Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of ℂ n

We give a Schwarz-Pick estimate for bounded holomorphic functions on unit ball in Cn, and generalize some early work of Schwarz-Pick estimates for bounded holomorphic functions on unit disk in C.     

Interpolation with gradient in the ball and polydisk

We consider an interpolation problem with gradient for the Banach algebras of bounded holomorphic functions in the ball and polydisk. Sufficient conditions for the solvability of this problem are obtained. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 407–416, September, 1999.     

Summability of Lagrange type interpolation series

We consider certain aspects of the theory of interpolation via entire functions of exponential type, which include sampling node sequences χ which can be highly irregular and data sequences {y(_x)}_xn∈χ which are not necessarily bounded. Under appropriate conditions, we show that there is an entire function of a suitable exponential type which uniquely interpolates the data and indicate the validity of certain summabilty methods for the corresponding Lagrange type interpolation series. Some of our results significantly extend the work of Schoenberg,Cardinal interpolation and spline functions VII: The behavior of cardinal spline interpolation as their degree tends to infinity, J. Analyse Math.27 (1974), 205–229.     

On an extreme point conjecture for concave functions

Let CO(A), A ∈ (1; 2], denote the family of concave univalent functions in the unit disk ⅅ with opening angle at infinity bounded by πA. We prove a weak form of a conjecture on the extreme points of clco CO(A) from the paper in Indian J. Math. 50, 339–349 (2008).     

Minimal dimension families of complex lines sufficient for holomorphic extension of functions

We consider continuous functions given on the boundary of a bounded domain D in ℂ ⁿ , n > 1, with the one-dimensional holomorphic extension property along families of complex lines. We study the existence of holomorphic extensions of these functions to D depending on the dimension and location of the families of complex lines.     

Integral representation with weights II, division and interpolation

Let f be a r×m-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic ψ such that ϕ=f ψ, provided that ϕ is holomorphic and annihilates a certain residue current with support on the set where f is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previously known for the case r=1. The author was partially supported by the Swedish Research Council     

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.     

Inverses of Generators of Integrated or Regularized Semigroups

It is shown that, for any bounded, injective operator C, the class of injective, densely defined operators with dense range and nonempty resolvent that generate bounded holomorphic C-regularized semigroups is closed under inversion, but, for any n ∈ N, the class of injective, densely defined operators with dense range that generate bounded holomorphic n-times integrated semigroups is very far from being closed under inversion: it is shown that, if both A and A^-1 generate bounded holomorphic n-times integrated semigroups of sufficiently large angle θ, then they both generate strongly continuous bounded holomorphic semigroups of angle θ.     

Representing systems for the space of holomorphic functions in a (ρ,α)-convex domain

This article is a natural supplement to Chapter 5 ofAsymptotic Characteristics of Entire Functions and Their Applications [Nauka, Novosibirsk (1991)]. Earlier, for the functions holomorphic in the closure of a (ρ,α)-convex bounded domain D, the author found a criterion of the existence of a series expansion in a special system of entire functions. Here it is shown that the same criterion applies to the functions holomorphic in D. Moreover, a new integral representation of entire functions is described, which enables one to construct new representing systems for the spaces of holomorphic functions and H(D). Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 91–196. Translated by D. V. Yakubowich.     

Weierstrass factorizations in compact Riemann surfaces

Letν′ be the complementary in a compact Riemann surface ν of a point (or a finite set). In this paper are characterized the subfields, of the field of meromorphic functions inν′, containing sufficient functions to verify a factorization property, similar to that of the classical Weierstrass theorem. It is also seen that the field generated by the Baker functions is not of this type, and the problem is solved of determining the divisors, inν′, of the holomorphic functions admiting Weierstrass factorizations with Baker functions as factors. As an application, a theorem is obtained characterizing the infinite products, of meromorphic functions in ν with bounded degree, which converge normally inν′. The first-named author is partially supported by PB91-0188.     

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.     

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     

Sharp Lebesgue constants for bounded cubic interpolation ℒ-splines

We construct the Lebesgue function and find sharp Lebesgue constants for bounded cubic interpolation ℒ-splines with equally spaced interpolation nodes and discontinuities of the second derivative chosen so that the cubic ℒ-splines satisfy a certain extremal property with respect to the functions under interpolation.     

Kergin approximation in Banach spaces

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges.     

Operator Valued Functions with Pick Operators Having Negative Subspaces of Bounded Dimensions

Functions whose values are bounded linear Hilbert space operators (each operator may be defined on its own subspace of the ambient Hilbert space), the domain of definition is contained in the open unit disc, and having the following property κ, are studied. (κ): All Pick operators associated with the function have the dimensions of their spectral subspace corresponding to the negative part of the spectrum bounded above by a fixed nonnegative integer κ, and the bound κ is attained. No a priori hypotheses concerning regularity of the functions are assumed. A particular class of functions, called standard functions, is introduced, and the corresponding nonnegative integer κ is identified for standard functions. It is proved that every function with property (κ) can be extended to a standard function with property (κ), for the same κ. This result is interpreted as a result on interpolation. As an application, maximal (with respect to the extension relation) functions with the property κ, for a fixed κ, are studied in terms of standard functions. Received: August 5, 2007., Accepted: October 24, 2007.     

A holomorphic extension theorem from a fractal hypersurface in ℂ<Superscript>2</Superscript>

We consider the problem of identifying boundary values of holomorphic functions on bounded domains in ℂ². We use the quaternionic analysis techniques to extending the CR structure to a pure function theoretical nature. The advantage of our procedure lies in the fact that it also runs for domains with fractal boundary.     

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.     

A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.     

Multiplicative structure in weighted BMOA spaces

We study the multiplicative structure of holomorphic functions in the space BMOA_φ, defined by a decrease condition on the mean oscillation. The inner and outer factors occurring in the canonical factorization of such functions are completely characterized by explicit quantitative conditions.     

Interpolation inequalities and decay estimates for derivatives

Under the only assumption of the cone property for a given domain Ω⊂R ⁿ, it is proved that interpolation inequalities for intermediate derivatives of functions in the Sobolev spaces W^m,p (Ω) or even in some weighted Sobolve spaces W _w ^m,p (Ω) still hold. That is, the usual additional restrictions that Ω is bounded or has the uniform cone property are both removed. The main tools used are polynomial inequalities, by which it is also obtained pointwise version interpolation inequalities for smooth and analytic functions. Such pointwise version inequalities give explicit decay estimates for derivatives at infinity in unbounded domains which have the cone property. As an application of the decay estimates, a previous result on radial basis function approximation of smooth functions is extended to the derivative-simultaneous approximation.