|
In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the complex plane to several complex variables. A strip can be thought of as the collection of complex numbers whose real part lie in a given subset of the real line and whose imaginary part is unconstrained; likewise, a tube is the set of complex vectors whose real part is in some given collection of real vectors, and whose imaginary part is unconstrained. Tube domains are domains of the Laplace transform of a function of several real variables (see multidimensional Laplace transform). Hardy spaces on tubes can be defined in a manner in which a version of the Paley–Wiener theorem from one variable continues to hold, and characterizes the elements of Hardy spaces as the Laplace transforms of functions with appropriate integrability properties. Tubes over convex sets are domains of holomorphy. The Hardy spaces on tubes over convex cones have an especially rich structure, so that precise results are known concerning the boundary values of ''H''''p'' functions. In mathematical physics, the future tube is the tube domain associated to the interior of the past null cone in Minkowski space, and has applications in relativity theory and quantum gravity. Certain tubes over cones support a Bergman metric in terms of which they become bounded symmetric domains. One of these is the Siegel half-space which is fundamental in arithmetic. ==Definition== Let R''n'' denote real coordinate space of dimension ''n'' and C''n'' denote complex coordinate space. Then any element of C''n'' can be decomposed into real and imaginary parts: : Let ''A'' be an open subset of R''n''. The tube over ''A'', denoted ''T''''A'', is the subset of C''n'' consisting of all elements whose real parts lie in ''A'':〔. Some conventions instead define a tube to be a domain such that the imaginary part lies in ''A''; see .〕 : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「tube domain」の詳細全文を読む スポンサード リンク
|