The Poisson Integral Formula and Representations of SU ( 1 , 1 )
نویسنده
چکیده
We present a new proof of the Poisson integral formula for harmonic functions using the methods of representation theory. In doing so, we exhibit the irreducible subspaces and unitary structure of a representation of the group SU(1, 1) of 2 × 2 complex generalized special unitary matrices. Our arguments illustrate a technique that can be used to prove similar reproducing formulas in higher dimensions and for other classes of functions. Our paper should be accessible to readers with minimal knowledge of complex analysis. Acknowledgements: The author would like to thank Professor Matvei Libine for his support and mentorship during the development of this paper. The author would also like to thank Professors Kevin Pilgrim and Bruce Solomon, as well as the National Science Foundation, for providing the organization and funding for the REU program at Indiana University that made this paper possible. This REU program was supported by the NSF grant DMS-0851852. Page 2 RHIT Undergrad. Math. J., Vol. 12, No. 2
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Realizations of su ( 1 , 1 ) and U q ( su ( 1 , 1 ) ) and generating functions for orthogonal polynomials
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