For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ℝn. Due to the conditions on M each function of this space admits a holomorphic extension in ℂn. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions...

The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces $C^{\infty }\left(G\right)$ and $C^{\infty }\left(K\right)$ of infinitely differentiable functions where G is an arbitrary domain in ${ℝ}^p$, p≥1, while K is a compact set in ${ℝ}^p$ with non-void interior K̇ such that $\overline{K}̇=K$. Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G\subseteq {ℂ}^p$ are also investigated.

Let H(B) denote the space of all holomorphic functions on the unit ball B of ℂⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. We study the integral-type operator $C_{\phi }^{g}f\left(z\right)={\int }_0^1\Re f\left(\phi \left(tz\right)\right)g\left(tz\right)dt/t$, f ∈ H(B). The boundedness and compactness of $C_{\phi }^g$ from Privalov spaces to Bloch-type spaces and little Bloch-type spaces are studied

It is shown that on strongly pseudoconvex domains the Bergman projection maps a space $L{v}_k$ of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character. Let Ω be a bounded, pseudoconvex set with C³ boundary. We show that if the Bergman projection is continuous on a space $E\supset L^{\infty }\left(\Omega \right)$ defined by weighted-sup seminorms and equipped with the topology...

Consider a family of integral operators and a related family of differential operators, both defined on a class of analytic functions holomorphic in the unit disk, distortion properties of the real part are derived from a general aspect.

We give in this paper conditions for a mapping to be globally injective in a topological vector space.

We define locally convex spaces LW and HW consisting of measurable and holomorphic functions in the unit ball, respectively, with the topology given by a family of weighted-sup seminorms. We prove that the Bergman projection is a continuous map from LW onto HW. These are the smallest spaces having this property. We investigate the topological and algebraic properties of HW.

Si K es un compacto no vacío en Rr, damos una condición suficiente para que la inyección canónica de ε{M},b(K) en ε{M},d(K) sea nuclear. Consideramos el caso mixto y obtenemos la existencia de un operador de extensión nuclear de ε{M1}(F)A en ε{M2}(Rr)D donde F es un subconjunto cerrado propio de Rr y A y D son discos de Banach adecuados. Finalmente aplicamos este último resultado al caso Borel, es decir cuando F = {0}.

Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f\left(z\right)={\sum }_{k=0}^{\infty }{a}_k{z}^{-k-1}$ be its Taylor expansion at ∞, and $H_s\left(f\right)=det{\left(a_{k+l}\right)}_{k,l=0}^s$ the sequence of Hankel determinants. The classical Pólya inequality says that $limsu{p}_{s\to \infty }{\left|H_s\left(f\right)\right|}^{1/s²}\le d\left(K\right)$, where d(K) is the transfinite diameter of K. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Pólya’s inequality, considered by the second author in Math. USSR Sbornik 25 (1975), 350-364.

In this paper, we establish some density theorems in the setting of particular locally convex vector lattices of continuous functions de ned on a locally compact Hausdorff space, which we introduced and studied in [3,4] and which we named regular vector lattices. In this framework, by using properties of the subspace of the so-called generalized af ne functions, we give a simple description of the closed vector sublattice, the closed Stone vector sublattice and the closed subalgebra generated by...