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We develop a theory of quasi post-critically finite degenerations of Blaschke products. This gives us tools to study the boundaries of hyperbolic components of rational maps in higher-dimensional moduli spaces. We use it to obtain a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic component Hd, i.e., the hyperbolic component in the space of monic and centered polynomials that contains zd. We also show that the closure Hd is not a topological manifold with boundary for d ≥ 4 by constructing self-bumps on its boundary. [ABSTRACT FROM AUTHOR]

For a bounded analytic function $\varphi $ on the unit disk $\mathbb {D}$ with $\|\varphi \|_\infty \le 1$ , we consider the defect operators $D_\varphi $ and $D_{\overline \varphi }$ of the Toeplitz operators $T_{\overline \varphi }$ and $T_\varphi $ , respectively, on the weighted Bergman space $A^2_\alpha $. The ranges of $D_\varphi $ and $D_{\overline \varphi }$ , written as $H(\varphi)$ and $H(\overline \varphi)$ and equipped with appropriate inner products, are called sub-Bergman spaces. We prove the following three results in the paper: for $-1 , the space $H(\varphi)$ has a complete Nevanlinna–Pick kernel if and only if $\varphi $ is a Möbius map; for $\alpha>-1$ , we have $H(\varphi)=H(\overline \varphi)=A^2_{\alpha -1}$ if and only if the defect operators $D_\varphi $ and $D_{\overline \varphi }$ are compact; and for $\alpha>-1$ , we have $D^2_\varphi (A^2_\alpha)= D^2_{\overline \varphi }(A^2_\alpha)=A^2_{\alpha -2}$ if and only if $\varphi $ is a finite Blaschke product. In some sense, our restrictions on $\alpha $ here are best possible. [ABSTRACT FROM AUTHOR]

In this paper, we prove the uniqueness theorem for a special class of meromorphic functions on the complex plane C. In particular, we study the class of meromorphic functions f in the domain C\K′, where K′ is the finite set of limit points of simple poles of the function f. In this class, we describe non-trivial subclasses in which every function f can be uniquely determined by the residues of the function f at its poles. The result covered in this paper is a part of a problem in a spectral operator theory. [ABSTRACT FROM AUTHOR]

We simplify the proofs of Colwell's theorems on Blaschke products. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the precise behavior of orbits inside attracting basins. Let f be a holomorphic polynomial of degree m ≥ 2 in C , A (p) be the basin of attraction of an attracting fixed point p of f, and Ω i (i = 1 , 2 , ⋯) be the connected components of A (p) . Assume Ω 1 contains p and { f - 1 (p) } ∩ Ω 1 ≠ { p } . Then there is a constant C so that for every point z 0 inside any Ω i , there exists a point q ∈ ∪ k f - k (p) inside Ω i such that d Ω i (z 0 , q) ≤ C , where d Ω i is the Kobayashi distance on Ω i. In paper Hu (Dynamics inside parabolic basins, 2022), we proved that this result is not valid for parabolic basins. [ABSTRACT FROM AUTHOR]

We show the structure of the parameter space for a family of rational maps containing Blaschke products. Through numerical simulations using the orbit of a single critical point, we reveal the existence of infinitely many Mandelbrot-like sets along the unit circle, as well as eight-like structures in other regions of parameter space. We pose some open questions related to the parameter space of these functions.

We study the decomposability of a finite Blaschke product B of degree 2^n into n degree-2 Blaschke products, examining the connections between Blaschke products, Poncelet's theorem, and the monodromy group. We show that if the numerical range of the compression of the shift operator, W(S_B), with B a Blaschke product of degree n, is an ellipse, then B can be written as a composition of lower-degree Blaschke products that correspond to a factorization of the integer n. We also show that a Blaschke product of degree 2^n with an elliptical Blaschke curve has at most n distinct critical values, and we use this to examine the monodromy group associated with a regularized Blaschke product B. We prove that if B can be decomposed into n degree-2 Blaschke products, then the monodromy group associated with B is the wreath product of n cyclic groups of order 2. Lastly, we study the group of invariants of a Blaschke product B of order 2^n when B is a composition of n Blaschke products of order 2. [ABSTRACT FROM AUTHOR]

We present results concerning the cyclic behaviour of weighted shifts and composition operators on the disc algebra. Most of our proofs use the representations of the dual space of the disc algebra as Banach spaces of analytic functions on the unit disc. Taking into account that no composition operator on the disc algebra can be hypercyclic, the results we obtain show that the composition operators induced by linear fractional maps with an interior fixed points have the same cyclic behaviour on the disc algebra as they do on the Hardy space H 2 , as described in Bourdon and Shapiro [Cyclic phenomena for composition operators. Mem Am Math Soc. 1997;125(596)], while the operators induced by linear fractional maps with boundary fixed points have the same cyclic behaviour on the disc algebra as they do on the Dirichlet space, as described in Gallardo-Gutiérrez and Montes-Rodríguez [The role of the spectrum in the cyclic behavior of composition operators. Mem Am Math Soc. 2004;167(791)]. [ABSTRACT FROM AUTHOR]

We derive new estimates on analytic capacities of finite sequences in the unit disc in Besov spaces with zero smoothness, which sharpen the estimates obtained by N.K. Nikolski in 2005 and, for a range of parameters, are optimal. The work is motivated both from the perspective of complex analysis by the description of sets of zeros/uniqueness, and from the one of matrix analysis/operator theory by estimates on norms of inverses. [ABSTRACT FROM AUTHOR]

In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂn cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n.

Analytic functions, despite exhibiting very satisfactory behavior inside their domains of definition, e.g., being infinitely differentiable and having power series representations, could be quite wild and show bizarre performance at the boundary points. Not being continuous at a single boundary point is the simplest of such phenomena. However, much more pathological behaviors are foreseeable in the frontiers. That is why boundary behavior studies have faced several obstacles since early days of complex analysis, even though in many cases the final outcome is very appealing and usually ends with profound and beautiful mathematical results. Dilation is an old technique that mathematicians have exploited to deviate from the frontiers and go deep inside the domain. Then, standing within the domain and coming back from inside toward the frontiers, since our footsteps are in a solid territory on which the function behaves well and is fully under the control, one tries to better understand the boundary behavior at the frontiers. No wonder the story has many ups and downs and is filled with numerous successes and failures. Starting with a pioneering theorem of F. Riesz in 1923, we provide a wide range of results which are uniformly spread over a century to tell parts of the dilation history in the complex plane. We finish with some recent results and open questions, obtained in the course of the last decade, to highlight the need for novel tools in studying newly emerged function spaces and the boundary behavior of their elements. [ABSTRACT FROM AUTHOR]

In this article, we briefly describe nearly T - 1 invariant subspaces with finite defect for a shift operator T having finite multiplicity acting on a separable Hilbert space H as a generalization of nearly T - 1 invariant subspaces introduced by Liang and Partington in Complex Anal. Oper. Theory15(1) (2021) 17 pp. In other words, we characterize nearly T - 1 invariant subspaces with finite defect in terms of backward shift invariant subspaces in vector-valued Hardy spaces by using Theorem 3.5 in Int. Equations Oper. Theory92 (2020) 1–15. Furthermore, we also provide a concrete representation of the nearly T B - 1 invariant subspaces with finite defect in a scale of Dirichlet-type spaces D α for α ∈ [ - 1 , 1 ] corresponding to any finite Blashcke product B, as was done recently by Liang and Partington for defect zero case (see Section 3 of Complex Anal. Oper. Theory15(1) (2021) 17 pp). [ABSTRACT FROM AUTHOR]

In this paper, we show that there always exists reducing subspace M for Sψ(z) such that the restriction of Sψ(z) on M is unitarily equivalent to the Bergman shift when ψ(z) is a finite Blaschke product. Moreover, we will show that only if ψ(z) is a finite Blaschke product can Sψ(z) has distinguished reducing subspaces. We also give the form of these distinguished reducing subspaces when ψ(z) is a finite Blaschke product. Finally, we show that every nontrivial minimal reducing subspace S of Sψ(z) is orthogonal to the direct sum of all distinguished subspaces when S is not a distinguished subspace of Sψ(z). [ABSTRACT FROM AUTHOR]

AVe survey a number of recent generalizations and sharpenings of Ne-hari's extension of Schwarz' lemma for holomorphic self-maps of the unit disk. In particular, we discuss the case of infinitely many critical points and its relation to the zero sets and invariant subspaces for Bergman spaces, as well as the case of equality at the boundary. [ABSTRACT FROM AUTHOR]

In this paper, we show that there always exists reducing subspace M for Sψ(z) such that the restriction of Sψ(z) on M is unitarily equivalent to the Bergman shift when ψ(z) is a finite Blaschke product. Moreover, we will show that only if ψ(z) is a finite Blaschke product can Sψ(z) has distinguished reducing subspaces. We also give the form of these distinguished reducing subspaces when ψ(z) is a finite Blaschke product. Finally, we show that every nontrivial minimal reducing subspace S of Sψ(z) is orthogonal to the direct sum of all distinguished subspaces when S is not a distinguished subspace of Sψ(z). [ABSTRACT FROM AUTHOR]

We show that in a scale of weighted Dirichlet spaces D α , including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in D α such that B satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell et al. (Indiana Univ Math J 51(4):931–961, 2002). As a particular instance, when B (z) = z k and | α | ≤ log (2) log (k + 1) , the chosen norm is the usual one in D α . [ABSTRACT FROM AUTHOR]

In this note, we prove that the uni‐modular mono‐components from Blaschke products in the unit disc are eigenvectors of the Sturm‐Liouville operators and investigate a kind of wave equations with some initial conditions, which give rise to mono‐components. [ABSTRACT FROM AUTHOR]

One important problem in the theory of Hardy space is to find the best rational approximation of a given order to a function in the Hardy space H 2 on the unit disk. It is equivalent to finding the best Blaschke form with free poles. The cyclic adaptive Fourier decomposition method is based on the grid search technique. Its approximative precision is limited by the grid spacing. This paper proposes two enhanced methods of the cyclic adaptive Fourier decomposition. The proposed algorithms utilize the gradient descent optimization to tune the best pole-tuple on the mesh grids, reaching higher precision. Their performances are confirmed by several examples. [ABSTRACT FROM AUTHOR]

We prove an estimate for the capacity of the condenser (D,Kr), r ∈ (0, 1), where D is the open unit disc and {Kr} is a compact exhaustion of the inverse image of a compact set under a Blaschke product B, involving weighted logarithmic integral means of the Frostman shifts of B. Also, we describe the asymptotic behavior of the capacity of condensers (D, Er), where Er is a connected component of the inverse image of a closed disc with radius r under universal covering maps as r → 0. [ABSTRACT FROM AUTHOR]

There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [16]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the theory of OPUC. [ABSTRACT FROM AUTHOR]

Abstract Let I be an inner function in the unit disk D and let N denote the Nevanlinna class. We prove that under natural assumptions, Bézout equations in the quotient algebra N / I N can be solved if and only if the zeros of I form a finite union of Nevanlinna interpolating sequences. This is in contrast with the situation in the algebra of bounded analytic functions, where being a finite union of interpolating sequences is a sufficient but not necessary condition. An analogous result in the Smirnov class is proved as well as several equivalent descriptions of Blaschke products whose zeros form a finite union of interpolating sequences in the Nevanlinna class. [ABSTRACT FROM AUTHOR]

This paper presents a novel signal compression algorithm based on the Blaschke unwinding adaptive Fourier decomposition (AFD). The Blaschke unwinding AFD is a newly developed signal decomposition theory. It utilizes the Nevanlinna factorization and the maximal selection principle in each decomposition step, and achieves a faster convergence rate with higher fidelity. The proposed compression algorithm is applied to the electrocardiogram signal. To assess the performance of the proposed compression algorithm, in addition to the generic assessment criteria, we consider the less discussed criteria related to the clinical needs—for the heart rate variability analysis purpose, how accurate the R-peak information is preserved is evaluated. The experiments are conducted on the MIT-BIH arrhythmia benchmark database. The results show that the proposed algorithm performs better than other state-of-the-art approaches. Meanwhile, it also well preserves the R-peak information. [ABSTRACT FROM AUTHOR]

We consider several seemingly unrelated theorems, each with the word ellipse in the statement: Siebeck's theorem, the elliptical range theorem, Poncelet's theorem, and the Blaschke ellipse theorem. Though Siebeck's theorem is a geometric statement about complex functions, we use linear algebra and the numerical range of a matrix to provide a proof of the theorem. Poncelet's theorem, from projective geometry, and rational functions known as Blaschke products provide some surprising additional connections. [ABSTRACT FROM AUTHOR]

For a Blaschke product B of degree d and λ on ∂ D , let ℓ λ be the set of lines joining each distinct two preimages in B − 1 ( λ ) . The envelope of the family of lines { ℓ λ } λ ∈ ∂ D is called the interior curve associated with B . In 2002, Daepp, Gorkin, and Mortini proved the interior curve associated with a Blaschke product of degree 3 forms an ellipse. While let L λ be the set of lines tangent to ∂ D at the d preimages B − 1 ( λ ) and the trace of the intersection points of each two elements in L λ as λ ranges over the unit circle is called the exterior curve associated with B . In 2017, the author proved the exterior curve associated with a Blaschke product of degree 3 forms a non-degenerate conic. In this paper, for a Blaschke product of degree d , we give some geometrical properties that lie between the interior curve and the exterior curve. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms to an Orlicz setting and an Orlicz-Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms is established. The new Orlicz-Brun-Minkowski inequality in special case yields the Lp-Brunn-Minkowski inequality for the radial mixed Blaschke-Minkowski homomorphisms and the mixed intersection bodies, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we prove a famous harmonic majorant lemma, and by applying this lemma to log+ [f], we claim that log+ [f], has indeed a harmonic majorant for every function f in the Nevanlinna space in a strip instead of the usual assumption on log+ [f], having a harmonic majorant in the same setting. By using the conformal mapping from a strip onto the unit disk and the inner-outer factorization theorem on the Nevanlinna space in the unit disk, we obtain an inner-outer factorization theorem on the Nevanlinna space in such a strip. [ABSTRACT FROM AUTHOR]

In this paper, we examine the motion of a robot end-effector by using the Blaschke approach of a ruled surface generated by a line fixed in the robot end-effector. In this way, we determine time dependent linear and angular differential properties of motion such as velocity and acceleration which play important roles in robot trajectory planning. Moreover, motion of a robot end-effector which can be represented by a right conoid and an additional parameter called spin angle is investigated as a practical example. [ABSTRACT FROM AUTHOR]

LetΩbe an open subset of the complex plane and let f be an injective holomorphic self-map of Ω such that the sequence of iterates of f is a run-away sequence. We prove that the composition operator Cf with symbol f is spherically universal on a suitable function space consisting of sphere-valued functions - in contrast to the known fact that, in general, Cf is not hypercyclic on H(Ω) in case that Ω is multiply connected. Moreover, concrete open sets which support spherically universal functions will explicitly be determined in case that the symbol of the composition operator is given by a finite Blaschke product of degree two that has an attracting fixed point at the origin. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we classify umbilic-free hypersurfaces of the unit sphere that have constant para-Blaschke eigenvalues and possess parallel Möbius form. To achieve the classification, we first of all show that, under the condition of having constant para-Blaschke eigenvalues, an umbilic-free hypersurface of the unit sphere is of parallel Möbius form if and only if its Möobius form vanishes identically. [ABSTRACT FROM AUTHOR]

In [SIAM J. Matrix Anal. Appl., 38 (2017), pp. 649–655], Crouzeix and Palencia show that the closure of the numerical range of a square matrix or linear operator A is a (1 +√2)spectral set for A; that is, for any function f analytic in the interior of the numerical range W(A) and continuous on its boundary, the inequality ‖f(A)‖ ≤ (1 +√2)‖f‖ W(A) holds, where the norm on the left is the operator 2-norm and ‖f‖ W(A) on the right denotes the supremum of ∣f(z)∣ over z ∈ W(A). In this paper, we show how the arguments in their paper can be extended to show that other regions in the complex plane that do not necessarily contain W(A) are K-spectral sets for a value of K that may be close to 1 +√2. We also find some special cases in which the constant (1 +√2) for W(A) can be replaced by 2, which is the value conjectured by Crouzeix. [ABSTRACT FROM AUTHOR]

We study the family of singular perturbations of Blaschke products B a , λ ( z ) = z 3 z − a 1 − a ‾ z + λ z 2 . We analyse how the connectivity of the Fatou components varies as we move continuously the parameter λ . We prove that all possible escaping configurations of the critical point c − ( a , λ ) take place within the parameter space. In particular, we prove that there are maps B a , λ which have Fatou components of arbitrarily large finite connectivity within their dynamical planes. [ABSTRACT FROM AUTHOR]

In this paper we propose a different (and equivalent) norm on which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with 3-isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of for a class of composition operators. We completely characterize multiplication operators which are m-isometries. As an application of the 3-isometry, we describe the reducing subspaces of on when is a finite Blaschke product of order 2. [ABSTRACT FROM AUTHOR]

A compact metric space with a bounded Borel measure is considered. Any measurable set of diameter not exceeding r is called r-cluster. The existence of a collection consisting of a fixed number of 2r-clusters possessing the following properties is investigated: the clusters are located at the distance r from each other and the collection measure (the total measure of the clusters in the collection) is close to the measure of the entire space. It is proved that there exists a collection with a maximum measure among such collections. The concept of r-parametric discretization of the distribution of distances into short, medium, and long distances is defined. In terms of this discretization, a lower bound on the measure of the maximum-measure collection is obtained. [ABSTRACT FROM AUTHOR]

In the present paper we give a structure of a removable set to Privalov's theorem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the problem under what conditions the equation f″ + Af = 0 possesses an entire (meromorphic) solution with given sequences of zeros (poles) and critical points. The results are extended to equations of higher orders. [ABSTRACT FROM AUTHOR]