By Swanhild Bernstein, Uwe Kähler, Irene Sabadini, Frank Sommen
Hypercomplex research is the extension of advanced research to raised dimensions the place the concept that of a holomorphic functionality is substituted by means of the concept that of a monogenic functionality. In contemporary many years this conception has come to the vanguard of upper dimensional research. There are numerous techniques to this: quaternionic research which in basic terms makes use of quaternions, Clifford research which depends on Clifford algebras, and generalizations of advanced variables to better dimensions resembling split-complex variables. This e-book contains a number of papers offered on the consultation on quaternionic and hypercomplex research on the ISAAC convention 2013 in Krakow, Poland. the subjects coated signify new views and present traits in hypercomplex research and functions to mathematical physics, photo research and processing, and mechanics.
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Extra info for Hypercomplex Analysis: New Perspectives and Applications
Funct. , 189(2):283–324, 2002.  N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. , 68:337–404, 1950. Ya. S. Iohvidov. Foundations of the theory of linear operators in spaces with indeﬁnite metric. Nauka, Moscow, 1986 (Russian). English translation: Linear operators in spaces with an indeﬁnite metric. John Wiley, New York, 1989.  J. Ball, T. Trent, and V. Vinnikov. Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces. A. Kaashoek, volume 122 of Operator Theory: Advances and Applications, pages 89–138.
Mathematics Subject Classiﬁcation (2010). MSC: 30G35, 30H20. Keywords. Fock space, slice hyperholomorphic functions, quaternions, Cliﬀord algebras. 1. Introduction Fock spaces are a very important tool in quantum mechanics, and also in its quaternionic formulation; see the book of Adler  and the paper . Roughly speaking, they can be seen as the completion of the direct sum of the symmetric or anti-symmetric, or full tensor powers of a Hilbert space which, from the point of view of Physics, represents a single particle.
3. Let S be a Hr×s -valued function slice hypermeromorphic in an axially symmetric s-domain Ω which intersects the positive real line. Then, S ∈ Sκ (H+ ) if and only if it can be written as S = B0− S0 , where B0 is a Hr×r valued ﬁnite Blaschke product of degree κ, and S0 ∈ S0 (H+ ). References  K. Abu-Ghanem, D. Alpay, F. P. Kimsey, I. Sabadini. 3352.  J. Agler. On the representation of certain holomorphic functions deﬁned on a polydisk, volume 48 of Operator Theory: Advances and Applications, pages 47–66.