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In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
to
several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
. 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 In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
of a function of several real variables (see multidimensional Laplace transform).
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In ...
s on tubes can be defined in a manner in which a version of the
Paley–Wiener theorem In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (189 ...
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 In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
s 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 In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
, and has applications in relativity theory and
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
. 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 Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
.


Definition

Let R''n'' denote
real coordinate space In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vecto ...
of dimension ''n'' and C''n'' denote complex coordinate space. Then any element of C''n'' can be decomposed into real and imaginary parts: :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. Let ''A'' be an
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ...
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'': :T_A = \.


Tubes as domains of holomorphy

Suppose that ''A'' is a connected open set. Then any complex-valued function that is holomorphic in a tube ''T''''A'' can be extended uniquely to a holomorphic function on the convex hull of the tube , which is also a tube, and in fact :\operatorname \, T_A = T_. Since any convex open set is a
domain of holomorphy In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain. Formal ...
( holomorphically convex), a convex tube is also a domain of holomorphy. So the
holomorphic envelope In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex der ...
of any tube is equal to its convex hull.


Hardy spaces

Let ''A'' be an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
in R''n''. The
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In ...
''H'' ''p''(''T''''A'') is the set of all
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s ''F'' in ''T''''A'' such that :\int_ , F(x+iy), ^p\, dy < \infty for all ''x'' in ''A''. In the special case of ''p'' = 2, functions in ''H''2(''T''''A'') can be characterized as follows. Let ''ƒ'' be a complex-valued function on R''n'' satisfying :\sup_\int_, f(t), ^2e^\,dt < \infty. The Fourier–Laplace transform of ''ƒ'' is defined by :F(x+iy) = \int_e^f(t)\,dt. Then ''F'' is well-defined and belongs to ''H''2(''T''''A''). Conversely, every element of ''H''2(''T''''A'') has this form. A corollary of this characterization is that ''H''2(''T''''A'') contains a nonzero function if and only if ''A'' contains no straight line.


Tubes over cones

Let ''A'' be an open convex cone in R''n''. This means that ''A'' is an
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ...
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
such that, whenever ''x'' lies in ''A'', so does the entire ray from the origin to ''x''. Symbolically, :x\in A \implies tx\in A\ \ \ \text\ t>0. If ''A'' is a cone, then the elements of ''H''2(''T''''A'') have ''L''2 boundary limits in the sense that :\lim_ F(x+iy) exists in ''L''2(''B''). There is an analogous result for ''H''''p''(''T''''A''), but it requires additional regularity of the cone (specifically, the dual cone ''A''* needs to have nonempty interior).


See also

* Reinhardt domain * Siegel domain


Notes


Citations


Sources

*. *. *. *. *. {{refend Fourier analysis Harmonic analysis Several complex variables