Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem

Abstract

[EN] Let (X, Sigma, mu) be a finite measure space and consider a Banach function space Y(mu). Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Kothe Bochner (vectorvalued) function spaces. We analyze operator-valued kernels Gamma that define integration maps L-Gamma between Kothe-Bochner spaces of Hilbert-valued functions Y(mu; kappa). We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces L-P(gd mu; kappa) and L-P (hd mu; kappa) - where 1/p + 1/p' = 1 - under the assumption of p-concavity of Y(mu). Equivalently, a new kernel obtained by multiplying Gamma by scalar functions can be given in such a way that the kernel operator is defined from L-P (mu; kappa) to L-P (mu; kappa) in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels.The second author acknowledges the support of the Ministerio de Economia y Competitividad (Spain), under project MTM2014-53009-P (Spain). The third author acknowledges the support of the Ministerio de Ciencia, Innovacion y Universidades (Spain), Agencia Estatal de Investigacion, and FEDER under project MTM2016-77054-C2-1-P (Spain).Agud Albesa, L.; Calabuig, JM.; Sánchez Pérez, EA. (2020). Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem. Analysis and Applications. 18(3):359-383. https://doi.org/10.1142/S0219530519500179S359383183Agud, L., Calabuig, J. M., & Sánchez Pérez, E. A. (2011). The weak topology on q-convex Banach function spaces. Mathematische Nachrichten, 285(2-3), 136-149. doi:10.1002/mana.201000030CARMELI, C., DE VITO, E., & TOIGO, A. (2006). VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES OF INTEGRABLE FUNCTIONS AND MERCER THEOREM. Analysis and Applications, 04(04), 377-408. doi:10.1142/s0219530506000838CARMELI, C., DE VITO, E., TOIGO, A., & UMANITÀ, V. (2010). VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES AND UNIVERSALITY. Analysis and Applications, 08(01), 19-61. doi:10.1142/s0219530510001503Cerdà, J., Hudzik, H., & Mastyło, M. (1996). Geometric properties of Köthe–Bochner spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 120(3), 521-533. doi:10.1017/s0305004100075058Chavan, S., Podder, S., & Trivedi, S. (2018). Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces. Journal of Mathematical Analysis and Applications, 466(2), 1337-1358. doi:10.1016/j.jmaa.2018.06.062Christmann, A., Dumpert, F., & Xiang, D.-H. (2016). On extension theorems and their connection to universal consistency in machine learning. Analysis and Applications, 14(06), 795-808. doi:10.1142/s0219530516400029Defant, A. (2001). Positivity, 5(2), 153-175. doi:10.1023/a:1011466509838Defant, A., & Sánchez Pérez, E. A. (2004). Maurey–Rosenthal factorization of positive operators and convexity. Journal of Mathematical Analysis and Applications, 297(2), 771-790. doi:10.1016/j.jmaa.2004.04.047De Vito, E., Umanità, V., & Villa, S. (2013). An extension of Mercer theorem to matrix-valued measurable kernels. Applied and Computational Harmonic Analysis, 34(3), 339-351. doi:10.1016/j.acha.2012.06.001Eigel, M., & Sturm, K. (2017). Reproducing kernel Hilbert spaces and variable metric algorithms in PDE-constrained shape optimization. Optimization Methods and Software, 33(2), 268-296. doi:10.1080/10556788.2017.1314471Fasshauer, G. E., Hickernell, F. J., & Ye, Q. (2015). Solving support vector machines in reproducing kernel Banach spaces with positive definite functions. Applied and Computational Harmonic Analysis, 38(1), 115-139. doi:10.1016/j.acha.2014.03.007Galdames Bravo, O. (2014). Generalized Kӧthe $p$-dual spaces. Bulletin of the Belgian Mathematical Society - Simon Stevin, 21(2). doi:10.36045/bbms/1400592625Lin, P.-K. (2004). Köthe-Bochner Function Spaces. doi:10.1007/978-0-8176-8188-3Lindenstrauss, J., & Tzafriri, L. (1979). Classical Banach Spaces II. doi:10.1007/978-3-662-35347-9Meyer-Nieberg, P. (1991). Banach Lattices. Universitext. doi:10.1007/978-3-642-76724-1Okada, S., Ricker, W. J., & Sánchez Pérez, E. A. (2008). Optimal Domain and Integral Extension of Operators. doi:10.1007/978-3-7643-8648-1Zhang, H., & Zhang, J. (2013). Vector-valued reproducing kernel Banach spaces with applications to multi-task learning. Journal of Complexity, 29(2), 195-215. doi:10.1016/j.jco.2012.09.00