# 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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just \$49/month

### Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

### Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

### Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

### Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
PubMed

Create folders to

Export folders, citations

Abstract access only

18 million full-text articles

Print

20 pages / month

PDF Discount

20% off