On the Inner-Product Spaces of Complex Interval Sequences

Year 2022, Volume: 5 Issue: 4, 180 - 188, 30.12.2022

Halise Levent , Yılmaz Yılmaz , Hacer Bozkurt

https://doi.org/10.33434/cams.1200557

Abstract

In recent years, there has been increasing interest in interval analysis. Thanks to interval numbers, many real world problems have been modeled and analyzed. Especially, complex intervals have an important place for interval-valued data and interval-based signal processing. In this paper, firstly we introduce the notion of a complex interval sequence and we present the complex interval sequence spaces $\mathbb{I}(w)$ and $\mathbb{I}(l_{p})$, $1\leq p<\infty$. Secondly, we show that these sequence spaces have an algebraic structure called quasilinear space. Further, we construct an inner-product on $\mathbb{I}(l_{2})$ and we show that $\mathbb{I}(l_{2})$ is an inner-product quasilinear space.

Keywords

Complex interval, Complex interval sequence, Inner-product quasilinear space, Consolidate space

References

• [1] R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to Interval Analysis, Philadelphia, SIAM, 2009.
• [2] G. Alefeld, J. Herzberger, Einf¨uhrung in die Intervallrechnung, Bibliographhisches Institut, Mannheim, 1974.
• [3] G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic Press, New York, 1983.
• [4] U. Kulisch, Grundzu¨ge der Intervallrechnung, in: Jahrburch U¨ berblicke Mathematik, Bibliographhisches Institut, Mannheim, 1969.
• [5] H. Bozkurt, Y. Yılmaz, New inner product quasilinear spaces on interval numbers, J. Function Spaces, 2 (2016), Article ID 2619271, 9 pages, doi:10.1155/2016/2619271.
• [6] M. S¸eng¨on¨ul, A. Eryılmaz, On the sequence spaces of interval numbers, Tai J. Math., 8 (2010), 503-510.
• [7] H. Levent, Y. Yılmaz, An application: Representations of some systems on non-deterministic EEG signals, J. Biostat Biometric. App., 2 (2017), 101.
• [8] S. M. Aseev, Quasilinear operators and their application in the theory of multivalued mappings, Proceedings of the Steklov Institute of Mathematics, 2 (1969), 23-52.
• [9] V. Lakshmikantham, T. Gana Bhaskar, J. Vasandura Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006.
• [10] J. P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.
• [11] H. Levent, Y. Yılmaz, Translation, modulation and dilation systems in set-valued signal processing, Carpathian Math.Publ., 10 (2018), 10-31.
• [12] Y. Yılmaz, H. Levent, Inner-product quasilinear spaces with applications in signal processing, Advanced Studies: Euro-Tbilisi Mathematical Journal, 14(4) (2021), 125–146.
• [13] H. Bozkurt, Y. Yılmaz, Some new results on inner product quasilinear spaces, Cogents Math., 3 (2016), Article ID 1194801, 11 pages, doi:10.1080/23311835.2016.1194801.
• [14] Y. Yılmaz, H. Bozkurt, S. C¸ akan, On orthonormal Sets in Inner Product Quasilinear Spaces, Creat. Math. Inform., 25 (2016), 229-239.