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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]

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]

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]

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]

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]

It is proved that M ϕ with 2-Blaschke product ϕ is reducible on Sobolev disk algebra R ( D ) if and only if ϕ = β α − z 2 1 − α ‾ z 2 for some α ∈ D and unimodular constant β . Also, an analytic multiplier M ϕ is unitary equivalent to M z n on R ( D ) if and only if ϕ = λ z n for some unimodular constant λ . [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]

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]

We revisit the classical rolling ball theorem of Blaschke for convex surfaces with positive curvature and show that it is linked to another inclusion principle in the optimal mass transportation theory due to Trudinger and Wang. We also discuss an application to reflector antennae design problem. [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]

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]

Abstract: In this paper we describe matrix representations of truncated Toeplitz operators on the model space , where B is an infinite Blaschke product satisfying some additional conditions. Our results are extensions of that obtained by Cima, Ross and Wogen in 2008. [Copyright &y& Elsevier]

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

Abstract: 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 products. [Copyright &y& Elsevier]

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]

Abstract: In this paper we study the -Minkowski problem for , which corresponds to the critical exponent in the Blaschke–Santalo inequality. We first obtain volume estimates for general solutions, then establish a priori estimates for rotationally symmetric solutions by using a Kazdan–Warner type obstruction. Finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow-up analysis. We also include an existence result for the -Minkowski problem which corresponds to the super-critical case of the Blaschke–Santalo inequality. [Copyright &y& Elsevier]

Abstract: We characterize the connected components of the subset of formed by the products bh, where b is Carleson–Newman Blaschke product and is an invertible function. We use this result to show that, except for finite Blaschke products, no inner function in the little Bloch space is in the closure of one of these components. Our main result says that every inner function can be connected with an element of within the set of products uh, where u is inner and h is invertible. We also study some of these issues in the context of Douglas algebras. [Copyright &y& Elsevier]

Abstract: Motivated by the Invariant Subspace Problem, we describe explicitly the closed subspace generated by the limit points in the norm of the orbit of a thin Blaschke product B under composition operators induced by non-elliptic automorphisms. This description exhibits a surprising connection to model spaces. Finally, we give a constructive characterization of the -eigenfunctions in for . [Copyright &y& Elsevier]

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]

Abstract: Let be the Banach algebra of bounded analytic functions on the open unit disk . Let G be the union set of all nontrivial Gleason parts in the maximal ideal space of . Let E be a nonvoid compact and totally disconnected subset of G and be a bounded numbering function on E. We characterize for which there is a closed ideal I in such that and for every . Let be closed ideals in satisfying for . We prove that is a closed ideal. A local ideal theory in plays an important role to prove our results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, using the group-like property of local inverses of a finite Blaschke product ϕ, we will show that the largest -algebra in the commutant of the multiplication operator by ϕ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of over the unit disk. If the order of the Blaschke product ϕ is less than or equal to eight, then every -algebra contained in the commutant of is abelian and hence the number of minimal reducing subspaces of equals the number of connected components of the Riemann surface of over the unit disk. [ABSTRACT FROM AUTHOR]

Abstract: A homogeneous Gibbons–Hawking ansatz is described, leading to 4-dimensional hyperkähler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkähler metrics that are, in a suitable sense, complete. Our construction also gives rise to incomplete metrics on 3-dimensional contact manifolds that induce complete Carnot–Carathéodory distances. [ABSTRACT FROM AUTHOR]

Abstract: A stability version of the Blaschke–Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established. [ABSTRACT FROM AUTHOR]

Abstract: Let E be a compact subset of G, the union set of nontrivial Gleason parts, and be the associate primary ideal of . We give a characterization of the numbering function , the zero''s order of at x in E, using the geometrical words of E. We also give some factorization theorems of Blaschke products. Using these, we give some descriptions of the higher order hulls of , and for a Carleson–Newman Blaschke product in it is proved that is generated by its subproducts as a closed ideal. When each is ρ-separated, we also prove that the tensor product is closed in . [Copyright &y& Elsevier]

Abstract: For a construction is given for convex bodies K and L in such that the orthogonal projection onto the subspace contains a translate of for every direction u, while the volumes of K and L satisfy . A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection onto a k-dimensional subspace ξ contains a translate of , while the mth intrinsic volumes of K and L satisfy for all . For each , we then define the collection to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms). It is subsequently shown that, if , and if the orthogonal projection contains a translate of for every k-dimensional subspace ξ of , then . The families , called k-cylinder bodies of , form a strictly increasing chain where is precisely the collection of centrally symmetric compact convex sets in , while is the collection of all compact convex sets in . Members of each family are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of are shown to satisfy certain geometric inequalities. Related open questions are also posed. [Copyright &y& Elsevier]

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]

Abstract: We generalize a well-known sufficient condition for interpolating sequences for the Hilbert Bergman spaces to other Bergman spaces with normal weights (as defined by Shields and Williams) and obtain new results regarding the membership of the derivative of a Blaschke product or a general inner function in such spaces. We also apply duality techniques to obtain further results of this type and obtain new results about interpolating Blaschke products. [Copyright &y& Elsevier]

Abstract: In this paper we study when two finite Blaschke products commute. We complete previous results by Chalendar and Mortini (when they have a fixed point in the unit disk) and by Arteaga (when they do not have a fixed point in the unit disk). [Copyright &y& Elsevier]

Abstract: Zero sets and uniqueness sets of the Dirichlet space are not completely characterized yet. There are sufficient conditions for zero sets and necessary conditions for uniqueness sets and there is a gap in between. Our goal is to take some preliminary steps to fill this gap. [Copyright &y& Elsevier]

Abstract: We prove that Jacobi, CMV, and Schrödinger operators, which are reflectionless on a homogeneous set (in the sense of Carleson), under the assumption of a Blaschke-type condition on their discrete spectra accumulating at , have purely absolutely continuous spectrum on . [Copyright &y& Elsevier]

Abstract: According to the notion of -affine surface area by Lutwak, in this paper, we introduce the concept of -dual affine surface area. Further, we establish the affine isoperimetric inequality and the Blaschke–Santaló inequality for -dual affine surface area. Besides, the dual Brunn–Minkowski inequality for -dual affine surface area is presented. [Copyright &y& Elsevier]

Abstract: In 1950, C.A. Rogers introduced and studied two simultaneous packing and covering constants for a convex body and obtained the first general upper bound. Afterwards, these constants have attracted the interests of many authors because, besides their own geometric significance, they are closely related to the packing densities and the covering densities of the convex body, especially to the Minkowski–Hlawka theorem. However, so far our knowledge about them is still very limited. In this paper we will determine the optimal upper bound of the simultaneous packing and covering constants for two-dimensional centrally symmetric convex domains, and characterize the domains attaining this upper bound. [Copyright &y& Elsevier]

For a finite Blaschke product B let TB denote the analytic multiplication operator (also called a Toeplitz operator) on the Bergman space of the unit disk. We show that the defect operators (I−TBTB&ast;)1/2 and (I−TB&ast;TB)1/2 both map the Bergman space to the Hardy space and the Hardy space to the Dirichlet space. [Copyright &y& Elsevier]

The context of much of the work in this paper is that of a backward-shift invariant subspace of the form KB≔H2(D)⊖BH2(D), where B is some infinite Blaschke product. We address (but do not fully answer) the question: For which B can one find a (convergent) sequence { fn}n=1∞ in KB such that the sequence of real measures {log | fn|dθ}n=1∞ converges weak-star to some nontrivial singular measure on ∂D? We show that, in order for this to hold, KB must contain functions with nontrivial singular inner factors. And in a rather special setting, we show that this is also sufficient. Much of the paper is devoted to finding conditions (on B) that guarantee that KB has no functions with nontrivial singular inner factors. Our primary result in this direction is based on the “geometry” of the zero set of B. [Copyright &y& Elsevier]

We show how to construct all finite Blaschke product solutions and the minimal scaled Blaschke product solution to the Nevanlinna–Pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data. Based on a result of Jones and Ruscheweyh we note that there always exists a finite Blaschke product of degree at most n−1 that maps n distinct points in the closed unit disc, of which at least one is on the unit circle, into n arbitrary points in the closed unit disc, provided that the points inside the unit circle form a positive semi-definite Pick matrix of full rank. Finally, we discuss a numerical limiting procedure. [Copyright &y& Elsevier]

In this paper, we consider smooth, closed, strictly convex curves contained inside the unit circle T and their associated curves of geodesic centers (or dual curves). We obtain a formula for the boundary and envelope of the union of the region bounded by the geodesic circles associated with our curves. This description includes a formula for the boundary of the hyperbolic convex hull of points identified by finite Blaschke products. We describe a setting and conditions under which one can produce an infinite chain of ellipses such that each ellipse is inscribed in a convex polygon that is itself inscribed in another polygon (a chain of Poncelet ellipses). We apply these results to operators that are compressions of the shift operator on model spaces and describe necessary and sufficient conditions for the numerical range to contain 0. [ABSTRACT FROM AUTHOR]

This paper continues the investigation of the relation between the geometry of a circle embedding and the values of its Fourier coefficients. First, we answer a question of Kovalev and Yang concerning the support of the Fourier transform of a starlike embedding. An important special case of circle embeddings are homeomorphisms of the circle onto itself. Under a one-sided bound on the Fourier support, such homeomorphisms are rational functions related to Blaschke products. We study the structure of rational circle homeomorphisms and show that they form a connected set in the uniform topology. [ABSTRACT FROM AUTHOR]

We prove some new results about the spacing between neighboring zeros of paraorthogonal polynomials on the unit circle. Our methods also provide new proofs of some existing results. The main tool we will use is a formula for the phase of the appropriate Blaschke product at points on the unit circle. [ABSTRACT FROM AUTHOR]