Konstantin M. Dyakonov
For an inner function $θ$ with $θ'\in\mathcal N$, where $\mathcal N$ is the Nevanlinna class, several problems are posed in connection with the canonical (inner-outer) factorization of $θ'$.
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Inner Functions, Möbius Distortion and Angular Derivatives
本文证明内函数有限$\mathcal{L}(p)$熵与累积莫比乌斯畸变属于$L^p$等价，并研究了使对应奇异内函数有有限$\mathcal{L}(p)$熵的正奇异测度支撑。
We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative $H^{\mathfrak{p}}$ membership purely in terms of contact order, a measure of the rate at which the zero set of a rational inner function approaches the distinguished boundary of the bidisk. We also show that derivatives of rational inner functions with singularities fail to be in $H^{\mathfrak{p}}$ for $\mathfrak{p}\ge\frac{3}{2}$ and that higher non-tangential regula…
Inner functions are an important and popular object of study in the field of complex function theory. We look at meromorphic inner functions with a given spectrum and provide sufficient conditions for them to have uniformly bounded derivatives on the real line. This question was first studied by Louis de Brange in 1968 and was later revived by Anton Baranov in 2011.
Let $\mathscr J$ be the set of inner functions whose derivatives lie in Nevanlinna class. In this note, we show that the natural map $F \to \text{Inn}(F'): \mathscr J/\text{Aut}(\mathbb{D}) \to \text{Inn}/S^1$ is is injective but not surjective. More precisely, we show that that the image consists of all inner functions of the form $BS_μ$ where $B$ is a Blaschke product and $S_μ$ is the singular factor associated to a measure $μ$ whose support is contained in a countable union of Beurling-Carleson sets. Our proof is based on extending the work of D. Kraus and O. Roth on maximal Blaschke produc…
For $1/2<p<1$, a description of inner functions whose derivative is in the Hardy space $H^p$ is given in terms of either their mapping properties or the geometric distribution of their zeros.
A one-component inner function $Θ$ is an inner function whose level set $$Ω_Θ(\varepsilon)=\{z\in \mathbb{D}:|Θ(z)|<\varepsilon\}$$ is connected for some $\varepsilon\in (0,1)$. We give a sufficient condition for a Blaschke product with zeros in a Stolz domain to be a one-component inner function. Moreover, a sufficient condition is obtained in the case of atomic singular inner functions. We study also derivatives of one-component inner functions in the Hardy and Bergman spaces. For instance, it is shown that, for $0<p<\infty$, the derivative of a one-component inner function $Θ$ is a member o…
We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular leve…
Higher-dimensional singularities of rational inner functions
本文研究多变量有理内函数的高维奇点，证明导数可积性条件沿零集分量不变且垂直直线奇点不影响全局可积性，并推广至n变量，给出新例。
Inner factors of Dirichlet space functions
本文研究狄利克雷空间函数的内因子分解，探讨了内函数的出现情况，并证明了奇异内因子的类比定理，同时给出了零点集的新充分条件。
Consider a scaled Nevanlinna-Pick interpolation problem and let $Π$ be the Blaschke product whose zeros are the nodes of the problem. It is proved that if $Π$ belongs to a certain class of inner functions, then the extremal solutions of the problem or most of them, are in the same class. Three different classical classes are considered: inner functions whose derivative is in a certain Hardy space, exponential Blaschke products and also the well known class of $α$-Blaschke products, for $0<α<1$.
For $\frac12<p<\infty$, $0<q<\infty$ and a certain two-sided doubling weight $ω$, we characterize those inner functions $Θ$ for which $$\|Θ'\|_{A^{p,q}_ω}^q=\int_0^1 \left(\int_0^{2π} |Θ'(re^{iθ})|^p dθ\right)^{q/p} ω(r)\,dr<\infty.$$ Then we show a modified version of this result for $p\ge q$. Moreover, two additional characterizations for inner functions whose derivative belongs to the Bergman space $A_ω^{p,p}$ are given.
Contractive analytic self-mappings of the disc
研究单位圆上收缩解析自映射的性质，通过Aleksandrov-Clark测度和内外因子分解描述，并探讨其边界行为。
Continuity of inner-outer factorization and cross sections from invariant subspaces to inner functions
本文研究内-外因子分解映射 $Q_{inn}$ 和 $Q_{out}$ 在 $H^{\infty}$ 子集上的 $H^2$ 连续性与 $H^{\infty}$ 不连续性，并证明从某些不变子空间到内函数无连续截面，同时探讨闭单位圆盘上解析函数的连续性。
Inner-outer factorization of analytic matrix-valued functions
本文研究解析矩阵值函数的内外因子分解，用乘法积分表示因子，并介绍Potapov定理，将压缩矩阵值函数分解为Blaschke-Potapov因子与乘法积分。
Clark measures for rational inner functions II: general bidegrees and higher dimensions
本文研究双圆盘二元及高维无奇性有理内函数的 Clark 测度，刻画其支撑、权重、酉嵌入条件，并揭示权重消失与奇性接触阶的关系。
In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges--Rovnyak spaces $\HH(b)$, where $b$ is in the unit ball of $H^\infty(\CC_+)$. In particular, we generalize a result of Ahern--Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel $k_{ω,n}^b$ of the evaluation of $n$-th derivative of elements of $\HH(b)$ a…
Let $\mathscr J$ be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on $\mathscr J$ where $F_n \to F$ if the critical structures of $F_n$ converge to the critical structure of $F$. We show that this occurs precisely when the critical structures of the $F_n$ are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic self-maps of the unit disk and solutions of the Gauss curvature equation. Building on the works of Korenblum and Roberts, we show that this topology also governs …
We study membership of rational inner functions in Dirichlet-type spaces in polydisks. In particular, we prove a theorem relating such inclusions to $H^p$ integrability of partial derivatives of a RIF, and as a corollary we prove that all rational inner functions on $\mathbb{D}^n$ belong to $\mathcal{D}_{1/n, \ldots ,1/n}(\mathbb{D}^n)$. Furthermore, we show that if $1/p \in \mathcal{D}_{α,...,α}$, then the RIF $\tilde{p}/p \in \mathcal{D}_{α+2/n,...,α+2/n}$. Finally we illustrate how these results can be applied through several examples, and how the Lojasiewicz inequality can sometimes be app…
Uniqueness theorems for meromorphic inner functions and canonical systems
本文研究上半平面亚纯内函数的唯一性，利用谱、导数值与Clark测度等数据，推广Borg-Levinson两谱定理并推动典范Hamilton系统逆谱理论。
We study representations of inner functions on the bidisc from a fractional linear transformation point of view, and provide sufficient conditions, in terms of colligation matrices, for the existence of two-variable inner functions. Here the sufficient conditions are not necessary in general, and we prove a weak converse for rational inner functions that admit one variable factorization. We present a complete classification of de Branges-Rovnyak kernels on the bidisc (which equally works in the setting of polydisc and the open unit ball of $\mathbb{C}^n$, $n \geq 1$). We also classify, in term…
Best approximations for the weighted combination of the Cauchy--Szegö kernel and its derivative in the mean
本文研究单位圆盘Hardy空间$H^1$中的极值逼近问题，给出由内函数和复数参数化的Cauchy-Szegö核及其导数加权组合在$H^1_0$子空间下最佳逼近的显式公式，并刻画了对应的极值函数。
We prove that, given a function $f$ in the Nevanlinna class $N$ and a positive integer $n$, there exist $g\in N$ and $h\in BMOA$ such that $f^{(n)}=gh^{(n)}$. We may choose $g$ to be zero-free, so it follows that the zero sets for the class $N^{(n)}:=\{f^{(n)}: f\in N\}$ are the same as those for $BMOA^{(n)}$. Furthermore, while the set of all products $gh^{(n)}$ (with $g$ and $h$ as above) is strictly larger than $N^{(n)}$, we show that the gap is not too large, at least when $n=1$. Precisely speaking, the class $\{gh': g\in N, h\in BMOA\}$ turns out to be the smallest ideal space containing …
The classical Gauss--Lucas theorem describes the location of the critical points of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role of polynomials is played by finite Blaschke products on the unit disk. We consider similar phenomena for generic inner functions, as well as for certain "locally inner" self-maps of the disk. More precisely, we look at a unit-norm function $f\in H^\infty$ that has an angular derivative on a set of positive measure (on the boundary) and we assume that its inner factor, $I$, is nontrivial. Under certain conditions to be discussed, i…
Critical values of inner functions
本文在有限熵内函数空间中证明内函数在单位圆周上几乎处处具有径向极限，并由此证明奇异值测度关于内函数连续变化，同时揭示Beurling-Carleson集与角导数之间的新联系。
We introduce the notion of {\it quasi-inner} function and show that the product $u=ρ_\infty\prod ρ_v$ of $m+1$ ratios of local {$L$-}factors {$ρ_v(z)=γ_v(z)/γ_v(1-z)$} over a finite set $F$ of places of the field of rational numbers {inclusive of} the archimedean place is {quasi-inner} on the left of the critical line $\Re(z)= \frac 12$ in the following sense. The off diagonal part $u_{21}$ of the matrix of the multiplication by $u$ in the orthogonal decomposition of the Hilbert space $L^2$ of square integrable functions on the critical line into the Hardy space $H^2$ and its orthogonal comple…
Functions in backward shift invariant subspaces have nice analytic continuation properties outside the spectrum of the inner function defining the space. Inside the spectrum of the inner function, Ahern and Clark showed that under some distribution condition on the zeros and the singular measure of the inner function, it is possible to obtain non-tangential boundary values of every function in the backward shift invariant subspace as well as for their derivatives up to a certain order. Here we will investigate, at least when the inner function is a Blaschke product, the non-tangential boundary…
Discrete random Clark measures and associated inner functions
研究一类随机内函数$\varphi$，其在$1$处克拉克测度为特定加权点质量之和。证明其几乎必然是布拉施克乘积，研究角导数情况并给出$0 - 1$律，还得到$\varphi$零点定量估计，其分布满足更强可和性条件。
We establish quantitative estimates for sampling (dominating) sets in model spaces associated with meromorphic inner functions, i.e. those corresponding to de Branges spaces. Our results encompass the Logvinenko-Sereda-Panejah (LSP) Theorem including Kovrijkine's optimal sampling constants for Paley-Wiener spaces. It also extends Dyakonov's LSP theoremfor model spaces associated with bounded derivative inner functions. Considering meromorphic inner functions allows us tointroduce a new geometric density condition, in terms of which the sampling sets are completely characterized. This, incompar…
We prove that the derivative $θ'$ of an inner function $θ$ is outer if and only if $θ$ is a Möbius transformation. An alternative characterization involving a reverse Schwarz--Pick type estimate is also given.
We continue our study of the set $\mathfrak I_c$ of inner functions $u$ in $H^\infty$ with the property that there is $η\in ]0,1[$ such that the level set $Ω_u(η):=\{z\in\mathbb D: |u(z)|<η\}$ is connected. These functions are called one-component inner functions. They play an important role in function theory and operator theory. Here we show that the composition of two one-component inner functions is again in $\mathfrak I_c$. We also give conditions under which a factor of one-component inner function belongs to $\mathfrak I_c$.
The convergence of DP Fourier series which are neither strongly convergent nor strongly divergent is discussed in terms of the Taylor series of the corresponding inner analytic functions. These are the cases in which the maximum disk of convergence of the Taylor series of the inner analytic function is the open unit disk. An essentially complete classification, in terms of the singularity structure of the corresponding inner analytic functions, of the modes of convergence of a large class of DP Fourier series, is established. Given a weakly convergent Fourier series of a DP real function, it i…
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