Volume 29, Issue 1

Suppose f is a C ∞ function or a power series in n variable belonging to a Carleman or a Beurling class F n . Let Γ be a family of polynomial maps P :C d → C n , d ≤ n , such that f ∘ P ∈ F d , ∀ P ∈Γ. Under what conditions can one conclude that f ∈ F n ? The F n -analogs of the following theorems are obtained: (i) The Bochnak–Siciak theorem: A C ∞ function on R n that is analytic on every line is analytic. (ii) Zorn´s theorem: If a double power series F(x,y) has the property that for all ξ , η ∈C the t -series F(ξt,ηt) converges, then F(x,y) is convergent as a double series. The methods applied here also yield new proofs of the above mentioned theorems as well as their improvements due to P. Lelong ([9]), Abhyankar, Moh, and Sathye ([1,19]), and Levenberg and Molzon ([10]).

In this paper we describe a net limit method for constructing topologies on given Boolean algebras. If functions in this limit procedure are monotone and approximately continuous in a generalized sense, then the obtained process is recursive.

Abelian Theorems for quasi-Nörlund and absolute quasi-Nörlund methods are known. A Tauberian theorem for quasi-Nörlund was established recently and in the present paper we prove a corresponding Tauberian theorem for absolute quasi-Nörlund.

This paper is concerned with a boundary Nevanlinna–Pick interpolation for Nevanlinna matrix functions and the Hamburger matrix moment problem but with specified constraints that the nonnegative matrix-valued measure has no mass distributions at a finite number of real points. Intrinsic connections between these two problems are established by the so-called Hankel vector approach. The connections, together with the theory of matrix moments and the theory of matrix polynomials with respect to a positive Hermitian block Hankel matrix due to H. Dym, enable us to get a solvability criterion and a parameterized description of all the solutions for each of these two problems.

Continuing work from [5], we use some normality criteria to show that certain sets of two analytic functions are not semidual. In particular, sets of the form V := {z ↦ 1/(1 - z) ; φ } where φ is a rational function analytic in the unit disk are studied, and sufficient conditions for V being not semidual are given.

It is well known that on Stein manifolds every analytic set is globally defined. In this note we give an elementary proof for the converse in the case of domains in ℂ n .

We exhibit several counterexamples showing that the famous Serrin´s symmetry result for semilinear elliptic overdetermined problems may not hold for partially overdetermined problems, that is when both Dirichlet and Neumann boundary conditions are prescribed only on part of the boundary. Our counterexamples enlighten subsequent positive symmetry results obtained by the first two authors for such partially overdetermined systems and justify their assumptions as well.

Given a probability space (Ω, A,P ), a Polish space X and a (B(X) ⊗ A) -measurable function f:X × Ω→ X we consider Borel and bounded solutions ψ:X →ℝ of the functional inequality ψ(x) ≤ ∫ Ω ψ(f(x,ω))P(dω) and a uniqueness-type problem for Borel and bounded solutions φ:X →ℝ of the functional equation φ(x) = ∫ Ω φ(f(x,ω))P(dω) .

In this paper, we use Nevanlinna theory to obtain a new bounded for Mason´s theorem and find a lower bounded for the number of the distinct simple roots of a polynomial on an algebraically closed field of characteristic zero. As an application of this result, we prove some results which improve a result of H. Davenport.