On a Bornological Structure in Infinite-Dimensional Holomorphy

The bornology (b) of bounded subsets with respect to continuous convergence is used on spaces of holomorphic functions. It is shown that Hom_coH_b(U) ? U for a circled convex open subset U of a complete nuclear space. Exponential laws for spaces of holomorphic functions with bornological structures are proved and the connection with Colombeau's Silva holomorphic functions is established.     

Extension results for slice regular functions of a quaternionic variable

In this paper we prove a new Representation Formula for slice regular functions, which shows that the value of a slice regular function f at a point q=x+yI can be recovered by the values of f at the points q+yJ and q+yK for any choice of imaginary units I,J,K. This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions.     

Vector-valued holomorphic functions revisited

Abstract. Let be open,X a Banach space and . We show that every is holomorphic if and only if every set inX is bounded. Things are different if we assume f to be locally bounded. Then we show that it suffices that is holomorphic for all , where W is a separating subspace of to deduce that f is holomorphic. Boundary Tauberian convergence and membership theorems are proved. Namely, if boundary values (in a weak sense) of a sequence of holomorphic functions converge/belong to a closed subspace on a subset of the boundary having positive Lebesgue measure, then the same is true for the interior points of , uniformly on compact subsets. Some extra global majorants are requested. These results depend on a distance Jensen inequality. Several examples are provided (bounded and compact operators; Toeplitz and Hankel operators; Fourier multipliers and small multipliers). Received January 29, 1998; in final form March 8, 1999 / Published online May 8, 2000     

Finely continuously differentiable functions

In Lávi?ka [A remark on fine differentiability, Adv. Appl. Clifford Algebras 17 (2007) 549–554], it is observed that finely continuously differentiable functions on finely open subsets of the plane are just functions which are finely locally extendable to usual continuously differentiable functions on the whole plane. In this note, it is proved that, under a mild additional assumption, this result remains true even in higher dimensions. Here the word “fine” refers to the fine topology of classical potential theory.     

Extension of holomorphic vector bundles across a boundary point

Let U ? C ⁿ , n ≥ 3, be a domain and P ∈ ?U such that U is 2-concave at P. Here we prove the existence of a holomorphic vector bundle on U which does not extend across P, but it extends across every Q ∈ ?U with Q ≠ P. We also prove a similar result taking a Stein space X instead of C ⁿ .     

Holomorphic liftings and Bergman kernel estimates for 𝒟ℱ𝒩‐domains

Let E be a 𝒟ℱ𝒩‐space and let U ⊂ E be open. By applying the nuclearity of the Fréchet space ℋ︁(U) of holomorphic functions on U we show that there are finite measures μ on U leading to Bergman spaces of μ ‐square integrable holomorphic functions. We give an explicit construction for μ by using infinite dimensional Gaussian measures. Moreover, we prove boundary estimates for the corresponding Bergman kernels K_μ on the diagonal and we give an application of our results to liftings of μ ‐square integrable Banach space valued holomorphic functions over U. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sets of Harmonicity for Finely Harmonic Functions

Given an open set U in R ⁿ (n3) and a dense open subset V of U, it is shown that there is a finely harmonic function u on U such that V is the largest open subset of U on which u is harmonic. This result, which establishes the sharpness of a theorem of Fuglede, is obtained following a consideration of fine cluster sets of arbitrary functions.     

Finely locally injective finely harmonic morphisms

We consider fine topology in the complex plane C and finely harmonic morphisms. We use oriented Jordan curves in the plane to prove that for a finely locally injective finely harmonic morphism f in a fine domain in C, either f or f is a finely holomorphic function. This partially extends result by Fuglede, who considered a kind of continuity for the fine derivatives of the finely harmonic morphism. As a consequence of this we obtain a both necessary and sufficient condition for a function f to be finely holomorphic or finely antiholomorphic. We do not know if the condition of finely local injectivity (q.e.) is automatically fulfilled by any non-constant finely harmonic morphism.     

Hartogs Extension Theorem for Functions with Values in Complex Clifford Algebras

A regular extension phenomenon of functions defined on Euclidean space with values in a Clifford algebra was studied by Le Hung Son in the 90’s using methods of Clifford analysis, a function theory which, is centred around the notion of a monogenic function, i.e. a null solution of the firstorder, vector-valued Dirac operator in . The isotonic Clifford analysis is a refinement of the latter, which arises for even dimension. As such it also may be regarded as an elegant generalization to complex Clifford algebra-valued functions of both holomorphic functions of several complex variables and two-sided biregular function theories. The aim of this article is to present a Hartogs theorem on isotonic extendability of functions on a suitable domain of . As an application, the extension problem for holomorphic functions and so for the two-sided biregular ones is discussed.      

Boundedness, regularity and smoothness of universal Taylor series

Let Ω be an unbounded simply connected domain in satisfying some topological assumptions; for example let Ω be an open half-plane. We show that there exists a bounded holomorphic function on Ω which extends continuously on and is a universal Taylor series in Ω in the sense of Luh and Chui–Parnes with respect to any center. Our proof uses Arakeljan’s Approximation Theorem. Further we strengthen results of G. Costakis [2] concerning universal Taylor series with respect to one center in the sense of Luh and Chui–Parnes in the complement G of a compact connected set. We prove that such functions can be smooth on the boundary of G and be zero at ∞. If the universal approximation is also valid on ∂G, then the function can not be smooth on ∂G, but it may vanish at ∞. Our results are generic in natural Fréchet spaces of holomorphic functions. Received: 29 September 2005; revised: 21 February 2006     

An example of a nuclear space in infinite dimensional holomorphy

LetU be an open subset of a complex locally convex spaceE, andH(U) the space of holomorphic functions fromU toC. If the dualE′ ofE is nuclear with respect to the topology generated by the absolutely convex compact subsets ofE, then it is shown thatH(U) endowed with the compact open topology is a nuclear space. In particular, ifE is the strong dual of a Fréchet nuclear space, thenH(U) is a Fréchet nuclear space.     

Finely holomorphic functions and quasi-analytic classes

We study the set of functions in quasi-analytic classes and the set of finely holomorphic functions. We show that no one of these two sets is contained in the other.LetI denote the set of complex functionsf: for which there exists a quasi-analytic classC{M _n} containingf. Let denote the set of complex functionsf: for which there exist a fine domainU containing the real line and a function finely holomorphic onU satisfyingf(x)= (x) for allx . The power of unique continuation is incomparable in these two cases (I\ is non-empty, \I is non-empty).Research supported by the grant No. 201/93/2174 of Czech Grant Agency and by the grant No. 354 of Charles University.     

An Extended Faber Operator

   Abstract. One of the basic tools in the theory of polynomial approximation in the uniform norm on compact plane sets is the Faber operator. Usually, the Faber operator is viewed as an operator acting on functions in the disk algebra, that is, functions which are holomorphic in the open unit disk D and continuous on D. We consider an extended Faber operator acting on arbitrary functions continuous on ; D.     

The bounded approximation property for spaces of holomorphic mappings on infinite dimensional spaces

We study the bounded approximation property for spaces of holomorphic functions. We show that if U is a balanced open subset of a Fréchet–Schwartz space or (DFM )‐space E , then the space ??(U ) of holomorphic mappings on U , with the compact‐open topology, has the bounded approximation property if and only if E has the bounded approximation property. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Biaxial monogenic functions from Funk‐Hecke's formula combined with Fueter's theorem

Funk‐Hecke's formula allows a passage from plane waves to radially invariant functions. It may be adapted to transform axial monogenics into biaxial monogenics that are monogenic functions invariant under the product group SO(p)× SO(q). Fueter's theorem transforms holomorphic functions in the plane into axial monogenics, so that by combining both results, we obtain a method to construct biaxial monogenics from holomorphic functions.     

An extension theorem for slice monogenic functions and some of its consequences

Slice monogenic functions were introduced by the authors in [6]. The central result of this paper is an extension theorem, which shows that every holomorphic function defined on a suitable domain D of a complex plane can be uniquely extended to a slice monogenic function defined on a domain U _D , determined by D, in a Euclidean space of appropriate dimension. Two important consequences of the result are a structure theorem for the zero set of a slice monogenic function (with a related corollary for polynomials with coefficients in Clifford algebras), and the possibility to construct a multiplicative theory for such functions. Slice monogenic functions have a very important application in the definition of a functional calculus for n-tuples of noncommuting operators.     

On the existence of polyanalytic functions

We give a sharp estimate for the codimension of the poly‐Bergman space in the poly‐Bergman space over the punctured domain. It is established the behaviour at the infinity point of polyanalytic Bergman functions on the complement of closed disks. In the main result of the paper, we prove that for and the j‐polyanalytic Bergman space over the domain U is trivial precisely when the complement of U has at most one point and at most two points or three points lying in a circle, respectively. We point out the differences between the domains over which the Bergman space and the non‐analytic poly‐Bergman space are trivial.     

The approximation property for spaces of holomorphic functions on infinite-dimensional spaces I

For an open subset U of a locally convex space E, let (H(U),τ₀) denote the vector space of all holomorphic functions on U, with the compact-open topology. If E is a separable Fréchet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that (H(U),τ₀) has the approximation property for every open subset U of E. These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra.