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| {{for|other uses of "tube" in mathematics|tube (mathematics)}}
| | I'm Bud and I live with my husband and our 2 children in Toronto, in the ON south area. My hobbies are Fantasy Football, Roller skating and Knapping.<br><br>My web-site - [http://tinyurl.com/ma8zrdo nike free run] |
| 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.
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| Tube domains are [[domain of a function|domains]] of the [[Laplace transform]] of a function of several [[real number|real]] variables (see [[multidimensional Laplace transform]]). [[Hardy space]]s 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 set]]s are [[domain of holomorphy|domains of holomorphy]]. The Hardy spaces on tubes over convex [[Monge cone|cone]]s have an especially rich structure, so that precise results are known concerning the boundary values of ''H''<sup>''p''</sup> 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]].<ref>{{harvnb|Gibbons|2000}}</ref> Certain tubes over cones support a [[Bergman metric]] in terms of which they become [[bounded symmetric domain]]s. One of these is the [[Siegel half-space]] which is fundamental in [[arithmetic]].
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| ==Definition==
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| Let '''R'''<sup>''n''</sup> denote [[real coordinate space]] of dimension ''n'' and '''C'''<sup>''n''</sup> denote [[complex number|complex]] coordinate space. Then any element of '''C'''<sup>''n''</sup> can be decomposed into real and imaginary parts:
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| :<math>a=(z_1,\dots,z_n) = (x_1+iy_1, \dots, x_n+iy_n) = (x_1,\dots,x_n) + i(y_1,\dots,y_n)=x+iy.</math>
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| Let ''A'' be an [[open set|open]] subset of '''R'''<sup>''n''</sup>. The '''tube over ''A''''', denoted ''T''<sub>''A''</sub>, is the subset of '''C'''<sup>''n''</sup> consisting of all elements whose real parts lie in ''A'':<ref>{{harvnb|Hörmander|1990}}. Some conventions instead define a tube to be a domain such that the imaginary part lies in ''A''; see {{harvnb|Stein|Weiss|1971}}.</ref>
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| :<math>T_A = \{ z=x+iy\in\mathbb{C}^n\mid x\in A\}.</math>
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| ==Tubes as domains of holomorphy==
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| Suppose that ''A'' is a connected open set. Then any complex-valued function that is [[holomorphic]] in a tube ''T''<sub>''A''</sub> can be extended uniquely to a holomorphic function on the [[convex hull]] of the tube {{nowrap|ch ''T''<sub>''A''</sub>}},<ref>{{harvnb|Hörmander|1990}}</ref> which is also a tube, and in fact
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| :<math>\operatorname{ch} \, T_A = T_{\operatorname{ch}\, A}.</math>
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| Since any convex open set is a [[domain of holomorphy]], a convex tube is also a domain of holomorphy. So the [[holomorphic envelope]] of any tube is equal to its convex hull.<ref>{{harvnb|Chirka|2001}}</ref>
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| ==Hardy spaces==
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| Let ''A'' be an [[open set]] in '''R'''<sup>''n''</sup>. The [[Hardy space]] ''H''<sup> ''p''</sup>(''T''<sub>''A''</sub>) is the set of all [[holomorphic function]]s ''F'' in ''T''<sub>''A''</sub> such that
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| :<math>\int_{\mathbb{R}^n} |F(x+iy)|^p\, dy < \infty</math>
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| for all ''x'' in ''A''.
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| In the special case of ''p'' = 2, functions in ''H''<sup>2</sup>(''T''<sub>''A''</sub>) can be characterized as follows.<ref>{{harvnb|Stein|Weiss|1971}}</ref> Let ''ƒ'' be a complex-valued function on '''R'''<sup>''n''</sub> satisfying
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| :<math>\sup_{x\in A}\int_{\mathbb{R}^n}|f(t)|^2e^{-4\pi x\cdot t}\,dt < \infty.</math>
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| The Fourier–Laplace transform of ''ƒ'' is defined by
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| :<math>F(x+iy) = \int_{\mathbb{R}^n}e^{2\pi z\cdot t}f(t)\,dt.</math>
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| Then ''F'' is well-defined and belongs to ''H''<sup>2</sup>(''T''<sub>''A''</sub>). Conversely, every element of ''H''<sup>2</sup>(''T''<sub>''A''</sub>) has this form.
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| A corollary of this characterization is that ''H''<sup>2</sup>(''T''<sub>''A''</sub>) contains a nonzero function if and only if ''A'' contains no straight line.
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| ===Tubes over cones===
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| Let ''A'' be an open convex cone in '''R'''<sup>''n''</sup>. This means that ''A'' is an [[open set|open]] [[convex set]] such that, whenever ''x'' lies in ''A'', so does the entire ray from the origin to ''x''. Symbolically,
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| :<math>x\in A \implies tx\in A\ \ \ \text{for all}\ t>0.</math>
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| If ''A'' is a cone, then the elements of ''H''<sub>2</sub>(''T''<sub>''A''</sub>) have ''L''<sup>2</sup> boundary limits in the sense that<ref>{{harvnb|Stein|Weiss|1971}}</ref>
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| :<math>\lim_{y\to 0} F(x+iy)</math> | |
| exists in ''L''<sup>2</sup>(''B''). There is an analogous result for ''H''<sup>''p''</sub>(''T''<sub>''A''</sub>), but it requires additional regularity of the cone (specifically, the [[dual cone]] ''A''* needs to have nonempty interior).
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| ==See also==
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| * [[Reinhardt domain]]
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| *[[Siegel domain]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{citation|last=Gibbons|first=G.W.|authorlink=Gary Gibbons|title=Holography and the future tube|journal=Classical and quantum gravity|year=2000|volume=17|pages=1071–1079}}.
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| * {{citation|last=Lars|first=Hörmander|authorlink=Lars Hörmander|title=Introduction to complex analysis in several variables|publisher=North-Holland|year=1990|location=New York|isbn=0-444-88446-7}}.
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| * {{citation|last1=Stein|first1=Elias|authorlink1=Elias Stein|first2=Guido|last2=Weiss|authorlink2=Guido Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=978-0-691-08078-9|location=Princeton, N.J.}}.
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| * {{springer|Tube domain|id=l/l057540|first=E.M.|last=Chirka}}
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| [[Category:Fourier analysis]]
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| [[Category:Harmonic analysis]]
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| [[Category:Several complex variables]]
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I'm Bud and I live with my husband and our 2 children in Toronto, in the ON south area. My hobbies are Fantasy Football, Roller skating and Knapping.
My web-site - nike free run