Document Type: Research Paper

Authors

1 Department of Mathematics, Payame Noor University (PNU), P.O. Box, 19395-3697, Tehran, Iran.

2 Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.

Abstract

This paper is devoted to the study of reproducing kernel Hilbert spaces. We focus on multipliers of reproducing kernel Banach and Hilbert spaces. In particular, we try to extend this concept and prove some related theorems. Moreover, we focus on reproducing kernels in vector-valued reproducing kernel Hilbert spaces. In particular, we extend reproducing kernels to relative reproducing kernels and prove some theorems in this subject.

Keywords

Main Subjects

[1] D. Alpay, P. Jorgensen, and D. Volok, Rlative reproducing kernel Hilbert spaces, Proc. Amer. Math. Soc., 142 (2014), pp. 3889-3895.

[2] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), pp. 337-404.

[3] A. Berlinet and C. Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics, Kluwer Academic Publishers, Boston, 2004.

[4] S.S. Dragomir, Semi- inner products and application, Nova Science Publishers, 2004.

[5] G.E. Fasshauer and Q. Ye, Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators, Numer. Math., 119 (2011), pp. 585-611.

[6] K. Fukumizu, G.R. Lanckriet, and B.K. Sriperumbudur, Learning in Hilbert vs. Banach Spaces: A measure embedding viewpoint, Advances in Neural Information Processing Systems, 24 (2011).

[7] J.R. Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129 (1967), pp. 436-446.

[8] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc., 100 (1961), pp. 29-43.

[9] B.D. Malviya, A note on semi-inner product algebras, Math.Nachr., 47 (1970), pp. 127-129.

[10] P.V. Pethe and N.K. Thakare, Applications of Riesz's representation theorem in semi-inner product spaces, Indian J. Pure Appl. Math., 7 (1976), pp. 1024-1031.

[11] B. Scholkopf and A.J. Smola, Learning with kernels, MIT Press, Cambridge, Massachusetts, 2002.

[12] A. Smola and S.V.N. Vishwanathan, Introduction to machine learning, Cambridge University Press, 2008.

[13] S. Tsui, Hilbert $C^*$-modules: a useful tool, Taiwanese Journal of Mathematics, 1 (1997), pp. 111-126.

[14] Y. Xu and Q. Ye, Constructions of reproducing kernel Banach spaces via generalized Mercer kernels, arXiv:1412.8663v1, 30 (2014).

[15] H. Zhang, Y. Xu, and J. Zhang, Reproducing kernel Banach spaces for machine learning, Journal of Machine Learning Research, 10 (2009), pp. 2741-2775.

[16] D.X. Zhou, Capacity of reproducing kernel spaces in learning theory, IEEE Trans. Inform. Theory, 49 (2003), pp. 1743-1752.