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Positive kernels in Lie group representations and Fourier interpolation
Abstract
For a given function f on a discrete subspace Γ of a noncompact Riemannian symmetric space G/K, we construct analytic, noncompactly supported functions W on G/K with W|Γ=f that, in addition, satisfy an invariant differential equation on G/K. The key step is to regard W as an orthogonal projection onto a space of coherent states in a representation of the Lie group G in a Hilbert space that admits a strictly positive definite reproducing kernel. The techniques are implemented for the heat kernel on G/K, and W is expressed in terms of certain closed formulas for linearly independent coherent states.
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BibTeX
@misc{uttenthal2026positive,
title = {{Positive kernels in Lie group representations and Fourier interpolation}},
author = {Peter Vang Uttenthal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03167},
eprint = {2609.03167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03167},
}
RIS
TY - UNPB
TI - Positive kernels in Lie group representations and Fourier interpolation
AU - Peter Vang Uttenthal
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.03167
UR - https://arxiv.org/abs/2609.03167
AB - For a given function f on a discrete subspace Γ of a noncompact Riemannian symmetric space G/K, we construct analytic, noncompactly supported functions W on G/K with W|_Γ=f that, in addition, satisfy an invariant differential equation on G/K. The key step is to regard W as an orthogonal projection onto a space of coherent states in a representation of the Lie group G in a Hilbert space that admits a strictly positive definite reproducing kernel. The techniques are implemented for the heat kernel on G/K, and W is expressed in terms of certain closed formulas for linearly independent coherent states.
ER -