63 research outputs found
Cauchy-type integrals in several complex variables
We present the theory of Cauchy-Fantappi\'e integral operators, with emphasis
on the situation when the domain of integration, , has minimal boundary
regularity. Among these operators we focus on those that are more closely
related to the classical Cauchy integral for a planar domain, whose kernel is a
holomorphic function of the parameter . The goal is to prove
estimates for these operators and, as a consequence, to obtain estimates
for the canonical Cauchy-Szeg\"o and Bergman projection operators (which are
not of Cauchy-Fantappi\'e type).Comment: This is an electronic reprint of the original article published in
the Bulletin of the Mathematical Sciences, vol. 3 (2) (2013) 241-285. This
reprint differs from the original in pagination and typographical detai
Conditioning moments of singular measures for entropy optimization. I
In order to process a potential moment sequence by the entropy optimization
method one has to be assured that the original measure is absolutely continuous
with respect to Lebesgue measure. We propose a non-linear exponential transform
of the moment sequence of any measure, including singular ones, so that the
entropy optimization method can still be used in the reconstruction or
approximation of the original. The Cauchy transform in one variable, used for
this very purpose in a classical context by A.\ A.\ Markov and followers, is
replaced in higher dimensions by the Fantappi\`{e} transform. Several
algorithms for reconstruction from moments are sketched, while we intend to
provide the numerical experiments and computational aspects in a subsequent
article. The essentials of complex analysis, harmonic analysis, and entropy
optimization are recalled in some detail, with the goal of making the main
results more accessible to non-expert readers.
Keywords: Fantappi\`e transform; entropy optimization; moment problem; tube
domain; exponential transformComment: Submitted to Indagnationes Mathematicae, I. Gohberg Memorial issu
A residue criterion for strong holomorphicity
We give a local criterion in terms of a residue current for strong
holomorphicity of a meromorphic function on an arbitrary pure-dimensional
analytic variety. This generalizes a result by A Tsikh for the case of a
reduced complete intersection
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