Korean J. Math. Vol. 28 No. 4 (2020) pp.973-984
DOI: https://doi.org/10.11568/kjm.2020.28.4.973

On 2-inner product spaces and reproducing property

Main Article Content

Saeed Hashemi Sababe

Abstract

This paper is devoted to study the reproducing property on 2-inner product Hilbert spaces. We focus on a new structure to produce reproducing kernel Hilbert and Banach spaces. According to multi variable computing, this structures play the key role in probability, mathematical finance and machine learning.


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References

[1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. Google Scholar

[2] A. Berlinet and C. Tomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics, Springer Science, Business, Media New York, 2004. Google Scholar

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

[4] S. Hashemi Sababe and A. Ebadian, Some properties of reproducing kernel Banach and Hilbert spaces, SCMA, 12 (1) (2018), 167–177. Google Scholar

[5] S. Hashemi Sababe, A. Ebadian and Sh. Najafzadeh, On reproducing property and 2-cocycles, Tkjm, 49 (2018), 143–153. Google Scholar

[6] P.E.T. Jorgensen and F. Tian, On reproducing kernels, and analysis of measures, Markov Process. Related Fields, 25 (3) (2019), 445–482. Google Scholar

[7] P.E.T. Jorgensen and F. Tian, Realizations and factorizations of positive definite kernels, J. Theoret. Probab. 32 (4) (2019), 1925–1942 Google Scholar

[8] P.E.T. Jorgensen and F. Tian, Positive matrices in the Hardy space with pre-scribed boundary representations via the Kaczmarz algorithm, J. Anal. Math. 138 (1) (2019), 209–234. Google Scholar

[9] P.E.T. Jorgensen and F. Tian, Decomposition of Gaussian processes, and factorization of positive definite kernels, Opuscula Math. 39 (4) (2019), 497-–541. Google Scholar

[10] U. Konca, M. Idris and H. Gunawan, A new 2-inner product on the space of p-summable sequences, Joems, 24 (2016), 244–249. Google Scholar

[11] Z. Lewandowska, Bounded 2-linear operators on 2-normed sets, Glasnik mathematicki, 39 (2004), 303-–314. Google Scholar

[12] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc, 100 (1961), 29-–43. Google Scholar

[13] J. Mercer, Functions of positive and negative type and their connection with the theorey of integral equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 209 (1909), 415-–446. Google Scholar

[14] V.N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis (2007), Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India. Google Scholar

[15] 2. V.N. Mishra, L.N. Mishra, Trigonometric Approximation of Signals (Functions) in Lp norm, Int. J. Contemp. Math. Sci., 7 (19) (2012), 909-918. Google Scholar

[16] L.N. Mishra, On existence and behavior of solutions to some nonlinear integral equations with applications, Ph.D. Thesis (2017), National Institute of Technology, Silchar 788 010, Assam, India. Google Scholar

[17] M. Gunaseelan, L.N. Mishra, V.N. Mishra, Generalized Coupled Fixed Point Results on Complex Partial Metric Space Using Contractive Condition, Nonlinear Anal-Model, 3 (1) (2021), 97–108. Google Scholar

[18] Deepmala, L.N. Mishra, Differential operators over modules and rings as a path to the generalized differential geometry, Facta Universitatis, 30 (5) (2015), 753– 764. Google Scholar