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Geometric Perspectives on Reproducing Kernels

In: Operator Theory

Author

Listed:
  • Daniel Beltiţǎ

    (Institute of Mathematics “Simion Stoilow” of the Romanian Academy)

  • José E. Galé

    (Universidad de Zaragoza and IUMA)

Abstract

It is shown how reproducing kernels, in a wide class, define in a very natural manner differential geometric objects like linear connections, covariant derivatives, and curvatures. The correspondence from kernels to connections is achieved through a pullback operation from the tautological universal bundle, using a suitable classifying morphism for the given kernel. The theory is illustrated by several examples including classical kernels in function spaces, kernels occurring in dilation theory for completely positive maps, and kernels on homogeneous vector bundles.

Suggested Citation

  • Daniel Beltiţǎ & José E. Galé, 2015. "Geometric Perspectives on Reproducing Kernels," Springer Books, in: Daniel Alpay (ed.), Operator Theory, edition 127, chapter 7, pages 127-148, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-0667-1_62
    DOI: 10.1007/978-3-0348-0667-1_62
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