# Bogolyubov theorem

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Bogolyubov's edge-of-the-wedge theorem is a generalization of the principle of analytic continuation, in particular to the case of several complex variables. It was obtained in 1956 by N.N. Bogolyubov in the justification of the dispersion relations in quantum field theory (, Appendix A). The modern formulation is as follows. Let a function , , be holomorphic in an open set , where is an open cone in with apex at zero such that , let the open set be contained in the ball and suppose that for any test function from the limit exists, independent from the way in which , ; then can be analytically continued into the domain : where is a complex neighbourhood of the set , is a constant which depends only on the cone , and is the distance from the point to the boundary of . Bogolyubov's edge-of-the-wedge theorem also remains valid if . In such a case, and under certain assumptions regarding the growth of the function , one obtains the original formulation of Bogolyubov (cf. ; the light cone in plays the role of the cone). There exist various proofs and generalizations of this theorem , . In particular, one can mention the generalizations to hyperfunctions  and to holomorphic cocycles .

Bogolyubov's edge-of-the-wedge theorem is extensively employed in axiomatic quantum field theory, in the theory of partial differential equations and in the theory of boundary values of holomorphic functions (especially of functions of several complex variables). A useful completion of the theorem is the -convex hull theorem . Let, under the conditions of Bogolyubov's theorem, , , where is a convex sharp cone; then where is the envelope of holomorphy (cf. Holomorphic envelope) of , is the real section of the domain , and is the -convex hull of the set , i.e. the smallest open set containing and having the following property: If the points and of can be connected by a -similar curve that is totally contained in , then all -similar curves homotopic to it are located in .

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How to Cite This Entry:
Bogolyubov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bogolyubov_theorem&oldid=16527
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article