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Positive Definite Functions and Kernels, and Reproducing Kernel Hilbert Spaces

In: An Advanced Complex Analysis Problem Book

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  • Daniel Alpay

    (Ben-Gurion University of the Negev)

Abstract

Positive definite kernels (and the associated reproducing kernel Hilbert spaces) play an important role in various fields in mathematics, and Dieudonné’s judgment is very harsh, and somewhat unjustified. Besides representation theory and harmonic analysis, they appear in function theory (the kernel function associated with a domain), in stochastic processes (every positive definite function is a correlation function, and vice versa; see Michel Loève’s book [223]), in infinitedimensional analysis, in learning theory (see for instance [276, 232, 291, 181]), and in linear system theory (positivity translates into dissipativity of some underlying linear system), to name a few. In this chapter we present exercises which reflect some of this diversity. We refer to the books [267, 268] of Saitoh and to the papers [179] by Hille and [302, 301] by Szafraniec for background information and applications.

Suggested Citation

  • Daniel Alpay, 2015. "Positive Definite Functions and Kernels, and Reproducing Kernel Hilbert Spaces," Springer Books, in: An Advanced Complex Analysis Problem Book, chapter 0, pages 331-403, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-16059-7_7
    DOI: 10.1007/978-3-319-16059-7_7
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