Related papers: Boundary interpolation by finite Blaschke products

We consider the interpretation and the numerical construction of the inverse branches of $n$ factor Blaschke-products on the disk and show that these provide a generalization of the $n$-th root function. The inverse branches can be defined…

It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general points. We prove a generalization of this to higher dimensional varieties, showing that smooth varieties of minimal degree can be interpolated…

We consider quantum interpolation of polynomials. We imagine a quantum computer with black-box access to input/output pairs (x_i, f(x_i)), where f is a degree-d polynomial, and we wish to compute f(0). We give asymptotically tight quantum…

We give a necessary and sufficient condition for the convergence of an infinite product of rational inner functions on the polydisk, and explore generalization to the polydisk of Malmquist- Takenaka bases and various versions of unwinding

We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc…

We consider radially symmetric capillary surfaces that are described by bounded generating curves. We use the arc-length representation of the differential equations for these surfaces to allow for vertical points and inflection points…

In this dissertation, it is first shown that, when the radial basis function is a $p$-norm and $1 < p < 2$, interpolation is always possible when the points are all different and there are at least two of them. We then show that…

For a circle $ C $ contained in the unit disk, the necessary and sufficient condition for the existence of a triangle inscribed in the unit circle and circumscribed about $ C $ is known as Chapple's formula. The geometric properties of…

Let $f=B_1/B_2$ be a ratio of finite Blaschke products having no critical points on $\partial\mathbb{D}$. Then $f$ has finitely many critical level curves (level curves containing critical points of $f$) in the disk, and the non-critical…

We give an elementary proof of a solvability criterion for the {\em boundary Carath\'{e}odory-Fej\'{e}r problem}: given a point $x \in \R$ and, a finite set of target values, to construct a function $f$ in the Pick class such that the first…

Consider the space $\mathcal{F}$ of all inner functions on the unit open disk under the uniform topology, which is a metric topology induced by the $H^{\infty}$-norm. In the present paper, a class of Blaschke products, denoted by…

Let $n \in \mathbb{Z}_{>0}$. We prove that there exist a finite set $V$ and finitely many algebraic curves $T_1, \ldots, T_k$ with the following property: if $(x_1, \ldots, x_n, y)$ is an $(n+1)$-tuple of pairwise distinct singular moduli…

Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell^{\infty}$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty$. We provide constructive examples which show that our…

By Northcott's Theorem there are only finitely many algebraic points in affine $n$-space of fixed degree over a given number field and of height at most $X$. For large $X$ the asymptotics of these cardinalities have been investigated by…

In this work we study the intersection properties of a finite disk system in the euclidean space. We accomplish this by utilizing subsets of spheres with varying dimensions and analyze specific points within them, referred to as poles.…

The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two…

We prove a general result on the factorization of matrix-valued analytic functions. We deduce that if $(E_0,E_1)$ and $(F_0,F_1)$ are interpolation pairs with dense intersections, then under some conditions on the spaces $E_0$, $E_1$, $F_0$…

Characterization of Schur-class functions (analytic and bounded by one in modulus on the open unit disk) in terms of their Taylor coefficients at the origin is due to I. Schur. We present a boundary analog of this result: necessary and…

Fractal interpolation technique is an alternative to the classical interpolation methods especially when a chaotic signal is involved. The logic behind the formulation of an iterated function system for the construction of fractal…

The classical Gauss--Lucas theorem describes the location of the critical points of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role of polynomials is played by finite Blaschke products on the unit disk. We…