Interpolation Sequences for the Class of Functions of Finite  -Type Analytic in the Unit Disk

Vynnyts’kyi, B. V.; Sheparovych, Iryna

doi:10.1023/b:ukma.0000045694.09930.44

Cited by 3 publications

(7 citation statements)

References 6 publications

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“…Combining Theorem 4 with the aforementioned result from [29] we are able to prove the following criterion, which essentially coincides with a result from [11].…”

Section: Resultssupporting

confidence: 74%

“…where ln γ(t) is a convex function in ln t and ln t = o(ln γ(t)) (t → ∞), was found by B. Vynnytskyi and I. Sheparovych in 2001 ( [28]). Consider the class of analytic functions such that…”

mentioning

confidence: 76%

“…where η : [1, +∞) → (0, +∞) is an increasing convex function in ln t such that ln t = o(η(t)) (t → ∞). In 2001 in the PhD thesis of the second author [23, Theorem 3.1] (see also [29]) it was proved that given a sequence (z n ) in D, in order that for every (b n ) such that ∃C > 0 ∀n ∈ N : log |b n | ≤ Cη C 1 − |z n | there exist an analytic function from the class (4) satisfying (1), it is necessary that…”

mentioning

confidence: 99%

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Interpolation of analytic functions of moderate growth in the unit disc and zeros of solutions of a linear differential equation

Chyzhykov

Sheparovych

2014

Journal of Mathematical Analysis and Applications

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In 2002 A. Hartmann and X. Massaneda obtained necessary and sufficient conditions for interpolation sequences for classes of analytic functions in the unit disc such that log M (r, f ) = O((1 − r) −ρ ), 0 < r < 1, ρ ∈ (0, +∞), where M (r, f ) = max{|f (z)| : |z| = r}. Using another method, we give an explicit construction of an interpolating function in this result. As an application we describe minimal growth of the coefficient a such that the equation f ′′ + a(z)f = 0 possesses a solution with a prescribed sequence of zeros.MathSubjClass 2010: 30C15, 30H05, 30H99, 30J99.

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“…Combining Theorem 4 with the aforementioned result from [29] we are able to prove the following criterion, which essentially coincides with a result from [11].…”

Section: Resultssupporting

confidence: 74%

mentioning

confidence: 76%

mentioning

confidence: 99%

See 1 more Smart Citation

Interpolation of analytic functions of moderate growth in the unit disc and zeros of solutions of a linear differential equation

Chyzhykov

Sheparovych

2014

Journal of Mathematical Analysis and Applications

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“…Let g(z) be an entire function of form (5), that is a solution of (1) with b k,1 ≡ 0, b k,2 = − L ′′ (λ k ) 2L ′ (λ k ) . So, if f (z) = L(z)e g(z) , then a 0 of the form ( 11) is an entire function and by Theorem 1 it follows that f (z) satisfies condition (12). Corollary 1.…”

Section: Note That Condition (2) Ensures the Convergence Of L(z)mentioning

confidence: 92%

“…Various interpolation problems in spaces of entire functions were investigated in papers of many authors (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). A. Gel'fond ( [1]) and Y. Kaz'min ( [4]) considered the interpolation problem g(λ k ) = b k , k ∈ N, where λ k = β k−1 , |β| > 1.…”

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confidence: 99%

On interpolation problem with derivative in a space of entire functions with fast-growing interpolation knots

Vynnyts’kyi¹,

Sheparovych²

2019

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We obtaine the conditions on a sequence (bhas the unique solution in a subspace of entire functions g satisfying the condition ln M g (r) ≤ c 1 exp (N (r) + N (ρ 1 r)), where |λ k /λ k+1 | ≤ ∆ < 1, and N (r) is the Nevanlinna counting function of the sequence (λ k ). These results have been applied to describe solutions of the differential equation f ′′ + a 0 f = 0 with a coefficient a 0 from the some space of entire functions.

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On the interpolation in some classes of holomorphic in the unit disk functions

Sheparovych

2024

Mat. Stud.

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There is considered an interpolation problem $f(\lambda_n )=b_n$ in the class of holomorphic in the unit disk $U(0;1)=\{z\in\mathbb{C}\colon |z|<1\}$functions of finite $\eta$-type, i.e such that $\displaystyle (\exists A>0)(\forall z\in U(0;1))\colon \quad |f(z)|\leq\exp\Big(A\eta\Big(\frac A{1-|z|}\Big)\Big),$ where $\eta\colon [1;+\infty)\to [0;+\infty)$ is an increasing convex function with respect to $\ln{t}$ and $\ln{t}=o\left(\eta ( t)\right)$ $(t\to+\infty)$.There were received sufficient conditions of the interpolation problem solvability in terms of the counting functions $\displaystyle N(r)=\int\nolimits_{0}^{r}\frac{\left(n(t)-1\right)^+}{t}dt$ and $\displaystyle N_{\lambda_n} (r)=\int\nolimits_{0}^{r}{\frac{{{(n}_{\lambda_n}\left(t\right)-1)}^+}{t}dt}$. Earlier, in 2004, necessary conditions were obtained (Ukr. Math. J., {\bf 56} (2004), \No 3) in these terms.For the moderate growth of $f$ (when the majorant $\eta=\psi$ satisfies the condition $\psi\left(2x\right)=O\left(\psi\left(x\right)\right),\ x\rightarrow+\infty$) that problem was solved in J. Math. Anal. Appl., {\bf 414} (2014), \No 1.In this paper, we remove any restrictions on the growth of $\eta$ and construct an interpolation function $f$ such that$\displaystyle (\exists A'>0)(\forall z\in U(0;1))\colon \quad |{f}(z)|\leq\exp\Big(\frac{A'}{(1-|z|)^{3/2}}\eta\Big(\frac{A'}{1-|z|}\Big)\Big)$.

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Interpolation Sequences for the Class of Functions of Finite -Type Analytic in the Unit Disk

Cited by 3 publications

References 6 publications

Interpolation of analytic functions of moderate growth in the unit disc and zeros of solutions of a linear differential equation

Interpolation of analytic functions of moderate growth in the unit disc and zeros of solutions of a linear differential equation

On interpolation problem with derivative in a space of entire functions with fast-growing interpolation knots

On the interpolation in some classes of holomorphic in the unit disk functions

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