Abstract
We study the Laplacian from an analytic viewpoint as a self-adjoint operator with discrete eigenvalues given by the Casimir spectrum. This leads naturally to a study of Sobolev spaces, which are also characterised from a Fourier analytic viewpoint. We introduce Sugiura’s zeta function as a tool to study regularity of Fourier series on groups. In particular, we find conditions for absolute and uniform convergence, and for smoothness. Smoothness is characterised by means of the Sugiura space of rapidly decreasing functions defined on the space of highest weights, and we will utilise this in the next chapter to study probability measures on groups that have smooth densities.
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Notes
- 1.
Technically speaking, the heat kernel is the mapping \(\tilde{k} \in C^{\infty }((0, \infty ) \times G \times G, \mathbb {R})\) given by \(\tilde{k}(t, g, h) = k(t, g^{-1}h)\).
- 2.
This is a generalisation of the celebrated Gauss circle problem.
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© 2014 Springer International Publishing Switzerland
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Applebaum, D. (2014). Analysis on Compact Lie Groups. In: Probability on Compact Lie Groups. Probability Theory and Stochastic Modelling, vol 70. Springer, Cham. https://doi.org/10.1007/978-3-319-07842-7_3
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DOI: https://doi.org/10.1007/978-3-319-07842-7_3
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07841-0
Online ISBN: 978-3-319-07842-7
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