Related papers: Interpolation and sampling for analytic selfmappin…
We introduce Nevanlinna classes associated to non radial weights in the unit disc in the complex plane and we get Blaschke type theorems relative to these classes by use of several complex variables methods. This gives alternative proofs…
We prove a high order Schwarz-Pick lemma for mappings between unit balls in complex spaces in terms of the Bergman metric. From this lemma, Schwarz-Pick estimates for partial derivatives of arbitrary order of mappings are deduced.
For an inner function $\theta$ with $\theta'\in\mathcal N$, where $\mathcal N$ is the Nevanlinna class, several problems are posed in connection with the canonical (inner-outer) factorization of $\theta'$.
We present and discuss several old and new methods for mapping a circular disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside the circular disc to a point (x,y) inside a square region.…
The well-known Schwarz-Pick lemma states that any analytic mapping $\phi$ of the unit disk $U$ into itself satisfies the inequality $$|\phi'(z)|\leq \frac{1-|\phi(z)|^2}{1-|z|^2}, \quad z\in U.$$ This estimate remains the same if we…
It is known (implicit in [HMNT]) that when $\Lambda$ is an interpolating sequence for the Nevanlinna or the Smirnov class then there exist functions $f_\lambda$ in these spaces, with uniform control of their growth and attaining values 1 on…
In this work we blend interpolation theory with numerical integration, constructing an interpolator based on integrals over $n$-dimensional balls. We show that, under hypotheses on the radius of the $n$-balls, the problem can be treated as…
Given a collection $K$ of positive integers, let $H^{\infty}_K(\mathbb{D})$ denote the set of all bounded analytic functions defined on the unit disk $\mathbb{D}$ in $\mathbb{C}$ whose $k^{\text{th}}$ derivative vanishes at zero, for all $k…
We study the Schwarz lemma for harmonic functions and prove sharp versions for the cases of real harmonic functions and the norm of harmonic mappings.
Assuming the existence of an analytic interpolant mapping a two-point data from the unit disc $\mathbb{D}$ to $\widetilde{\mathbb{G}}_n$, we describe a class of such interpolating functions where $$\widetilde{\mathbb{G}}_n := \{ (y_1,\dots,…
This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of…
In this paper we study the consequences of overinterpolation, i.e., the situation when a function can be interpolated by polynomial, or rational, or algebraic functions in more points that normally expected. We show that in many cases such…
We give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is…
We introduce the notion of a \lambda-nonisotropically balanced domain and show that the symmetrized polydisc in C^n, n \geq 2, is an example of such a domain. Given a \lambda-nonisotropically balanced domain \Omega, we derive effective…
In this article, we establish a connection between Pick bodies and invariant functions. We demonstrate that an invariant function can be associated with any Pick body, which determines the solvability of a given Pick interpolation problem…
In this paper we would like to show the interrelation between the different mathematical theories concerning the Schur interpolation problem, contractions in Hilbert spaces, pseudocontinuation and Darlington synthesis. The main objects of…
An analytical method is advanced for constructing interpolation formulae for complicated problems of statistical mechanics, in which just a few terms of asymptotic expansions are available. The method is based on the self-similar…
Based on the recently proved Khavinson conjecture, we establish an inequality of Schwarz-Pick type for harmonic functions on the unit ball of $\mathbb{R}^n$.
Let ${\mathcal S}$ be the class of analytic functions $f$ in the unit disk ${\mathbb D}$ with $f({\mathbb D}) \subset \overline{\mathbb D}$. Fix pairwise distinct points $z_1,\ldots,z_{n+1}\in \mathbb{D}$ and corresponding interpolation…
This paper introduces the fractal interpolation problem defined over domains with a nonlinear partition. This setting generalizes known methodologies regarding fractal functions and provides a new holistic approach to fractal interpolation.…