×

Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem. (English) Zbl 1116.46019

The authors study reproducing kernel Hilbert spaces such that their elements are \(p\)-integrable functions. It is shown that this is equivalent to the fact that the integral operator whose kernel is the reproducing kernel is a bounded operator from \(L^{p/(p-1)}\) to \(L^p\).

MSC:

46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B34 Kernel operators
47G10 Integral operators
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/978-1-4612-1258-4
[2] DOI: 10.1090/S0002-9947-1950-0051437-7
[3] DOI: 10.1016/S0723-0869(03)80014-2 · Zbl 1037.22009
[4] DOI: 10.1007/978-1-4419-9096-9
[5] Borel A., Représentations de Groupes Localement Compacts (1972) · Zbl 0242.22007
[6] Bourbaki N., Elements of Mathematics, in: Integration. I Chapters 1–6 (2004) · Zbl 1095.28001
[7] Burbea J., Banach and Hilbert Spaces of Vector-Valued Functions (1984) · Zbl 0555.46012
[8] Conway J. B., A Course in Functional Analysis (1990) · Zbl 0706.46003
[9] DOI: 10.1090/S0273-0979-01-00923-5 · Zbl 0983.68162
[10] DOI: 10.1137/1.9781611970104 · Zbl 0776.42018
[11] Führ H., Abstract Harmonic Analysis of Continuous Wavelet Transforms (2005) · Zbl 1060.43002
[12] Godement R., Trans. Amer. Math. Soc. 63 pp 1–
[13] DOI: 10.1007/978-3-642-67016-9
[14] DOI: 10.1216/RMJ-1972-2-3-321 · Zbl 0266.30009
[15] Hochstadt H., Wiley Classics Library, in: Integral Equations (1989)
[16] A. N. Kolmogorov, Selected Works. Probability Theory and Mathematical Statistics II (Kluwer, 1992) pp. 228–271.
[17] Kreĭn M. G., Ukrain. Mat. Žurnal 1 pp 64–
[18] Kreĭn M. G., Ukrain. Nat. Žurnal 2 pp 10–
[19] Michelli C. A., J. Mach. Learn. Res. 6 pp 615–
[20] Novitskii I. M., Far Eastern Mathematical Reports 7 pp 123–
[21] Pedrick G., Theory of Reproducing Kernels of Hilbert Spaces of Vector Valued Functions (1957)
[22] DOI: 10.1090/S0002-9947-1938-1501970-8 · JFM 64.0371.02
[23] Saitoh S., Integral Transforms, Reproducing Kernels and Their Applications (1997) · Zbl 0891.44001
[24] Saitoh S., Theory of Reproducing Kernels and its Applications (1988) · Zbl 0652.30003
[25] DOI: 10.1090/S0002-9947-1938-1501980-0 · JFM 64.0617.02
[26] DOI: 10.1007/BF02786620 · Zbl 0124.06504
[27] DOI: 10.1016/j.jco.2004.09.002 · Zbl 1094.46021
[28] Young R. M., An Introduction to Nonharmonic Fourier Series (2001) · Zbl 0981.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.