Related papers: Approximating the modulus of an inner function

We study uniform estimates for the family of fundamental Lagrange polynomials associated with any Leja sequence for the complex unit disk. The main result claims that all these polynomials are uniformly bounded on the disk, i.e.…

A sequence which is a finite union of interpolating sequences for $H^\infty$ have turned out to be especially important in the study of Bergman spaces. The Blaschke products $B(z)$ with such zero sequences have been shown to be exactly…

A proof for the maximum modulus principle (in the unit disc) is presented. This proof is unusual in that it is based on linear algebra.

An asymmetric operator of generalised translation is introduced in this paper. Using this operator, we define a generalised modulus of smoothness and prove direct and inverse theorems of approximation theory for it.

We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the…

Bounded holomorphic interpolation problems associated to finitely many data have, in general, distinct solutions. Uniqueness arises only in some convex extreme configurations. Rational inner functions in a polydisk are the best understood…

Here we research the univariate quantitative approximation of real and complex valued continuous functions on a compact interval or all the real line by quasi-interpolation, Baskakov type and quadrature type neural network operators. We…

Differential subordination and superordination preserving properties for univalent functions in the open unit disk with an operator involving generalized Bessel functions are derived. Some particular cases involving trigonometric functions…

In this paper we establish asymptotically best possible interpolation Lebesgue-type inequalities for $2\pi$-periodic functions $f$, which are representable as generalized Poisson integrals of the functions $\varphi$ from the space $L_p$,…

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| =…

We present a method that allows efficient and safe approximation of model predictive controllers using kernel interpolation. Since the computational complexity of the approximating function scales linearly with the number of data points, we…

We study a concept of inner function suited to Dirichlet-type spaces. We characterize Dirichlet-inner functions as those for which both the space and multiplier norms are equal to 1.

Let be F a family of curves in the unit disc. We show that the set of all functions f holomorphic on the unit disc, which satisfy the following condition, is G-delta and dense in the space of all functions holomorphic on the unit disc: For…

We characterize the connected components of the subset $\cni$ of $H^\infty$ formed by the products $bh$, where $b$ is Carleson-Newman Blaschke product and $h\in H^\infty$ is an invertible function. We use this result to show that, except…

Herglotz's representation of holomorphic functions with positive real part and Carath\'eodory's theorem on approximation by inner functions are two well-known classical results in the theory of holomorphic functions on the unit disc. We…

We prove a Bernstein-type inequality involving the Bergman and the Hardy norms, for rational functions in the unit disc \mathbb{D} having at most n poles all outside of \frac{1}{r}\mathbb{D}, 0

In the author's PhD thesis (2019) universal envelopes were introduced as a tool for studying the continuously obtainable information on discontinuous functions. To any function $f \colon X \to Y$ between $\operatorname{qcb}_0$-spaces one…

This paper develops a fully discrete soft thresholding polynomial approximation over a general region, named Lasso hyperinterpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation under the same…

In this paper, we introduce a subclass of close-to-convex functions defined in the open unit disk. We obtain the inclusion relationships, coefficient estimates and Fekete-Szego inequality. The results presented here would provide extension…

For a weight function in the unit disk which is the modulus of a finite product of powers of Blaschke factors, we give a canonical representation for the reproducing kernel of the corresponding weighted Bergman space in terms of the values…