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...

This paper studies a two-variable zeta function $Z_K(w,s)$ attached to an algebraic number field $K$, introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. When $w=1$ this function becomes the completed Dedekind zeta function ${\widehat{\zeta }}_K\left(s\right)$ of the field $K$. The function is a meromorphic function of two complex variables with polar divisor $s(w-s)$, and it satisfies the functional equation $Z_K(w,s)=Z_K(w,w-s)$. We consider the special case $K=\mathbb{Q}$, where for $w=1$ this function...

We give a proof of an integral formula of Berndtsson which is related to the inversion of Fourier-Laplace transforms of $\overline{\partial }$-closed $(n,n-1)$-forms in the complement of a compact convex set in ${ℂ}^n$.

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

For bounded logarithmically convex Reinhardt pairs "compact set - domain" (K,D) we solve positively the problem on simultaneous approximation of such a pair by a pair of special analytic polyhedra, generated by the same polynomial mapping f: D → ℂⁿ, n = dimΩ. This problem is closely connected with the problem of approximation of the pluripotential ω(D,K;z) by pluripotentials with a finite set of isolated logarithmic singularities ([23, 24]). The latter problem has been solved recently for arbitrary...

A characterization of a generalized order of analytic functions of several complex variables by means of polynomial approximation and interpolation is established.

Sufficient conditions for the existence of an analytic solution of analytic equations in the complex and real cases are given.

We study the residue current $R^f$ of Bochner-Martinelli type associated with a tuple $f=(f_1,\cdots ,f_m)$ of holomorphic germs at $0\in {\mathbf{C}}^n$, whose common zero set equals the origin. Our main results are a geometric description of $R^f$ in terms of the Rees valuations associated with the ideal $\left(f\right)$ generated by $f$ and a characterization of when the annihilator ideal of $R^f$ equals $\left(f\right)$.

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...