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.
Instituto de Matemática e Estatística, Universidade de São Paulo（圣保罗大学数学与统计学院）；Instituto de Matemática Pura e Aplicada（纯数学与应用数学研究所）；Shenzhen International Center for Mathematics, SUSTech（南方科技大学深圳国际数学中心）
A Blaschke product has no radial limits on a subset $E$ of the unit circle $T$ but has unrestricted limit at each point of $T \setminus E$ if and only if $E$ is a closed set of measure zero.
Given $n$ distinct points $t_1,\ldots,t_n$ on the unit circle $\T$ and equally many target values $\f_1,\ldots,\f_n\in\T$, we describe all Blaschke products $f$ of degree at most $n-1$ such that $f(t_i)=\f_i$ for $i=1,\ldots,n$. We also describe the cases where degree $n-1$ is the minimal possible.
Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation and let $f^{n}$ be its $n$-th iterate. Given a sequence $\{a_{n}\}$ of complex numbers consider $F= \sum a_n f^{n}$. If $\{a_n\}$ tends to $0$ but $\sum |a_n| = \infty$, we prove that for any complex number $w$ there exists a point $ξ$ in the unit circle such that $\sum a_{n}f^{n}(ξ)$ converges and its sum is $w$. If $\sum |a_n| < \infty$ and the convergence is slow enough in a certain precise sense, then the image of the unit circle by $F$ has a non empty interior. The proofs are based on inductive constructions which …
A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.
Given two compact sets, $E$ and $F$, on the unit circle, we study the class of subharmonic functions on the unit disk which can grow at the direction of $E$ and $F$ (sets of singularities) at different rate. The main result concerns the Blaschke-type condition for the Riesz measure of such functions. The optimal character of such condition is demonstrated.
For any finite Blaschke product $B$, there is an injective analytic map $φ:\mathbb{D}\to\mathbb{C}$ and a polynomial $p$ of the same degree as $B$ such that $B=p\circφ$ on $\mathbb{D}$. Several proofs of this result have been given over the past several years, using fundamentally different methods. However, even for low-degree Blaschke products, no method has hitherto been developed to explicitly compute the polynomial $p$ or the associated conformal map $φ$. In this paper, we show how these functions may be computed for a Blaschke product of degree at most three, as well as for Blaschke produ…
We study the Fourier series of circle homeomorphisms and circle embeddings, with the emphasis on Blaschke product approximation and the vanishing of Fourier coefficients. The analytic properties of the Fourier series are related to the geometry of the circle embeddings, and have implications for the curvature of minimal surfaces.
Let \[ Γ= \{(z+w, zw): |z|\leq 1, |w|\leq 1\} \subset \mathbb{C}^2. \] A $Γ$-inner function is defined to be a holomorphic map $h$ from the unit disc $\mathbb{D}$ to $Γ$ whose boundary values at almost all points of the unit circle $\mathbb{T}$ belong to the distinguished boundary $bΓ$ of $Γ$. A rational $Γ$-inner function $h$ induces a continuous map $h|_\mathbb{T}$ from the unit circle to $bΓ$. The latter set is topologically a Möbius band and so has fundamental group $\mathbb{Z}$. The {\em degree} of $h$ is defined to be the topological degree of $h|_\mathbb{T}$. In a previous paper the aut…
It is well known that the bounding curve of the central hyperbolic component of the Multibrot set in the parameter space of unicritical degree $d$ polynomials is an epicycloid with $d-1$ cusps. The interior of the epicycloid gives the polynomials of the form $z^d+c$ which have an attracting fixed point. We prove an analogous result for unicritical Blaschke products: in the parameter space of degree $d$ unicritical Blaschke products, the parabolic functions are parameterized by an epicycloid with $d-1$ cusps and inside this epicycloid are the parameters which give rise to elliptic functions hav…
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 here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) Through the realization matrix of Schur stable systems. (ii) The Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters. (iii) Through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infi…
Suppose that $f$ is a transcendental entire function, $V \subsetneq \mathbb{C}$ is a simply connected domain, and $U$ is a connected component of $f^{-1}(V)$. Using Riemann maps, we associate the map $f \colon U \to V$ to an inner function $g \colon \mathbb{D} \to \mathbb{D}$. It is straightforward to see that $g$ is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of $f$ in $V$ lie away from the boundary, there is a strong relationship between singularities of $g$ and accesses to inf…
For $0<s<1$, let $\{z_n\}$ be a sequence in the open unit disk such that $\sum_n (1-|z_n|^2)^s δ_{z_n}$ is an $s$-Carleson measure. In this paper, we consider the connections between this $s$-Carleson measure and the theory of Möbius invariant $F(p, p-2, s)$ spaces by the Volterra type operator, the reciprocal of a Blaschke product, and second order complex differential equations having a prescribed zero sequence.
We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szegő polynomials. Our results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions. Thi…
Let p(z) be a complex polynomial of degree n. Let C be a circle containing its n-1 zeros, having its center in the centroid of these zeros. We prove that C must contain at least int((n-1):2) zeros of its derivative.
Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$ having no zeros in the unit disk. ~Then it is well known that for $R\geq 1,$ $\displaystyle{\max_{|z|=R}|p(z)|}\leq \Big(\dfrac{R^n+1}{2}\Big)\displaystyle{\max_{|z|=1}|p(z)|}.$ In this paper, we consider polynomials with gaps, having all its zeros on the circle $S(0, K):=\{z: |z|=K\}, ~0<K\le 1,$~ and estimate the value of $\Big(\dfrac{{\max_{|z|=R}|p(z)|}}{{\max_{|z|=1}|p(z)|}}\Big)^s$ for any positive integer $s.$
In this paper, we examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $Λ$ in the sense of Baire category. Notably, we prove that for any open subset $U$ of the unit disk $D$ with a non-real boundary point on the unit circle, $f(U)$ is a dense set of $\CC$. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.
Following Gorkin, Mortini, and Nikolski, we say that an inner function $I$ in $H^\infty$ of the unit disc has the WEP property if its modulus at a point $z$ is bounded from below by a function of the distance from $z$ to the zero set of $I$. This is equivalent to a number of properties, and we establish some consequences of this for $H^\infty/IH^\infty$. The bulk of the paper is devoted to "wepable" functions, i.e. those inner functions which can be made WEP after multiplication by a suitable Blaschke product. We prove that a closed subset $E$ of the unit circle is of finite entropy (i.e. is a…
In 1955, Lehto showed that, for every measurable function $ψ$ on the unit circle $\mathbb T,$ there is function $f$ holomorphic in the unit disc $\mathbb D,$ having $ψ$ as radial limit a.e. on $\mathbb T.$ We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point $p$ is replaced by (asymptotically) total approach to $p.$
In 1955, Lehto showed that, for every measurable function $ψ$ on the unit circle $\mathbb T,$ there is a function $f$ holomorphic in the unit disc, having $ψ$ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions $f$ of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.
Given a natural number $n\geq3$ and two points $a$ and $b$ in the unit disk $\mathbb D$ in the complex plane, it is known that there exists a unique elliptical disk having $a$ and $b$ as foci that can also be realized as the intersection of a collection of convex cyclic $n$-gons whose vertices fill the whole unit circle $\mathbb T$. What is less clear is how to find a convenient formula or expression for such an elliptical disk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful tool for finding such a formula for some values of $n$. The main idea is to real…
