Integral representations in tube domains over polyhedra and problems of approximation of functions
Let \(P\) be a polyhedron in \(\mathbb{R}^ n\), \(T_ P = \{z : z = x + iy \in \mathbb{C}^ n\), \(x \in \mathbb{R}^ n\), \(y \in P\}\). Let \(\gamma (y)\), \(y \in P\), be an arbitrary positive continuous function of the space \(L^ 1(P)\). For \(s,p \in (0, + \infty)\) the author denotes by \(H^ p_{s, \gamma} (T_ P)\) the class of holomorphic functions \(f(x + iy)\), \(x + iy \in T_ P\), such that \[ \int_ P \left( \int_{\mathbb{R}^ n} \bigl | f(x + iy) \bigr |^ p dx \right)^ s \gamma (y)dy < + \infty. \] The main result of this paper is Theorem 1.2 about the integral representation of a function \(f(z) \in H^ p_{s, \gamma} (T_ P)\), \(1 \leq p \leq 2\), \(1/p \leq s < + \infty\), in the form \[ f(z) = {1 \over (2 \pi)^ n} \int_{T_ P} f(w) \Phi(z,w) \gamma (v)dudv, \quad z \in T_ P,\;w = u + iv, \] where \[ \Phi (z,w) = \int_{\mathbb{R}^ n} {\exp \Bigl\{ \sum^ n_{k=1} (z_ k - \overline w_ k)t_ k \Bigr\} \over \exp \Bigl\{ - \sum^ n_{k=1} y_ k2t_ k \Bigr\} \gamma (y)dy} dt,\;z, w \in T_ P. \] Besides if the polyhedron \(P\) is the intersection of polyhedron angles \(Y_ m\) then the author receives the approximation theorem (Theorem 2.1) for functions \(f \in H^ 2_{s,t} (T_ P)\) by the sums of type \(\sum^ M_ 1 f_ m\), where \(f_ m \in H^ 2_{s,1} (T_{Y_ m})\), \(1 \leq m \leq M\).
scientific article; zbMATH DE number 713875
scientific article