Related papers: Holomorphic Approximation in Banach Spaces: A surv…

Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each {\epsilon}>0,…

We consider hilbert spaces of holomorphic functions in Cartan domains (in particular in ball and polydisk) and operator of restriction of holomorphic function to a submanifold in Shilov boundary. We discuss conditions when this operator…

The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Hilbert and Banach spaces. Initially, we extend the classical Schwarz lemmas of…

We prove that every complex analytic set X in a Runge domain D can be approximated by Nash sets on relatively compact subdomains of D. We give a necessary and sufficient condition for a complex analytic set X to admit a Nash approximation…

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed:…

Let $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$. Considering the Gelfand transform image $\widehat{\mathcal{A}}$ of the Banach algebra $\mathcal{A}$, which is…

We give conditions in order to approximate locally uniformly holomorphic covering mappings of the unit ball of $\mathbb{C}^n$ with respect to an arbitrary norm, with entire holomorphic covering mappings. The results rely on a generalization…

A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…

A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $\Cn$ for $0<p<\infty$ is given. When $0<p\leq1$, the result is new even in the case of the unit disk.

We compute correlation functions, specifically 1-point and 2-point functions, in holographic boundary conformal field theory (BCFT) using geodesic approximation. The holographic model consists of a massive scalar field coupled to a…

We obtain rates of convergence of numerical approximations of abstract linear parabolic evolution equations in Banach spaces. Our estimates extend known results from the literature of finite element approximations of parabolic equations to…

We analyze the convergence and approximation error of the inverse Born series, obtaining results that hold under qualitatively weaker conditions than previously known. Our approach makes use of tools from geometric function theory in Banach…

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane which can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in $\SL(2,\ZZ)$. It is…

The purpose of this article is to characterize the quasi-isometry type of a proper metric space via the Banach algebra of Higson functions on it.

We obtain sharp approximation results for into nearisometries between Lp spaces and nearisometries into a Hilbert space. Our main theorem is the optimal approximation result for nearsurjective nearisometries between general Banach spaces.

Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions which…

We discuss the geometry of Banach spaces whose norm is octahedral or, more generally, locally or weakly octahedral. Our main results characterize these spaces in terms of covering of the unit ball.

We introduce a property of Banach spaces called uniform convex-transitivity, which falls between almost transitivity and convex-transitivity. We will provide examples of uniformly convex-transitive spaces. This property behaves nicely in…

We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…

Previously unknown estimates of uniform continuity of projection operators in Banach space have been obtained. They can be used in the investigations of approximation methods, in particular, the method of quasisolutions, methods of…