Christer O. Kiselman

On entire functions of exponential type and indicators of analytic functionals

SummaryWe shall be concerned with the indicatorp of an analytic functional μ on a complex manifoldU:

Regularity Properties of Distance Transformations in Image Analysis

p(\varphi ) = \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {\mu (e^{t\varphi } )} \right|,

A semigroup of operators in convexity theory

where ϕ is an arbitrary analytic function onU. More specifically, we shall consider the smallest upper semicontinuous majorantpJ of the restriction ofp to a subspace £ of the analytic functions. An obvious problem is then to characterize the set of functionspJ which can occur as regularizations of indicators. In the case whenU=Cn and £ is the space of all linear functions onCn, this set can be described more easily as the set of functions(0.1)

Functions on discrete sets holomorphic in the sense of Isaacs, or monodiffric functions of the first kind

\mathop {\lim }\limits_{\theta \to \zeta } \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {u(t\theta )} \right|

Characterizing digital straightness and digital convexity by means of difference operators

ofn complex variables ζ∈Cn whereu is an entire function of exponential type inCn. We hall prove that a function inCn is of the form (0.1) for some entire functionu of exponential type if and only if it is plurisubharmonic and positively homogeneous of order one (Theorem 3.4). The proof is based on the characterization given by Fujita and Takeuchi of those open subsets of complex projectiven-space which are Stein manifolds.Our objective in Sections 4 and 5 is to study the relation between properties ofpJ and existence and uniqueness of £-supports of μ, i.e. carriers of μ which are convex with respect to £ in a certain sense and which are minimal with this property (see Section 1 for definitions). An example is that under certain regularity conditions,pJ is convex if and only if μ has only one £-support.Section 2 contains a result on plurisubharmonic functions in infinite-dimensional linear spaces and approximation theorems for homogeneous plurisubharmonic functions inCn.The authors original proof of Theorem 3.1 was somewhat less direct than the present one (see the remark at the end of Section 3). It was suggested by Professor Lars Hörmander that a straightforward calculation of the Levi form might be possible. I wish to thank him also for other valuable suggestions and several discussions on the subject.

Plurisubharmonic functions and their singularities

Efim Khalimskys digital Jordan curve theorem states that the complement of a Jordan curve in the digital plane equipped with the Khalimsky topology has exactly two connectivity components. We present a new, short proof of this theorem using induction on the Euclidean length of the curve. We also prove that the theorem holds with another topology on the digital plane but then only for a restricted class of Jordan curves.