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

Chyzhykov, Igor; Sheparovych, Iryna

doi:10.1016/j.jmaa.2013.12.066

Cited by 10 publications

(10 citation statements)

References 18 publications

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“…The rest of the proof of Theorem 2.1 repeats that of Theorem F [3]. It consists in estimating of the interpolating function…”

Section: Lemma I (Chyzhykov and Sheparovych [3 Proposition 11]) Given A Functionmentioning

confidence: 75%

“…Its proof literally repeats that of Theorem C [3] with the only difference that we apply Theorem 2.1 instead of Theorem F. The same scheme is used in the proof of Theorem 2.5. In particular, after the substitution f (z) = P (z)e g (z) where g(z) is analytic in D, P (z) is the canonical product (2.2), the construction of an analytic function a reduces to the interpolation problem of finding an analytic function…”

Section: Resultsmentioning

confidence: 94%

“…Proof of Theorem 2.1. We follow the scheme of the proof of Theorem F from [3]. It follows from the estimate (2.3) and [3,Lemma 9]…”

Section: Proofs Of the Resultsmentioning

confidence: 99%

“…Note that the proof of sufficiency in [11] uses a non-constructive method of L 2 -estimate for the solution to a∂-equation. On the other hand, in [3], an interpolating function is constructed explicitly. The following theorem gives sufficient conditions for interpolation sequences in classes of analytic functions of moderate growth in the unit disc.…”

Section: Interpolation In the Unit Discmentioning

confidence: 99%

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On non-separated zero sequences of solutions of a linear differential equation

Chyzhykov¹,

Long²

2021

Proceedings of the Edinburgh Mathematical Society

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Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb {D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb {D}$ and such that possesses a solution having zeros precisely at the points $z_k$ , and the resulting function $a(z)$ has ‘minimal’ growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$ for any $p > 0$ . We established a new estimate for the maximum modulus of $a(z)$ in terms of the functions $n_z(t)=\sum \nolimits _{|z_k-z|\le t} 1$ and $N_z(r) = \int_0^r {{(n_z(t)-1)}^ + } /t{\rm d}t.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem.

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“…The rest of the proof of Theorem 2.1 repeats that of Theorem F [3]. It consists in estimating of the interpolating function…”

Section: Lemma I (Chyzhykov and Sheparovych [3 Proposition 11]) Given A Functionmentioning

confidence: 75%

Section: Resultsmentioning

confidence: 94%

“…Proof of Theorem 2.1. We follow the scheme of the proof of Theorem F from [3]. It follows from the estimate (2.3) and [3,Lemma 9]…”

Section: Proofs Of the Resultsmentioning

confidence: 99%

Section: Interpolation In the Unit Discmentioning

confidence: 99%

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On non-separated zero sequences of solutions of a linear differential equation

Chyzhykov¹,

Long²

2021

Proceedings of the Edinburgh Mathematical Society

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“…This result was developed in many papers [18][19][20][21][22][23][24]. S. Bank ([19]) proved the following theorem.…”

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

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 Zeros of Solutions of a Linear Differential Equation

Chyzhykov

Long

2021

Trends in Mathematics

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

Cited by 10 publications

References 18 publications

On non-separated zero sequences of solutions of a linear differential equation

On non-separated zero sequences of solutions of a linear differential equation

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

On Zeros of Solutions of a Linear Differential Equation

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