Here and in the sequel, unless otherwise noted, we denote by c, c,...,c n,β,... some arbitrary positive constants depending on, β,..., whose specific values are immaterial. It is clear that the space l coincides with the space of sequences {γ + { sup ln + γ < +, such that where { + D. For the further osition of the results we introduce metrics in spaces S and l as follows: f,g S f,g = sup { r ln + fre iθ gre iθ dθ, 0<r< 237

4 Faizo A.Shamoyan, Eugenia G.Rodiova On Interpolation in the Class of Analytic Functions... a = {a,b = {b l { ρ l a,b = sup ln + a b. It is easy to chec that S and l are complete metric spaces with respect to these metrics, and the space S is invariant regarding the shift. Lemma.2. If the operator Rf = f,...,f n,... maps space S there exists the sequence of the functions {g n z + S such that onto space l, then and where,n =,2,... g n = γ n sup g n,0 C, C > 0 n,where γn = 0, for all n,, for = n, Proof. Let S L, L > 0, be the set of the sequences c = {c + from the space l such that c = F, =,2,... for certain function F S, satisfying condition F,0 L, that is sup { r ln + Fre iθ dθ L. 0<r< By hypothesis, RS = l, equivalently to l = + L= S L. Let prove that the sets S L are closed for all L in l, in this terms if c m = {c m + S L and ρ l c m,c 0 0 as m +, then c 0 S L. By assumption, c m = F m, =,2,..., F m S with F m,0 L 9 and ρ l c m,c 0 0 as m +. In this notation, we need to prove that there exists function F 0 S, such that F 0 = c 0, =,2,... and F 0,0 L. From 9 it follows that for any m =,2,... ln + F m re iθ CL r, where CL is independent on m. By Montel s theorem we can choose the subsequence of functions {F m z, uniformly convergent to function F 0 on compact subsets of the unit dis. Let chec that F 0 S. Using inequality 9, we get: sup { r ln + F m re iθ dθ L. 0<r R< Taing in this inequality the limit as +, we obtain: { r sup 0<r R< ln + F 0 re iθ dθ 238 L.

5 Faizo A.Shamoyan, Eugenia G.Rodiova On Interpolation in the Class of Analytic Functions... Whence guiding R 0, we have: F 0,0 L. 0 Thus, F 0 S. Since F m z F 0 z as + for all z D, we have c m n = F m n F 0 n = c 0 n as + for all n <, n =,2,... Taing into account the estimate 0, we conclude: c 0 = {c 0 n + S L. Thus, we prove that the set S L is closed in l and l = + L= S L. By Baire s theorem there exists a number L 0 such that the set S L0 includes the ball with the center at one of its interior points, for example, BF 0 n,d := {c = {c : ρ l c,f 0 n = sup ln + c F 0 d, where the sequence {F 0 n = {c 0 n is the center of the ball, d > 0 is its radius. It means that for any sequence c = {c B there exists a function F S such that F = c, =,2,... So, if ρ l γ,0 d for some γ = {γ, then there exists a function g S such that g,0 2L 0 and g = γ for all. It is sufficient to put γ = c F 0, =,2,..., gz = Fz F 0 z, where F is the function with the above properties. Let γ n = {γ n, n N, where γn = c n F 0 =,2,..., c n B. Then ρ l γ n,0 d. We put c n = F 0 for n, c n = F 0 + for = n. Arguing as above, we see that there exists a function g n z = F n z F 0 z such that g n,0 2L 0 and g n = γ n = 0 for n, g n = γ n = for = n. The proof of Lemma.2 is complete with the constant C = 2L 0. Remar.. The idea of proving Lemma.2 is adopted from P.Koosis see [2, p.200], who first applied this for solving the interpolation problem in the class of the bounded analytic functions. Lemma.3. see [3] If members of the sequence { + are contained in a finite union of Stolz angles, i.e. { n Γ δ θ s for certain 0 < δ <, then for any function + gz = n C ze iθs, z D, > the following estimate is valid g s c 0 where c 0,C are some positive constants. 2. Proof of main result C, s =,2,...,n, s Let prove the implication i ii. We assume that { + D is an interpolation sequence in the class S, > 0, i.e. for any {γ l there exists a function f S such that f = γ, =,2,... Let consider the sequence {γ + : γ =,γ 2 = γ 3 =... = 0. Evidently, {γ + l. Since { + =2 is zero-sequence for the function f S, > 0, we have nr by Theorem A. The estimate 6 is established. c r 239

6 Faizo A.Shamoyan, Eugenia G.Rodiova On Interpolation in the Class of Analytic Functions... In order to show 7 we fix n N and tae the sequence {γ n + as follows: γn = 0, n, γ n =, = n. By Lemma.2 there exists a function g n S such that g n,0 C and g n = γ n for all =,2,..., where the constant C > 0 is independent on n. In particular, g n n = γ n n. According to Theorem A, any function g n S, > 0 can be represented as g n z = c λn z λn π β,n z, {h n z, z D, where h n z = 2π So, ψ n e iθ ze iθ dθ, β >. β+2 g n n = γ n n = n = c λ n λ π β,n n, {h n. Since {h n z S, then taing into account the estimate we have: g n n = where c 2 is independent on n according to Lemma.2. From the last inequality we obtain: n c c 2 π β,n n, n, π β,n n, M n, M > 0. 2 We show, that 2 7. Really, differentiating the function π β, we get: π βz, = j= U β z, j zj U β z, j U βz, j π β,j z,, j where U β z, j defined by 3. have: Since π β,j n, = +, j n U β n, = 0 for all j =,2,..., j n, we π β n, = n U β n, n π β,n n,. 3 Using now estimate 2, from 3 and 3 we obtain: π β n, M, M > 0. n Thus, 2 7. The implication i ii is established. Now we prove that ii i. Suppose that { + 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 Ψ S such that Ψ = γ, =,2,... for each {γ + l. We construct a function Ψz as follows: Ψz = γ π β z, j z π β, j 240 m fz z f, 4

7 Faizo A.Shamoyan, Eugenia G.Rodiova On Interpolation in the Class of Analytic Functions... with β >, m > +, and fz = n C, z D. 5 ze iθs It is obvious, that Ψ n = γ n, n =,2,... Now we need to prove that function Ψz is analytic in D and Ψ S. First, from the estimate 6 we conclude: m < + 6 for all m > +. Taing into account the convergence of the series 6 and Lemma.3, we obtain that the infinite product π β z, j and the series 4 are absolutely and uniformly convergent in D. Now we get an upper estimate for the function Ψz. Since {γ + l and condition 7 is valid, we have: c Ψz γ π βz, j z λ π β z, j z π β, j z M z m fz f m fz f. We estimate the factor π βz, j. Using the well-nown estimate for the Djrbashian product z see [3]: + β+2 ln + π β, z, j c β, 7 z we get: π β z, j = z z π β,z, j z π β, z, j U β z, c β, z π β, z, j c β z z + β+2 n c β n z for all β >. Therefore + β+2 n Ψz c β fz c β n z λ + M m f z m+. Now we consider the last factor in the product:we obtain the following estimate We split the sum into n parts: n Γ δ θ s λ + M λ + M m f z m+. m f z m+. 24

8 Faizo A.Shamoyan, Eugenia G.Rodiova On Interpolation in the Class of Analytic Functions... Since { n Γ δ θ s for certain 0 < δ <, we can apply Lemma.3 for each part of the + sum. Thus we have: n λ + M C m z m+. Γ δ θ s Choosing the positive constant C such that λ + M C < 0, we obtain the following estimate: for all =,2,... Thus we have: Ψz c β + λ + M C, β+2 n n z fz c β Taing into account the convergence of the series 6, we have: m z m+. m z m+ c z m+ m c z m+ for all m > +. The estimate of function Ψz taes form: + β+2 n c 2 Ψz c β fz. 8 n z z m+ Now we show that Ψz S, in this notation Tr,Ψ = 2π where > 0, C > 0. Using 8, we get Tr,Ψ c β ln + Ψre iθ dθ β+2 n π n re iθ dθ + ln + fre iθ dθ + 2π ln C r, c r m+. We estimate each of summands in this sum separately. As established in [] see also [4], β+2 n c n re iθ dθ r, 9 for all β >. Further, from 5 we have: ln + fre iθ dθ Applying elementary estimate, we obtain: n ln + fre iθ dθ C re iθ θs. c 3 r. 20 Combining 9, 20, we obtain that function Ψz belongs to the class S. This shows that ii i. The proof of Theorem. is complete. Remar 2.. Note that if = 0, then condition 5 is necessary, as established in [4]. The wor is supported by the Russian Foundation for Basic Research, project

9 Faizo A.Shamoyan, Eugenia G. Rodiova On Interpolation in the Class of Analytic Functions... References [] R.Nevanlinna, Eindeutige analytische Funtionen, 2nd ed., Springer-Verlag, 953. [2] L.Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math., 80958, [3] S.A.Vinogradov, V.P.Havin, Free interpolation in H and in some other classes of functions. I, Zap. Nauchn. Sem. LOMI, 47974, 5 54 in Russian. [4] A.G.Naftalevic, On interpolation by functions of bounded characteristic, Vilniaus Valst. Univ. Moslu Darbai. Mat. Fiz. Chem. Moslu Ser., 5956, 5 27 in Russian. [5] A.Hartmann, X.Massaneda, A.Nicolau, P.Thomas, Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants, J. Funct. Anal., , 37. [6] J.Shapiro, A.Shields, On some interpolation problems for analytic functions, Amer. J. Math., 8396, [7] K.Seip, Interpolation and sampling in spaces of analytic functions, University Lecture Series, Amer. Math. Soc., Providence, [8] M.M.Djrbashian, On the representation problem of analytic functions, Soob. Inst. Mat. i Meh. AN ArmSSR, 2948, 3 40 in Russian. [9] I.I.Privalov, Boundary properties of analytic functions, Gostehizdat, Moscow i Leningrad, 950 in Russian. [0] E.M.Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 970. [] F.A.Shamoyan, E.N.Shubabo, Parametrical representations of some classes of holomorphic functions in the dis, Operator Theory: Advances and Applications, Birhauser Verlag, Basel, 32000, [2] P.Koosis, Introduction to H p Spaces, Cambridge University Press, 998. [3] F.A.Shamoyan, M.M.Dzhrbashyan s factorization theorem and characterization of zeros of functions analytic in the dis with a majorant of bounded growth, Izv. Aad. Nau ArmSSR, Matematia, 3978, no. 5 6, in Russian. [4] F.A.Shamoyan, E.N.Shubabo, Introduction to the theory of weighted L p -classes of meromorphic functions, Bryansiy Gosudarstven. Universitet, Bryans, 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. 243