On functional reproducing kernels-Reference-Cited by-同舟云学术

On functional reproducing kernels

Published:2023-01-01 Issue:1 Volume:21 Page:
ISSN:2391-5455
Container-title:Open Mathematics
language:en
Short-container-title:

Author:

Zhou Weiqi¹

Affiliation:

1. School of Mathematics and Statistics, Xuzhou University of Technology, Lishui Road 2, Yunlong District , Xuzhou , Jiangsu Province, 221111 , China

Abstract

Abstract We show that even if a Hilbert space does not admit a reproducing kernel, there could still be a kernel function that realizes the Riesz representation map. Constructions in spaces that are the Fourier transform of weighted

L 2

{L}^{2}

spaces are given. With a mild assumption on the weight function, we are able to reproduce Riesz representatives of all functionals through a limit procedure from computable integrals over compact sets, despite that the kernel is not necessarily in the underlying Hilbert space. Distributional kernels are also discussed.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/math-2023-0102/pdf

Reference12 articles.

1. S. Bergman, Über die Entwicklung der harmonischen Funktionen der Ebene und des Raumes nach Orthogonalfunktionen, Math. Ann. 86 (1922), 238–271, DOI: https://doi.org/10.1007/BF01457987.

2. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404, DOI: http://doi.org/10.1090/S0002-9947-1950-0051437-7.

3. R. Schaback, Native Hilbert spaces for radial basis functions, in: M. Müller, M. Buhmann, D. Mache, M. Felten (Eds.), New Developments in Approximation Theory, Vol. 132, Birkhäuser, Basel, 1999, pp. 255–282.

4. M. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge Univ. Press, Cambridge, 2003.

5. H. Wendland, Scattered Data Approximation, Cambridge Univ. Press, Cambridge, 2004.