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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]

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]

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]

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]

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 [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]

For a Blaschke product B whose zeros lie in a Stolz domain, we find a condition regarding ω which guarantees that B' belongs to the Bergman space Apω. In addition, the sharpness of this condition is considered. [ABSTRACT FROM AUTHOR]

The aim of this paper is to consider the following two problems:(1) find sufficient conditions on the analytic coefficients of the differential equation for all analytic solutions to belong to the general family function space;(2) for a given Blaschke sequence satisfying certain conditions, find a function A(z), analytic in such that possesses a solution having zeros precisely at the points of this sequence. Estimate the growth of the resulting function A(z).The results we obtain are a generalization of some earlier results by Heittokangas, Korhonen and Rättyä. [ABSTRACT FROM PUBLISHER]

It is well known that the bounding curve of the central hyperbolic component of the Multibrot set in the parameter space of unicritical degreedpolynomials is an epicycloid withcusps. The interior of the epicycloid gives the polynomials of the formwhich have an attracting fixed point. We prove an analogous result for unicritical Blaschke products: in the parameter space of degreedunicritical Blaschke products, the parabolic functions are parameterized by an epicycloid withcusps and inside this epicycloid are the parameters which give rise to maps having an attracting fixed point in the unit disk. We further study in more detail the case whenin which every Blaschke product is unicritical in the unit disk. [ABSTRACT FROM AUTHOR]

We introduce and characterize interpolation sets in a weak sense for the Lipschitz class in the unit disc of the complex plane. Interpolation sets in the classical sense and in a strong sense for this space have already been examined. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let Γ = def { ( z + w , z w ) : | z | ≤ 1 , | w | ≤ 1 } ⊂ C 2 . A Γ -inner function is a holomorphic map h from the unit disc D to Γ whose boundary values at almost all points of the unit circle T belong to the distinguished boundary b Γ of Γ. A rational Γ-inner function h induces a continuous map h | T from T to b Γ. The latter set is topologically a Möbius band and so has fundamental group Z . The degree of h is defined to be the topological degree of h | T . In a previous paper the authors showed that if h = ( s , p ) is a rational Γ-inner function of degree n then s 2 − 4 p has exactly n zeros in the closed unit disc D − , counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational Γ-inner functions of degree n with the n zeros of s 2 − 4 p prescribed. [ABSTRACT FROM AUTHOR]

We prove that two finite endomorphisms of the unit disk with degree at least two have orbits with infinite intersections if and only if they have a common iterate. [ABSTRACT FROM AUTHOR]

In numerical analysis it is often necessary to estimate the condition number C N ( T ) = ‖ T ‖ ⋅ ‖ T − 1 ‖ and the norm of the resolvent ‖ ( ζ − T ) − 1 ‖ of a given n × n matrix T . We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the fact that the supremum of C N ( T ) over all matrices with ‖ T ‖ ≤ 1 and minimal absolute eigenvalue r = min λ ∈ σ ( T ) ⁡ | λ | > 0 is the Kronecker bound 1 r n . This result is subsequently generalized by computing for given ζ in the closed unit disc the supremum of ‖ ( ζ − T ) − 1 ‖ , where ‖ T ‖ ≤ 1 and the spectrum σ ( T ) of T is constrained to remain at a pseudo-hyperbolic distance of at least r ∈ ( 0 , 1 ] around ζ . We find that the supremum is attained by a triangular Toeplitz matrix. This provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically. The occurring Toeplitz matrices are so-called model matrices, i.e. matrix representations of the compressed backward shift operator on the Hardy space H 2 to a finite-dimensional invariant subspace. [ABSTRACT FROM AUTHOR]

We provide a new proof of a theorem of Fujimura characterizing Blaschke products of degree-4 that are compositions of two degree-2 Blaschke products, connect this result to the numerical ranges of certain operators, and characterize Poncelet ellipses that are inscribed in quadrilaterals geometrically and algebraically. [ABSTRACT FROM AUTHOR]

Let ƒ be a holomorphic function in the unit disc and |ƒ(z)-1| < 1 for |z| < 1. We generalize the uniqueness portion of Schwarz's lemma and provide sufficient conditions on the local behavior of ƒ near a finite set of boundary points that needed for ƒ to be a finite Blaschke product. [ABSTRACT FROM AUTHOR]

We consider a dynamics generated by families of maps whose invariant density depends on a parameteraand whereaitself obeys a stochastic or periodic dynamics. For slowly varyingathe long-term behaviour of iterates is described by a suitable superposition of local invariant densities. We provide rigorous error estimates how good this approximation is. Our method generalizes the concept of superstatistics, a useful technique in nonequilibrium statistical mechanics, to maps. Our main example is Blaschke products, for which we provide rigorous error estimates on the difference between Birkhoff density and the superstatistical approximation. [ABSTRACT FROM AUTHOR]

We address the question: Are the inner functions in the uniform closure in H ∞ of the indestructible Blaschke products? We show, in particular, that every inner function with countable spectrum is in the closure of the indestructible Blaschke products, that every Blaschke product is a product of two indestructible Blaschke products and we study approximation in modulus. [ABSTRACT FROM AUTHOR]

In digital signal processing, it is a well know fact that a causal signal of finite energy is front loaded if and only if the corresponding analytic signal, or the physically realizable signal, is a minimum phase signal, or an outer function in the complex analysis terminology. Based on this fact, a series expansion method, called unwinding adaptive Fourier decomposition (AFD), to give rise to positive frequency representations with rapid convergencewas proposed several years ago. It appears to be a promising positive frequency representation with great potential of applications. The corresponding algorithm, however, is complicated due to consecutive extractions of outer functions involving computation of Hilbert transforms. This paper is to propose a practical algorithm for unwinding AFD that does not depend on computation of Hilbert transform, but, instead, factorizes out the Blaschke product type of inner functions. The proposed method significantly improves applicability of unwinding AFD. As an application, we give the associated Dirac-type time-frequency distribution of physically realizable signals. [ABSTRACT FROM AUTHOR]

We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant. In domain dimension at least 2, we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite. By contrast, when the target dimension is at least twice the domain dimension, it is well known that there are uncountably many spherical equivalence classes. We generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps. We introduce Whitney sequences, a precise analogue (in higher dimensions) of the notion of finite Blaschke product (in one dimension). We show that terms in a Whitney sequence are homotopic to monomial mappings, and we establish an additional result about the target dimensions of such homotopies. [ABSTRACT FROM AUTHOR]

We present four algorithms to determine whether or not a Blaschke product is a composition of two non-trivial Blaschke products and, if it is, the algorithms suggest what the composition must be. The initial algorithm is a naive counting argument, the second considers critical values and the counting argument, the third is a geometric argument that exploits the relationship between Blaschke products and curves with the Poncelet property, and it can also be expressed in terms of a group associated with the Blaschke product. The final algorithm looks at inverse images under the Blaschke product. Our algorithms are accompanied by an applet that implements them. [ABSTRACT FROM AUTHOR]

Let φ be a holomorphic self-map of the unit disk U:={z∈C:|z|<1}, and the composition operator with symbol φ is defined by Cφf=f∘φ. In this paper we present formula for the adjoint of composition operators in some Hilbert spaces of analytic functions, in the case that φ is a finite Blaschke product or a rational univalent holomorphic self-map of the unit disk U. [ABSTRACT FROM AUTHOR]

We give estimates for the approximation numbers of composition operators on the Hp spaces, 1 ≤ p < ∞. [ABSTRACT FROM AUTHOR]

We give estimates for the approximation numbers of composition operators on the Hp spaces, 1 ≤ p < ∞. [ABSTRACT FROM AUTHOR]

The class of functions with finite Dirichlet-type integral is defined as the set of holomorphic functions f in the unit disk satisfying the following condition: These classes are usually denoted by D. In this paper, we prove the converse of Rudin's theorem and thus provide a necessary and sufficient condition for a Blaschke product to belong to the class D. [ABSTRACT FROM AUTHOR]

Abstract: It is shown that inner functions in weak Besov spaces are precisely the exponential Blaschke products. [Copyright &y& Elsevier]

It is proved that a countably generated closed ideal in H8 whose common zero set is contained in the union set of nontrivial Gleason parts of H8 is generated by two Carleson-Newman Blaschke products as a closed ideal. [ABSTRACT FROM AUTHOR]

Abstract: Using weighted reproducing kernels, we give a description of the orthocomplement of in when is a singly generated shift invariant subspace of the Bergman space and is a Blaschke product vanishing at the origin. We also obtain an approximation result for weighted kernels corresponding to Blaschke products. [Copyright &y& Elsevier]

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra H∞[b : b has finite angular derivative everywhere]. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we construct a class of new splines related to a Blaschke product. They emerge naturally when studying the filter functions of a class of linear time-invariant systems which are related to the boundary values of a Blaschke product for the purpose of sampling non-bandlimited signals using nonlinear Fourier atoms. The new splines generalize the well-known symmetric B-splines. We establish their properties such as integral representation property, a partition of unity property, a recurrence relation and difference property. We also investigate their random behaviour. Finally, our numerical experiments confirm our theories. [Copyright &y& Elsevier]

It is accepted knowledge that inner functions and outer functions in complex analysis correspond, respectively, to all-pass filters and signals of minimum phase. The knowledge, however, has not been justified for general inner and outer functions. In digital signal processing the correspondence and related results are based on studies of rational functions. In this paper, based on the recent result on positivity of phase derivatives of inner functions, we establish the theoretical foundation for all-pass filters and signals of minimum phase. We, in particular, deal with infinite Blaschke products and general singular inner functions induced by singular measures. A number of results known for rational functions are generalized to general inner functions. Both the discrete and continuous signals cases are rigorously treated. [ABSTRACT FROM PUBLISHER]

Abstract: Let be the Hardy space over the bidisk. For sequences of Blaschke products and satisfying some additional conditions, we may define a Rudin type invariant subspace . We shall determine the rank of for the pair of operators and . [Copyright &y& Elsevier]

This research is partially a continuation of a 2007 paper by the author. Growth estimates for generalized logarithmic derivatives of Blaschke products are provided under the assumption that the zero sequences are either uniformly separated or exponential. Such Blaschke products are known as interpolating Blaschke products. The growth estimates are then proven to be sharp in a rather strong sense. The sharpness discussion yields a solution to an open problem posed by E. Fricain and J. Mashreghi in 2008. Finally, several aspects are pointed out to illustrate that interpolating Blaschke products appear naturally in studying the oscillation of solutions of a differential equation f″+ A( z) f=0, where A( z) is analytic in the unit disc. In particular, a unit disc analogue of a 1988 result due to S. Bank on prescribed zero sequences for entire solutions is obtained, and a more careful analysis of a 1955 example due to B. Schwarz on the case $A(z)=\frac{1+4\gamma^{2}}{(1-z^{2})^{2}}$ reveals that an infinite zero sequence is always a union of two exponential sequences. [ABSTRACT FROM AUTHOR]

Abstract: By connecting Blaschke products, unitary dilations of matrices, numerical range, Poncelet''s theorem and interpolation, we extend and simplify Gau and Wu''s work (Gau and Wu (2004) ). We look at Blaschke products'' role in the Sendov conjecture. [Copyright &y& Elsevier]

Abstract: In this paper we give a function theoretic similarity classification for Toeplitz operators on weighted Bergman spaces with symbol analytic on the closure of the unit disk. [Copyright &y& Elsevier]

An analytic function f in the unit disc D belongs to F(p, q, s), if ∫D ∣f′(z)∣p(1 - ∣z∣2)qgs(z, a) dA(z) is uniformly bounded for all a ϵ D. Here g(z, a) = - log ∣φa(z)∣ is the Green function of D, and φa(z) = (a - z)/(1 - āz). It is shown that for 0 < γ < ∞ and ∣w∣ = 1 the singular inner function exp(γ(z + w)/(z-w)) belongs to F(p, q, s), 0 < s ⩽ 1, if and only if p ⩽ q + ½ (s+3). Moreover, it is proved that, if 0 < s < 1, then an inner function belongs to the Möbius invariant Besov-type space Bsp = F(p, p - 2, s) for some (equivalently for all) p > max{s, 1 - s} if and only if it is a Blaschke product whose zero sequence {zn} satisfies supaϵD Σn∞=1(1 - ∣φa(zn)∣²)s < ∞. [ABSTRACT FROM AUTHOR]

Let Fa,λ be the Blaschke product of the form Fa,λ = λz2((z - a)/(1 - āz)) and α denote an irrational number satisfying the Brjuno condition. Henriksen [1997] showed that for any α there exists a constant a0 ≧ 3 and a continuous function λ(a) such that Fa,λ(a) possesses an Herman ring and also that modulus M(a) of the Herman ring approaches 0 as a approaches a0. It is remarked that the question whether a0 = 3 holds or not is open. According to the idea of Fagella and Geyer [2003] we can show that for a certain set of irrational rotation numbers, a0 is strictly larger than 3. [ABSTRACT FROM AUTHOR]

Given 2n distinct points z1, z′1, z2, z′2, ..., zn, z′n (in this order) on the unit circle, and n points w1, ..., wn on the unit circle, we show how to construct a Blaschke product B of degree n such that B(zj) = wj for all j and, in addition, B(z′j) = B(z′k) for all j and k. Modifying this example yields a Blaschke product of degree n - 1 that interpolates the zj's to the wj's. We present two methods for constructing our Blaschke products: one reminiscent of Lagrange's interpolation method and the second reminiscent of Newton's method. We show that locating the zeros of our Blaschke product is related to another fascinating problem in complex analysis: the Sendov Conjecture. We use this fact to obtain estimates on the location of the zeros of the Blaschke product. [ABSTRACT FROM AUTHOR]

The article presents two theorems with proofs on elimination of singularities of meromorphic functions with finitely many poles through the maximum modulus principle for analytic functions. The spaces of bounded analytic functions are discussed. The author concludes that in the process, analogs of the second theorem are obtained for Smirnov's spaces and for spaces with the value of 'p' lying between 1 and infinity, the desired result follows from the M. Riesz generalized theorem.

The article presents two theorems with proofs on elimination of singularities of meromorphic functions with finitely many poles through the maximum modulus principle for analytic functions. The spaces of bounded analytic functions are discussed. The author concludes that in the process, analogs of the second theorem are obtained for Smirnov's spaces and for spaces with the value of 'p' lying between 1 and infinity, the desired result follows from the M. Riesz generalized theorem.

Abstract: In many cases, a real-valued signal χ(t) may be associated with a complex-valued signal a(t)e iθ(t), the analytic signal associated with χ(t) with the characteristic properties χ(t) = a(t) cosθ(t) and H(a(·)cosθ(·))(t) = a(t)sinθ(t). Using such obtained amplitude-frequency modulation the instantaneous frequency of χ(t) at the time t 0 may be defined to be θ′(t 0), provided θ′(t 0) ≥ 0. The purpose of this note is to characterize, in terms of analytic functions, the unimodular functions F(t) = C(t) + iS(t),C 2(t) + S 2 (t) = 1, a.e., that satisfy HC(t) = S(t). This corresponds to the case a(t) ≡ 1 in the above formulation. We show that a unimodular function satisfies the required condition if and only if it is the boundary value of a so called inner function in the upper-half complex plane. We also give, through an explicit formula, a large class of functions of which the parametrization C(t) = cosθ(t) is available and the extra condition θ′(t) ≥ 0, a.e. is enjoyed. This class of functions contains Blaschke products in the upper-half complex plane as a proper subclass studied by Picinbono in [1]. [Copyright &y& Elsevier]

Let A be a contraction on Hilbert space H and φ a finite Blaschke product. In this paper, we consider the problem when the norm of φ(A) is equal to 1. We show that (1) &par;φ(A)&par;=1 if and only if &par;Ak&par;=1, where k is the number of zeros of φ counting multiplicity, and (2) if H is finite-dimensional and A has no eigenvalue of modulus 1, then the largest integer l for which &par;Al&par;=1 is at least m/(n−m), where n=dim H and m=dim ker(I−A*A), and, moreover, l=n−1 if and only if m=n−1. [Copyright &y& Elsevier]

Various BMO-type characteristics are given for some class of holomorphic functions with a mixed norm. Some criteria are established for Blaschke products to belong to analytic Besov spaces. [ABSTRACT FROM AUTHOR]