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     

Universal functions for composition operators with non-automorphic symbol

For sequences (φ _n ) of eventually injective holomorphic self-maps of planar domains Ω, we present necessary and sufficient conditions for the existence of holomorphic functions f on whose orbits under the action of (φ _n ) are dense in H (Ω). It is deduced that finitely connected, but non-simply connected domains never admit such universal functions. On the other hand, if we allow arbitrary sequences of holomorphic self-maps (φ _n ), the situation changes dramatically.     

Universal Taylor series have a strong form of universality

We prove that a function f holomorphic in a simply connected domain Ω whose Taylor series at ξ ∈ Ω is universal with respect to overconvergence automatically has a strong kind of universality: its expansion in Faber series corresponding to any connected compact set Γ ⊂ Ω with connected is universal, and we may take a supremum over all such Γ’s in a compact set. The topology used here is the Carathéodory topology. This answers a question of Mayenberger and Müller. This research is co-funded by the European Social Fund and National Resources-(EPEAEK II) PYTHAGORASII.     

Generalization of theorems of Szász and Ruscheweyh on exact bounds for derivatives of analytic functions

Let Ω and Π be two domains in the extended complex plane equipped by the Poincaré metric. In this paper we obtain analogs of Schwarz-Pick type inequalities in the class A(Ω, gH) = {f: Ω → Π} of functions locally holomorphic in Ω; for the domain Ω we consider the exterior of the unit disk and the upper half-plane. The obtained results generalize the well-known theorems of Szász and Ruscheweyh about the exact estimates of derivatives of analytic functions defined on the disk |z| < 1.     

Universal matrix-overconvergence of noncontinuable holomorphic functions

Summary Assume that there are given a simply connected domain Gin the complex plane containing the open unit disk, but not the closed unit disk and an infinite triangular matrix A. Under suitable conditions on A, which are even weaker than its P-regularity, in this paper we construct a noncontinuable holomorphic function Φ on Gsuch that the A-transforms of its Taylor expansion around any point of Gtend to a scalar multiple of Φ compactly in Gand exhibit universal properties in the complement of Gwith respect to uniform approximation on compact sets and to almost everywhere approximation on measurable sets.     

Holomorphic Cliffordian functions

The aim of this paper is to put the foundations of a new theory of functions, called holomorphic Cliffordian, which should play an essential role in the generalization of holomorphic functions to higher dimensions. Let ℝ_0,2m+1 be the Clifford algebra of ℝ^2m+1 with a quadratic form of negative signature, be the usual operator for monogenic functions and Δ the ordinary Laplacian. The holomorphic Cliffordian functions are functionsf: ℝ^2m+2 → ℝ_0,2m+1, which are solutions ofDδ ^m f = 0. Here, we will study polynomial and singular solutions of this equation, we will obtain integral representation formulas and deduce the analogous of the Taylor and Laurent expansions for holomorphic Cliffordian functions. In a following paper, we will put the foundations of the Cliffordian elliptic function theory.     

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ^-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq. Work supported in part by an operating grant from NSERC (Canada).     

Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle

Motivated by problems arising from Arithmetic Geometry, in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric. In the case of a germ of holomorphic isometry f: (Δ, λ ds _Δ²;0) → (Ω, ds _Ω²;0) of the Poincaré disk Δ into a bounded symmetric domain Ω ⋐ ℂ ^N in its Harish-Chandra realization and equipped with the Bergman metric, f extends to a proper holomorphic isometric embedding F: (Δ, λ ds _Δ²;) → (Ω, ds _Ω²) and Graph(f) extends to an affine-algebraic variety V ⊂ ℂ × ℂ ^N . Examples of F which are not totally geodesic have been constructed. They arise primarily from the p-th root map ρ _p : H → H ^p and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H ₃ of genus 3. In the current article on the one hand we examine second fundamental forms σ of these known examples, by computing explicitly φ = |σ|². On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincaré disk into polydisks. For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ = s + it on the upper half-plane H, we have φ(τ) = t ² u(τ), where u| _t=0 ≢ 0. We show that u must satisfy the first order differential equation δu/δt| _t=0 ≡ 0 on the real axis outside a finite number of points at which u is singular. As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincaré disk into the polydisk must develop singularities along the boundary circle. The equation δu/δt| _t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G. Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article.     

Nonexistence of Proper Holomorphic Maps Between Certain Classical Bounded Symmetric Domains

The author,motivated by his results on Hermitian metric rigidity,conjectured in [4] that a proper holomorphic mapping f：Ω→Ω′from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ′is necessarily totally geodesic provided that r′：=rank（Ω′）≤rank（Ω）：=r.The Conjecture was resolved in the affirmative by I.-H.Tsai [8].When the hypothesis r′≤r is removed,the structure of proper holomorphic maps f：Ω→Ω′is far from being understood,and the complexity in studying such maps depends very much on the difference r′-r,which is called the rank defect.The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H.Tu [10],in which a rigidity theorem was proven for certain pairs of classical domains of type I,which implies nonexistence theorems for other pairs of such domains.For both results the rank defect is equal to 1,and a generaliza- tion of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of （Ω→Ω′） of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and L-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.     

An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2 ⁻ⁿ where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure μ on Ω, define Ω_μ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ⁻¹ (Ω_μ) in the metric d is

A Hartogs type theorem for separately Cr functions

The main step in the proof of Hartogs’ theorem on separate analyticity (see [3], [4], [5]) consists in showing that if a function f defined in Δ × Δ is holomorphic for |z ₂| < ε and separately holomorphic in z ₂ when z ₁ is kept fixed, then it is jointly holomorphic; the normal convergence of the Taylor series of f is obtained through the celebrated Hartogs’ lemma on subharmonic functions.     

Global Representatives of Colombeau Holomorphic Generalized Functions

It is proved that a holomorphic generalized function in the sense of Colombeau has a representative consisting of a net of holomorphic functions. More generally, the existence of global representatives of Colombeau solutions to elliptic partial differential equations is proved. Further, it is shown that a generalized holomorphic function on Ω, which is equal to zero in an open set L ì WL\subset{\rm{\Omega}} , is equal to zero.     

Cluster sets for sobolev functions and quasiminimizers

In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous quasiminimizers). We develop their basic theory with a particular emphasis on when they coincide and when they are connected. As a main result, we obtain that if a Sobolev function u on an open set Ω has boundary values f in Sobolev sense and f |_∂Ω is continuous at x ₀ ∈ ∂Ω, then the essential cluster set (u, x ₀,Ω) is connected. We characterize precisely in which metric spaces this result holds. Further, we provide some new boundary regularity results for quasiharmonic functions. Most of the results are new also in the Euclidean case.     

Approximation in weighted Hardy spaces

This paper is concerned with several approximation problems in the weighted Hardy spacesH ^p(Ω) of analytic functions in the open unit disc D of the complex plane ℂ. We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH ^p(Ω) coincides, moduloH ^p(Ω), with the space of uniformly continuous functions on a certain hull ofX which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets forH ^p(Ω), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spacesH ^p(D), 1 ⪯p < ∞. Dedicated to Professor Nácere Hayek on the occasion of his 75th birthday.     

Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

A holomorphic function on a planar domain \(\Omega \) is said to possess a universal Taylor series about a point \(\zeta \) of \(\Omega \) if subsequences of the partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compact sets in \(\mathbb {C}\backslash \Omega \) that have connected complement. In the case where \(\Omega \) is simply connected, such functions are known to be unbounded and to form a collection that is independent of the choice of \(\zeta \) . This paper uses tools from potential theory to show that, even for domains \(\Omega \) of arbitrary connectivity, such functions are unbounded whenever they exist. In the doubly connected case, a further analysis of boundary behaviour reveals that the collection of such functions can depend on the choice of \(\zeta \) . This phenomenon was previously known only for domains that are at least triply connected. Related results are also established for universal Laurent series.     

Boundary values of holomorphic functions, singular unitary representations ofO(p,q), and their limits asq→∞

Let Ω be a bounded circular domain in ℂ ^N , let M be a submanifold in the boundary of Ω, and let H be a Hilbert space of holomorphic functions in Ω. We show that, under certain conditions stated in terms of the reproducing kernel of the space H, the restriction operator to the submanifold M is well defined for all functions from H. We apply this result to constructing a family of “singular” unitary representations of the groups SO(p,q). The singular representations arise as discrete components of the spectrum in the decomposition of irreducible unitary highest weight representations of the groups U(p,q) restricted to the subgroups SO(p,q). Another property of the singular representations is that they persist in the limit as q→∞. Bibliography: 70 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 9–91. Translated by B. Bekker.     

Gaps and Fatou theorems for series in Schauder basis of holomorphic functions

We prove a gap theorem and the “Fatou change-of-sign theorem” [Fatou, P., 1906, Sèries trigonométriques e séries de Taylor. Acta Mathematica, 39, 335–400] for expansions in common Schauder basis of holomorphic functions.     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

Identities for the Riemann zeta function

In this paper, we obtain several expansions for ζ(s) involving a sequence of polynomials in s, denoted by α _k (s). These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some series expansions for the zeta function that are known for integer values of s. The expansions also give a different approach to the analytic continuation of the Riemann zeta function.