-- 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 . 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 , and there was only a single example for n > 1 concerning a very special case (see Theorem 16.7 in ). 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 of Inverse and Ill-Posed Problems – de Gruyter
Published: Jan 1, 1993
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