# Carleman formulae with holomorphic kernels and their uniqueness properties

Carleman formulae with holomorphic kernels and their uniqueness properties -- Carleman formulae with a holomorphic kernel and integration over a boundary set of maximum dimension are obtained. These formulae have a uniqueness property: if a limit in the formula exists, it gives exactly the function which was an integrand. The Cauchy formula and its multidimensional analogies lack this property. The Carleman formulae are proved by approximating the kernel (M.M. Lavrent'ev's method). L ONE-DIMENSIONAL CASE The Carleman formulae, which allow us to reconstruct the holomorphic functions in a domain using their values on part of the domain boundary, are systematically considered in the book [2]. A number of existence theorems for the Carleman formulae with a holomorphic kernel and integration over a boundary set of maximum dimension 2n -- 1, in a domain T> c C1, were given in Section 17 [2], and there was only a single example for n > 1 concerning a very special case (see Theorem 16.7 in [2]). However, there exist very simple Carleman formulae of this kind, which generalize the following fact (n = 1). Let be a piecewise-smooth arc whose end points belong to a circle 7 centered at zero ( belongs to the interior of 7). Suppose that T> is http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Inverse and Ill-Posed Problems de Gruyter

# Carleman formulae with holomorphic kernels and their uniqueness properties

Journal of Inverse and Ill-Posed Problems, Volume 1 (3) – Jan 1, 1993
8 pages

/lp/de-gruyter/carleman-formulae-with-holomorphic-kernels-and-their-uniqueness-wO2YaGo5hs
Publisher
de Gruyter
ISSN
0928-0219
eISSN
1569-3945
DOI
10.1515/jiip.1993.1.3.169
Publisher site
See Article on Publisher Site

### Abstract

-- Carleman formulae with a holomorphic kernel and integration over a boundary set of maximum dimension are obtained. These formulae have a uniqueness property: if a limit in the formula exists, it gives exactly the function which was an integrand. The Cauchy formula and its multidimensional analogies lack this property. The Carleman formulae are proved by approximating the kernel (M.M. Lavrent'ev's method). L ONE-DIMENSIONAL CASE The Carleman formulae, which allow us to reconstruct the holomorphic functions in a domain using their values on part of the domain boundary, are systematically considered in the book [2]. A number of existence theorems for the Carleman formulae with a holomorphic kernel and integration over a boundary set of maximum dimension 2n -- 1, in a domain T> c C1, were given in Section 17 [2], and there was only a single example for n > 1 concerning a very special case (see Theorem 16.7 in [2]). However, there exist very simple Carleman formulae of this kind, which generalize the following fact (n = 1). Let be a piecewise-smooth arc whose end points belong to a circle 7 centered at zero ( belongs to the interior of 7). Suppose that T> is

### Journal

Journal of Inverse and Ill-Posed Problemsde Gruyter

Published: Jan 1, 1993

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