Blaschke Condition Research Articles

On universal quotient Blaschke products

In this paper, we introduce a new member to the universal families, called universal quotient Blaschke product, which is a formal quotient of two formal infinite Blaschke products. A formal infinite Blaschke product is of the form B ( z ) = ∏ k = 1 ∞ z − z k 1 − z k ¯ z , where { z k } k = 1 ∞ is a sequence of points in the unit disk but may not satisfy the Blaschke condition: ∑ k = 1 ∞ ( 1 − | z k | ) < ∞ . A partial quotient of a universal quotient Blaschke product is the quotient of two finite Blaschke products. We show that the set of partial quotients of a universal quotient Blaschke product is dense in the set of continuous self-mappings on the unit circle in the complex plane. Meanwhile, subsequences of the partial quotients of a universal quotient Blaschke product can be used to approximate any holomorphic functions bounded by one on the unit disk. Moreover, we prove that the set of universal quotient Blaschke products is huge in the sense of Baire category.

Boundary behavior of Berezin symbols and related results

For a given function ? ? H? with |?(z)| &lt; 1 (z ? D), we associate some special operators subspace and study some properties of these operators including behavior of their Berezin symbols. It turns that such boundary behavior is closely related to the Blaschke condition of sequences in the unit disk D of the complex plane. In terms of Berezin symbols the trace of some nuclear truncated Toeplitz operator is also calculated.

An Analog of the Hyperbolic Metric Generated by a Hilbert Space with the Schwarz–Pick Kernel

It is proved that a Hilbert function space on a set X with the Schwarz–Pick kernel (this is a wider class than the class of Hilbert spaces with the Nevanlinna–Pick kernel) generates a metric on the set X which is an analog of the hyperbolic metric in the unit disk. For a sequence satisfying an abstract Blaschke condition, it is proved that the associated infinite Blaschke product converges uniformly on any fixed bounded set and in the strong operator topology of the multiplier space. Bibliography: 8 titles.

Solutions of Complex Differential Equation Having Pre-given Zeros in the Unit Disc

Behavior of solutions of $f''+Af=0$ is discussed under the assumption that $A$ is analytic in $\mathbb{D}$ and $\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty$, where $\mathbb{D}$ is the unit disc of the complex plane. As a main result it is shown that such differential equation may admit a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature. It is also proved that $\Lambda\subset\mathbb{D}$ is the zero-sequence of a non-trivial solution of $f''+Af=0$ where $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure if and only if $\Lambda$ is uniformly separated. As an application an old result, according to which there exists a non-normal function which is uniformly locally univalent, is improved.

ON THE COMPLEXITY OF FINDING A NECESSARY AND SUFFICIENT CONDITION FOR BLASCHKE-OSCILLATORY EQUATIONS

AbstractIf A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .

Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature

A classical result due to Blaschke states that for every analytic self-map f of the open unit disc of the complex plane there exists a Blaschke product B such that the zero sets of f and B agree. Indeed, a sequence is the zero set of an analytic self-map of the open unit disc if and only if it satisfies the simple geometric condition known as the Blaschke condition. In contrast, the critical sets of analytic self-maps of the open unit disc have not been completely described yet. In this paper, we show that for every analytic self-map f of the open unit disc 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 weighted Bergman space as well as to the Berger–Nirenberg problem from differential geometry. By solving the Berger–Nirenberg problem in a special case, we identify the critical sets of bounded analytic functions with the zero sets of the weighted Bergman space 𝒜12.

Blaschke-Type Conditions for Analytic and Subharmonic Functions in the Unit Disk: Local Analogs and Inverse Problems

We continue the study of analytic and subharmonic functions in the unit disk of finite order with an arbitrary set of singular points on the unit circle, introduced in [6]. We focus on a local analog of the main result from [6], which is similar to the standard local Blaschke condition for analytic functions bounded in a lune of the unit disk. We also discuss the inverse problem, i.e. the existence of a subharmonic function from this class with a given singular set on the circle and given Riesz measure subject to a certain Blaschke-type condition.

Blaschke condition and zero sets in weighted Dirichlet spaces

Let D(μ) be the Dirichlet space weighted by the Poisson integral of the positive measure μ. We give a characterization of the measures μ equal to a countable sum of atoms for which the Blaschke condition is a necessary and sufficient condition for a sequence to be a zero set for D(μ).

Strong asymptotics of extremal polynomials on the segment in the presence of denumerable set of mass points

The strong asymptotics of the monic extremal polynomials with respect to a \(L_{p}(\sigma )\) norm are studied. The measure \(\sigma \) is concentrated on the segment \([-1,1]\) plus a denumerable set of mass points which accumulate at the boundary points of \([-1,1]\) only. Under the assumptions that the mass points satisfy Blaschke's condition and that the absolutely continuous part of \(\sigma \) satisfies Szeg?'s condition.

Boundary Values and Boundary Uniqueness of J-Holomorphic Mappings

We establish a Fatou-type theorem for J-holomorphic mappings that are bounded in an appropriate sense and prove the Blaschke condition for their zero sets. We also prove a Privalov-type uniqueness theorem for pairs of J-holomorphic mappings.

Remark on Zero Sets of Holomorphic Functions in Convex Domains of Finite Type

AbstractWe prove that if the (1, 1)-current of integration on an analytic subvariety V ⊂ D satisfies the uniform Blaschke condition, then V is the zero set of a holomorphic function ƒ such that log |ƒ| is a function of bounded mean oscillation in bD. The domain D is assumed to be smoothly bounded and of finite d’Angelo type. The proof amounts to non-isotropic estimates for a solution to the -equation for Carleson measures.

CMV matrices with asymptotically constant coefficients. Szegö–Blaschke class, scattering theory

We develop a modern extended scattering theory for CMV matrices with asymptotically constant Verblunsky coefficients. We show that the traditional (Faddeev–Marchenko) condition is too restrictive to define the class of CMV matrices for which there exists a unique scattering representation. The main results are: (1) the class of twosided CMV matrices acting in l 2 , whose spectral density satisfies the Szegö condition and whose point spectrum the Blaschke condition, corresponds precisely to the class where the scattering problem can be posed and solved. That is, to a given CMV matrix of this class, one can associate the scattering data and the FM space. The CMV matrix corresponds to the multiplication operator in this space, and the orthonormal basis in it (corresponding to the standard basis in l 2 ) behaves asymptotically as the basis associated with the free system. (2) From the point of view of the scattering problem, the most natural class of CMV matrices is that one in which (a) the scattering data determine the matrix uniquely and (b) the associated Gelfand–Levitan–Marchenko transformation operators are bounded. Necessary and sufficient conditions for this class can be given in terms of an A 2 kind condition for the density of the absolutely continuous spectrum and a Carleson kind condition for the discrete spectrum. Similar conditions close to the optimal ones are given directly in terms of the scattering data.

On the zeros of functions in Bergman spaces and in some other related classes of functions

A well-known theorem of H.S. Shapiro and A.L. Shields implies that if f ≢ 0 is a function which belongs to the Bergman space A p ( 0 < p < ∞ ) and { z k } is a sequence of zeros of f which is contained in a Stolz angle, then { z k } satisfies the Blaschke condition. In this paper we improve this result. We consider a large class of regions contained in the unit disc D which touch ∂ D at a point ξ tangentially and we prove that the mentioned result remains true if we substitute a Stolz angle by any of these regions of tangential approach.

DECREASE OF BOUNDED HOLOMORPHIC FUNCTIONS ALONG DISCRETE SETS

AbstractWe provide uniqueness results for holomorphic functions in the Nevanlinna class which bridge those previously obtained by Hayman and by Lyubarskii and Seip. In particular, we propose certain classes of hyperbolically separated sequences in the disc, in terms of the rate of non-tangential accumulation to the boundary (the outer limits of this spectrum of classes being, respectively, the sequences with a non-tangential cluster set of positive measure, and the sequences satisfying the Blaschke condition). For each of those classes, we give a critical condition of radial decrease on the modulus which will force a Nevanlinna class function to vanish identically.AMS 2000 Mathematics subject classification: Primary 30D50; 30D55

Zero varieties for the Nevanlinna class on all convex domains of finite type

It is shown, that the so-called Blaschke condition characterizes in any bounded smooth convex domain of finite type exactly the divisors which are zero sets of functions of the Nevanlinna class on the domain. The main tool is a non-isotropic L1 estimate for solutions of the Cauchy-Riemann equations on such domains, which are obtained by estimating suitable kernels of Berndtsson-Andersson type.

Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points

Let a be a positive measure whose support is an interval E plus a denumerable set of mass points which accumulate at the boundary points of E only. Under the assumptions that the mass points satisfy Blaschke's condition and that the absolutely continuous part of a satisfies Szego's condition, asymptotics for the orthonormal polynomials on and off the support are given. So far asymptotics were only available if the set of mass points is finite.

Reproducing kernels and contractive divisors in bergman spaces

In the Hardy spaces Hp of holomorphic functions, Blaschke products are applied to factor out zeros. However, for Bergman spaces, the zero sets of which do not necessarily satisfy the Blaschke condition, the study of divisors is a more recent development. Hedenmalm proved the existence of a canonical contractive zero-divisor which plays the role of a Blascke product in the Bergman space\(L_\alpha ^2 \left( \mathbb{D} \right)\). Duren, Khavinson, Shapiro, and Sundberg later extended Hedenmalm's result to\(L_\alpha ^p \left( \mathbb{D} \right)\), 0<p<∞. In this paper, an explicit formula for the contractive divisor is given for a zero set that consists of two points of arbitrary multiplicities. There is a simple one-to-one correspondence between contractive divisors and reproducing kernels for certain weighted Bergman spaces. The divisor is obtained by calculating the associated reproducing kernel. The formula is then applied to obtain the contractive divisor for a certain regular zero set, as well as the contractive divisor associated with an inner function that has singular support on the boundary. Bibliography: 13 titles.

A remark on spaces of holomorphic maps to taut manifolds

We generalize the qualitative part of the Blaschke condition to holomorphic mappings to a given taut manifold. Namely, we prove that given two taut complex manifolds X and Y there exists a discrete subset \( S\subset X \) such that the natural restriction map induces a homeomorphism between Hol (X,Y ) and a closed subset of Map (S,Y ).

Zero Varieties for the Nevanlinna Class in Convex Domains of Finite Type in \Bbb C n

with X = {ak}, each ak being counted according to its multiplicity. In this case too, it is well known that conversely the Blaschke condition implies that the sequence X is the zero sequence of a function f E N(Q) or even a bounded holomorphic function. The question whether in dimension n > 1 the Blaschke condition (B) is sufficient for the divisor X = {Xk, nk} to be defined by a function f E N(Q) has been one of the main topics in this area. The first result was independently obtained by Henkin [11] and Skoda [14], who proved it for strictly pseudoconvex domains, with some obvious topological conditions. Also some partial results

On functions of bounded type*

We consider certain functions meromorphic in the plane that are of bounded type in suitable half planes. Let Z j be a sequence in a half plane not satisfying a Blaschke condition there. We deduce that conditions of being positive semi-definite on matrices whose elements depend on z j and the values that such a function f takes at the z j, determine f uniquely. This can be used to study the properties of f further. In particular, we give a function theoretic derivation for the fact that a certain set of inequalities is equivalent to Riemann's hypothesis. We also comment on iteration theoretic methods for studying such problems.