We continue the study of analytic functions in the unit disk of finite order with arbitrary set of singular points on the unit circle, introduced in \cite{FG}. The main focus here is made upon the inverse problem: the existence of a function from this class with a given singular set and zero set subject to certain Blaschke-type condition. We also discuss the local analog of the main result from \cite{FG} similar to the standard local Blaschke condition for analytic and bounded functions in the unit disk.
We prove that the composition of two indestructible Blaschke products is again an indestructible Blaschke product. We also show that if an indestructible Blaschke product is the composition of two bounded analytic functions, then both functions are indestructible Blaschke products.
We give a generalization of the notion of finite Blaschke products from the perspective of generalized inner functions in various reproducing kernel Hilbert spaces. Further, we study precisely how these functions relate to the so-called Shapiro--Shields functions and shift-invariant subspaces generated by polynomials. Applying our results, we show that the only entire inner functions on weighted Hardy spaces over the unit disk are multiples of monomials, extending recent work of Cobos and Seco.
We obtain an upper bound for the derivative of a Blaschke product, whose zeros lie in a certain Stolz-type region. We show that the derivative belongs to the space of analytic functions in the unit disk, introduced recently in \cite{FG}. As an outcome, we obtain a Blaschke-type condition for critical points of such Blaschke products.
We consider the classical problem of maximizing the derivative at a fixed point over the set of all bounded analytic functions in the unit disk with prescribed critical points. We show that the extremal function is essentially unique and always an indestructible Blaschke product. This result extends the Nehari--Schwarz Lemma and leads to a new class of Blaschke products called maximal Blaschke products. We establish a number of properties of maximal Blaschke products, which indicate that maximal Blaschke products constitute an appropriate infinite generalization of the class of finite Blaschke…
In this survey paper, we discuss the problem of characterizing the critical sets of bounded analytic functions in the unit disk of the complex plane. This problem is closely related to the Berger-Nirenberg problem in differential geometry as well as to the problem of describing the zero sets of functions in Bergman spaces. It turns out that for any non-constant bounded analytic function in the unit disk there is always a (essentially) unique "maximal" Blaschke product with the same critical points. These maximal Blaschke products have remarkable properties simliar to those of Bergman space inn…
Let $E$ be a closed set on the unit circle. We find a Blaschke-type condition, optimal in a sense of the order, on the Riesz measure of a subharmonic function $v$ in the unit disk with a certain growth at the direction of $E$. In particular case when $E$ is a finite set, and $v=\log|f|$ with an analytic function $f$, our result agrees with the recent one by A. Borichev, L. Golinskii and S. Kupin. An application to contractions close to unitary operators in the Hilbert space is given.
We show that a nonvanishing analytic function on a domain in the unit disc can be approximated by (a scalar multiple of) a Blaschke product whose zeros lie on a prescribed circle enclosing the domain. We also give a new proof of the analogous classical result for polynomials. A connection is made to universality results for the Riemann zeta function.
Recently, several papers have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either nonlinear phase unwinding or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of a complex function $F: \partial\mathbb{D}\to \mathbb{C}$, and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was studied by Coifman and Stei…
This is a sequel of a recent article by Borichev-Golinskii-Kupin, where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were applied to get Lieb-Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set. The first result of the present paper is an appropriate local version of the Blaschke-type condition from Borichev-Golinskii-Kupin. We apply it to obtain a similar condition for an analytic function in a finitely connected d…
We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation problem.
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.
The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space $\mathbf H_2$ of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of $\mathbb C^N$, and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury-Arveson space of functions analytic in the unit ball of $\mathbb C^N$. This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy a…
A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map $f$ of the open unit disk there is even an indestructible Blaschke product $B$ such that the critical sets of $f$ and $B$ coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighte…
We give a survey of results on zero distribution and factorization of analytic functions in the unit disc in classes defined by the growth of $\log|f(re^{iθ})|$ in the uniform and integral metrics. We restrict ourself by the case of finite order of growth. For a Blaschke product $B$ we obtain a necessary and sufficient condition for the uniform boundedness of all $p$-means of $\log|B(re^{iθ})|$, where $p>1$.
Answering a question of A.Zygmund in \cite{MR} G.MacLane and L.Rubel described boundedness of $L_2$-norm w.r.t. the argument of $\log |B|$, where $B$ is a Blaschke product. We generalize their results in several directions. We describe growth of $p$th means, $p\in(1, \infty)$, of subharmonic functions bounded from above in the unit disc. Necessary and sufficient conditions are formulated in terms of the complete measure (of a subharmonic function) in the sense of A.Grishin. We also prove sharp estimates of the growth of $p$th means of analytic and subharmonic functions of finite order in the u…
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…
Extending the results of Borichev--Golinskii--Kupin [2009], we obtain refined Blaschke-type necessary conditions on the zero distribution of analytic functions on the unit disk and on the complex plane with a cut along the positive semi-axis satisfying some non-radial growth restrictions.
We apply a recent result of Borichev-Golinskii-Kupin on the Blaschke-type conditions for zeros of analytic functions on the complex plane with a cut along the positive semi-axis to the problem of the eigenvalues distribution of the Fredholm-type analytic operator-valued functions.
In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. Via the Bers embedding, one may view the Weil-Petersson metric as a metric on the main cardioid of the Mandelbrot set. We prove that the metric completion attaches the geometrically finite parameters from the Euclidean boundary of the main cardioid and conjecture that this is the entire completion. For the upper bound, we estimate the intersection of a circle $S_r = \{z : |z| = r\}$, $r \approx 1$, with an invariant subset $\mathcal G \subset \mathbb{D}$ called a …
We answer affirmatively the problem left open in \cite{DSZ,GSZZ} and prove that for a finite Blaschke product $ϕ$, the minimal reducing subspaces of the Bergman space multiplier $M_ϕ$ are pairwise orthogonal and their number is equal to the number $q$ of connected components of the Riemann surface of $ϕ^{-1}\circ ϕ$. In particular, the double commutant $\{M_ϕ,M_ϕ^\ast\}'$ is abelian of dimension $q$. An analytic/arithmetic description of the minimal reducing subspaces of $M_ϕ$ is also provided, along with a list of all possible cases in degree of $ϕ$ equal to eight.
The classical theorems of Mittag-Leffler and Weierstrass show that when $\{λ_n\}$ is a sequence of distinct points in the open unit disk $\D$, with no accumulation points in $\D$, and $\{w_n\}$ is any sequence of complex numbers, there is an analytic function $ϕ$ on $\D$ for which $ϕ(λ_n) = w_n$. A celebrated theorem of Carleson \cite{MR117349} characterizes when, for a bounded sequence $\{w_n\}$, this interpolating problem can be solved with a bounded analytic function. A theorem of Earl \cite{MR284588} goes further and shows that when Carleson's condition is satisfied, the interpolating func…
Subharmonic functions associated with the stationary Schrodinger operator are its weak subsolutions under appropriate assumptions on the potential of the operator. We prove for these functions analogs of several classical results on analytic and subharmonic functions, such as the Phragmen-Lindelof theorem with the precise growth rate, the Blaschke theorem on bounded analytic functions, the Carleman formula, the Hayman-Azarin theorem on asymptotic behavior of special subharmonic functions in many-dimensional cones.
Let $E$ be a subset of the unit disc $U$ of the complex plane $\CC$. Recall that $H^p(U)$ is the space of all holomorphic functions $g$ on $U$ for which $\|g\|_{H^p}$ $<$ $\infty$. Put \begin{equation} C_p(ε, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, \|g\|_p\leq 1, |g(ζ)| \leq ε\forall ζ\in E\}, \end{equation} for positive $ε$ and $R$ in $(0, 1)$. It can be seen that $C_p(ε, R)$ is bounded from above by $(1-R^2)^{-1/p}$. \begin{theorem} If $\bar E$ $\subset$ $U$ then there exists $ε_0>0$ such that for $0<ε<ε_0$ there is correspondingly a finite Blaschke product $B_ε(z)$ whose zeros are in…
Functions of bounded characteristic in simply connected domains have a classical factorization to Blaschke, outer and singular inner parts. The latter has a singular measure on the boundary assigned to it. The exponential speed of change of a function when approaching a point of a boundary (mean type) corresponds to a point mass at this point. In this paper we consider the analogous relation for functions in arbitrary infinitely connected (Denjoy) domains. The factorization result holds of course with one important addition: all functions involved become multiple valued even though the initial…
We are concerned with extensions of the Mason--Stothers $abc$ theorem from polynomials to analytic functions on the unit disk $\mathbb D$. The new feature is that the number of zeros of a function $f$ in $\mathbb D$ gets replaced by the norm of the associated Blaschke product $B_f$ in a suitable smoothness space $X$. Such extensions are shown to exist, and the appropriate $abc$-type estimates are exhibited, provided that $X$ admits a "Garsia-type norm", i.e., a norm sharing certain properties with the classical Garsia norm on BMO. Special emphasis is placed on analytic Lipschitz spaces.
Let $\mathcal{N}$ be the Nevanlinna class and let $B$ be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra $\mathcal{N} / B \mathcal{N}$, that is, $|f| \ge e^{-H} $ on the set $B^{-1}\{0\}$ for some positive harmonic function $H$, holds if and only if the function $- \log |B|$ has a harmonic majorant on the set $\{z\in\mathbb{D}:ρ(z,Λ)\geq e^{-H(z)}\}$; at least for large enough functions $H$. We also study the corresponding class of positive harmonic functions $H$ in the unit disc such that the latter condition holds. We also discuss the analogou…
We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrised bidisc $Γ$. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence $\mathcal{C}_ν, ν\geq 0,$ of necessary conditions for solvability, prove that they are of strictly increasing strength and show…
In this paper, we continue the study of the polar analytic functions, a notion introduced in \cite{BBMS1} and successfully applied in Mellin analysis. Here we obtain another version of the Cauchy integral formula and a residue theorem for polar Mellin derivatives, employing the new notion of logarithmic pole. The identity theorem for polar analytic functions is also derived. As applications we obtain an analogue of Boas' differentiation formula for polar Mellin derivatives, and an extension of the classical Bernstein inequality to polar Mellin derivatives. Finally we give an analogue of the we…
Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed $q-$difference-differential equations in the complex domain are constructed. They stand for a $q-$analog of the singularly perturbed partial differential equations considered in our recent work [A. Lastra, S. Malek, Boundary layer expansions for initial value problems with two complex time variables, submitted 2019]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to …
We propose a definition of sampling set for the Nevanlinna class in the disk, i.e. a subset of the disk such that the analogue of the norm of a function in the Nevanlinna class can be recovered only from its values on the subset. We show it is equivalent with the notion of determination set for the same class, that is, subsets such that any function in the class which is bounded on the subset must be bounded everywhere (by the same bound, in fact); and more restrictive than the condition of being a determination set for the class of differences of positive harmonic functions (which had been st…
It was shown classically that matrix monotone and matrix convex functions must be real analytic by Löwner and Kraus respectively. Recently, various analogues have been found in several noncommuting variables. We develop a general framework for lifting automatic analyticity theorems in matrix analysis from one variable to several variables, the so-called "royal road theorem." That is, we establish the principle that the hard part of proving any automatic analyticity theorem lies in proving the one variable theorem. We use our main result to prove the noncommutative Löwner and Kraus theorems ove…
