By L.A. Aizenberg

Integral representations of holomorphic services play a huge half within the classical concept of capabilities of 1 advanced variable and in multidimensional com plex research (in the later case, along with integration over the total boundary advert of a site D we regularly come across integration over the Shilov boundary five = S(D)). They clear up the classical challenge of improving on the issues of a do major D a holomorphic functionality that's sufficiently well-behaved whilst drawing close the boundary advert, from its values on advert or on S. along with this classical challenge, it truly is attainable and average to contemplate the subsequent one: to get better the holomorphic functionality in D from its values on a few set MeaD now not containing S. after all, M is to be a collection of specialty for the category of holomorphic services into account (for instance, for the features non-stop in D or belonging to the Hardy classification HP(D), p ~ 1).

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PE«()a almost everywhere for ( E E 1, imply that < 00. 8) h < 00. 5). 4) is true. 5) is also true; namely, we can prove the following assertion. 0:' ~ 1. 3 (Videnskii-Gavurina-Khavin). 5) is true also for J E Hi, z E U, 0 < (} < l. 5) reconstructs the corresponding function also if the uniqueness set E C au has Lebesgue measure zero. PART II CARLEMAN FORMULAS IN MULTIDIMENSIONAL COMPLEX ANALYSIS CHAPTER III INTEGRAL REPRESENTATIONS OF HOLOMORPHIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES AND LOGARITHMIC RESIDUES 10.

GENERALIZATIONS 20 where that Zkj are the cofactors of the matrix W. 3) under the integral sign. 0 In case of the generalized unit disk TO, we can take for D (see [167, p. 109]) the ordinary disk U(O, 1). 2) holds, Ifml = m2 = ... = m. 2. 3. where Me au(o, 1). 2 from [114, p. 167]. 4 (Sylvester interpolation formula). Below, let D be a simply connected bounded domain in ([I and let A be a Banach algebra with one (not necessarily commutative). We set nD={x: xEA,spxCD}. For a function f E A(D) we define f(x), x EnD, as is usually done in the theory of Banach algebras ([48, § 4]).

Formula of Logarithmic Residue in the Spirit of Carleman Let D be a bounded domain in ((:1 with piecewise smooth boundary 8D. Then there is a function FE A(J(D)) such that F(O) = 1 and IF(w)1 < 1. 1) wEJ(8D\M) This function F can be used as "quenching" function to obtain the formula of logarithmic residue, following the idea of Carleman to introduce a "quenching" function. 1 (Alzenberg). 2) holds, where E J is the set of zeros of fill D. 2), each zero is counted according to its multiplicity. Proof.