Document Type : Research Paper

Author

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

Abstract

In this paper, a special class of relative reproducing kernel Banach spaces a semi-inner product is studied. We extend the concept of relative reproducing kernel Hilbert spaces to Banach spaces. We present these relative reproducing kernel Banach spaces  in terms of the feature maps and establish the separability of the domains when they are  separable. In addition, we prove some theorems concerning feature maps and reproducing kernel Banach spaces. And finally, the relative kernels are compared with the  semi-inner ones.

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Main Subjects

[1] D. Alpay, P. Jorgensen and D. Volok, Relative reproducing kernel Hilbert spaces, Proc. Amer. Math. Soc., 142(11), (2014), pp. 3889-3895.
[2] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math., Soc., 68.3 (1950), pp. 337-404.
[3] P.L. Duren, Theory of $ H^p $ spaces, New York and London, Academic Press, (1970).
[4] G.E. Fasshauera, F. Hickernella and Q. Ye, Solving support vector machines in reproducing kernel Banach spaces with positive definite functions, Appl. Comput. Harmon. Anal., 38 (2015), pp. 115-139.
[5] P. Georgiev, L. Sanchez-Gozalez and P. Pardalos, Construction of pair of reproducing kernel Banach spaces, Springer Optim. Appl. 87 (2014), pp. 39-57.
[6] E.F. Gregory, F.J. Hickernell, Q. Ye, Solving support vector machines in reproducing kernel Banach spaces with positive definite functions, Appl. Comp. Harm., Anal, 38 (2015), pp. 115-139.
[7] J.R. Giles, Classes of sem–inner-product, Trans. Amer. Math. Soc., 129 (1967), pp. 436-446.
[8] M. Hein, O. Bousquet and B. Scholkopf, Maximal margin classification for metric spaces, J. Comput. System Sci., 71 (2005), pp. 333-359.
[9] P.E.T. Jorgensen and M.S Song, Reproducing kernel Hilbert space vs. frame estimates, Math., 3(3) (2016), pp. 615-625.
[10] J. Mashreghi, Representation theorems in Hardy spaces, Cambridge University Press, Cambridge, 2009.
[11] H. Owhadi and C. Scovel, Separability of reproducing kernel Banach spaces, Proc. Amer. Math. Soc., 145 (2017), pp. 2131-2138.
[12] V. Paulsen, An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge University Press., vol 152, (2016).
[13] S. Saitoh, Integral transforms, reproducing kernels and their applications, Pitman Research Notes in Mathematics Series, 369 CRC Press, (1997).
[14] G. Song, H. Zhang and F. Hickernell, Reproducing kernel Banach spaces with the $l ^{1}$ norm, Appl. Comp. Har. Anal., 34 (2013), pp. 96-116.
[15] S. Zaremba, L'équation biharmonique et une classe remarquable de fonctions fondamentales harmoniques, Bulletin IntAcad des Sciences de Cracovie,3 (1907), pp. 147-196.
[16] H. Zhang, Y. Xu and J. Zhang, Reproducing kernel Banach spaces for machine learning, Mach. Learn. Res., 10 (2009), pp. 2741-2775.
[17] C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008), pp. 266-272.