Mathematics & Physics 2015, 8(2), 371–374 УДК 517.55 Carleman’s Formula for a Matrix Polydisk Bahodir A. Shoimkhulov Jorabek T.Bozorov∗ National University of Uzbekistan VUZ Gorodok, Tashkent, 100174 Uzbekistan Received 20.01.2015, received in revised form 24.03.2015, accepted 06.05.2015 In this work an integral formula for a matrix polydisk is obtained. <...> For a function from the Hardy class it allows to recover its value at any interior point from its values on a part of the Shilov boundary. <...> Statement of the problem and preliminaries Consider a class of holomorphic functions in a domain D that behave reasonably well near the boundary ∂D. Carleman’s formulas solve the problem of recovery of a function from such a class from its values on a set of uniqueness M ⊂ ∂D for this class, which does not contain the Shilov boundary of D. One-dimensional and multidimensional formulas were studied in the monograph [1]. <...> In the paper [2] a new method is proposed for finding Carleman’s formulas in homogeneous domains using a domain’s automorphisms. <...> In the present paper we use this method for a matrix polydisk. <...> Let Z = Z1,Z2, .,Zn ∈ Cn[mЧm] be a vector of quadratic matrices of order m over the field of complex numbers C. The unit matrix disk is defined as the set τ = {Z ∈ C[mЧm] : ZZ∗ < I} , [mЧm]-matrix) means that the Hermitian matrix I −ZZ∗ is positive definite. <...> To formulate the main result, we need the following Carleman formula for the unit matrix disk obtained in [2]. <...> All rights reserved c – 371 – (1) ′ is the conjugate transpose of the matrix Z, the notation ZZ∗ < I (I is the unit Bahodir A.Shoimkhulov, Jorabek T.Bozorov Carleman’s Formula for a Matrix Polydisc where SU(m) is the group of special unitary matrices, i.e. det(u) = 1,m1 is the normed Lebesgue measure on the unit circle ∂U. Further on, let ϕ0 = expψ0, where ψ0(ξ) = 1 2πi the class H1(D) if 0r<1 sup M0,u η +λ η −λ · S(D) |f(rξ)| dµ1 <∞, dη η . <...> The Hardy class H1(D <...>