Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

Function theory in several complex variables

Geometry of sets and measures in euclidean spaces fractals and rectifiability

Polynomials orthogonal on the unit circle

In what follows, we introduce the classical Gamma function in Sect. 2.1. It is essentially understood to be a generalized factorial. However, there are many further applications, e.g., as part of probability distributions (see, e.g., Evans et al. 2000). The main properties of the Gamma function are explained in this chapter (for a more detailed discussion the reader is referred to, e.g., Artin (1964), Lebedev (1973), Muller (1998), Nielsen (1906), and Whittaker and Watson (1948) and the references therein).

The Gamma Function

A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the position and orientation of the matter fields. In this manner all properties of the background spacetime are removed from physics and what remains are a set of ‘intrinsic’ relations between physical fields. For a wide range of phenomena, including all present experimental tests, the theory reproduces the predictions of general relativity. Differences do emerge, however, through the first–order nature of the equations and the global properties of the gauge fields and through the relationship with quantum theory. The properties of the gravitational gauge fields are derived from both classical and quantum viewpoints. Field equations are then derived from an action principle and consistency with the minimal coupling procedure selects an action which is unique up to the possible inclusion of a cosmological constant. This in turn singles out a unique form of spin–torsion interaction. A new method for solving the field equations is outlined and applied to the case of a time–dependent, spherically symmetric perfect fluid. A gauge is found which reduces the physics to a set of essentially Newtonian equations. These equations are then applied to the study of cosmology and to the formation and properties of black holes. Insistence on finding global solutions, together with the first–order nature of the equations, leads to a new understanding of the role played by time reversal. This alters the physical picture of the properties of a horizon around a black hole. The existence of global solutions enables one to discuss the properties of field lines inside the horizon due to a point charge held outside it. The Dirac equation is studied in a black hole background and provides a quick (though ultimately unsound) derivation of the Hawking temperature. Some applications to cosmology are also discussed and a study of the Dirac equation in a cosmological background reveals that the only models consistent with homogeneity are spatially flat. It is emphasized throughout that the description of gravity in terms of gauge fields, rather than spacetime geometry, leads to many simple and powerful physical insights. The language of ‘geometric algebra’ best expresses the physical and mathematical content of the theory and is employed throughout. Methods for translating the equations into other languages (tensor and spinor calculus) are given in appendices.

Gravity, gauge theories and geometric algebra

It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thoroughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.

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Lie-groups as Spin groups.

Recently several generalizations to higher dimension of the Fourier transform using Clifford algebra have been introduced, including the Clifford-Fourier transform by the authors, defined as an operator exponential with a Clifford algebra-valued kernel.

In this paper an overview is given of all these generalizations and an in depth study of the two-dimensional Clifford-Fourier transform of the authors is presented. In this special two-dimensional case a closed form for the integral kernel may be obtained, leading to further properties, both in the L 1 and in the L 2 context. Furthermore, based on this Clifford-Fourier transform Clifford-Gabor filters are introduced.

The Two-Dimensional Clifford-Fourier Transform

Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several c...

Fundaments of Hermitean Clifford analysis part II: splitting of h -monogenic equations

In this paper the classical theory of spherical harmonics in R^m is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of this operator, a new type of integration over the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti. This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk-Hecke theorem to superspace. Finally, this integration over the supersphere is used to define an integral over the whole superspace and it is proven that this is equivalent with the Berezin integral, thus providing a more sound definition of the Berezin integral.

Spherical harmonics and integration in superspace

In relation to the solution of the Vekua system for axial type monogenic functions, generalizations of Fueter’s Theorem are discussed. We show that if f is holomorphic function in one complex variable, then for any underlying space R1 the induced function ∆ k+(n−1)/2f(x0+x)Pk(x), where Pk(x) is left-monogenic and homogeneous of degree k, is left-monogenic whenever k + (n− 1)/2 is a non-negative integer. If the space dimension n + 1 is odd, then the above also holds for k being non-negative integers. §

https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2002/0009/0002/MAA-2002-0009-0002-a005.pdf

Frank Sommen

Papers

Lie-groups as Spin groups.

The Two-Dimensional Clifford-Fourier Transform

Fundaments of Hermitean Clifford analysis part II: splitting of h -monogenic equations

Spherical harmonics and integration in superspace

Generalizations of Fueter’s theorem