International Journal of Nonlinear Analysis and Applications
Equivalence relations on approximation theory
Document Type : Research Paper
Authors
Faculty of Mathematics, Yazd University, Yazd, Iran
Abstract
In this paper, we define relations between the best approximation and the worst approximation. We show that these relations are equivalence relations if the sets are Chebyshev or uniquely remotal. We obtain cosets sets of best approximation and cosets sets of worst approximation. We obtain some results on these sets, for example, compactness and weakly compactness. Finally, we consider the semi-inner products (Lumer-Giles) and semi-inner(usual).
Keywords
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[2] C. Franchetti and M. Furi, Some characteristic properties of real Hilbert spaces, Rev. Roumaine Math. Pures Appl. 17 (1972), 1045–1048.
[3] R.C. Buck, Applications of duality in approximation theory, In Approximation of Functions, (Proc. Symp. Gen. Motors Res. Lab. 1964, 1965, pp. 27–42.
[4] S. Elumalai and R. Vijayaragavan, Farthest points in normed linear spaces, Gen. Math. 14 (3) (2006), 9–22.
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[7] R. Khalil and Sh. Al-Sharif, Remotal sets in vector valued function spaces, Sci. Math. Japon. 3 (2006), 433–442.
[8] H.V. Machado, A characterization of convex subsets of normed spaces, Kodai Math. Sem. Rep. 25 (1973), 307–320.
[9] M. Martin and T.S.S.R.K. Rao, On remotality for convex sets in Banach spaces, J. Approx. Theory 162 (2010), 392–396.
[10] H. Mazaheri, T. D. Narang and H.R. Khademzadeh, Nearest and Farthest points in normed spaces, In Press Yazd University, 2015.
[11] H. Mazaheri, A characterization of weakly-Chebyshev subspaces of Banach spaces, J. Nat. Geom. 22 (2002), no. 1-2, 39–48.
[12] H. Mohebi, On quasi-Chebyshev subspaces of Banach spaces, J. Approx. Theory 107 (2000), no. 1, 87–95.
[13] A. Niknam, On uniquely remotal sets, Indian J. Pure Appl. Math. 15 (1984), 1079–1083.
[14] A. Niknam, Continuity of the farthest point map, Indian J. Pure Appl. Math. 18 (1987), 630–632.
[15] T.D. Narang, On the farthest points in convex metric spaces and linear metric spaces, Pub. Inst. Math. 95 (2014), no. 109, 229–238.
[16] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, New York-Berlin, 1970.
[6] O. Hadzic, A theorem on best approximations and applications, Univ. Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 22 (1992), 47–55.
[7] R. Khalil and Sh. Al-Sharif, Remotal sets in vector valued function spaces, Sci. Math. Japon. 3 (2006), 433–442.
[8] H.V. Machado, A characterization of convex subsets of normed spaces, Kodai Math. Sem. Rep. 25 (1973), 307–320.
[9] M. Martin and T.S.S.R.K. Rao, On remotality for convex sets in Banach spaces, J. Approx. Theory 162 (2010), 392–396.
[10] H. Mazaheri, T. D. Narang and H.R. Khademzadeh, Nearest and Farthest points in normed spaces, In Press Yazd University, 2015.
[11] H. Mazaheri, A characterization of weakly-Chebyshev subspaces of Banach spaces, J. Nat. Geom. 22 (2002), no. 1-2, 39–48.
[12] H. Mohebi, On quasi-Chebyshev subspaces of Banach spaces, J. Approx. Theory 107 (2000), no. 1, 87–95.
[13] A. Niknam, On uniquely remotal sets, Indian J. Pure Appl. Math. 15 (1984), 1079–1083.
[14] A. Niknam, Continuity of the farthest point map, Indian J. Pure Appl. Math. 18 (1987), 630–632.
[15] T.D. Narang, On the farthest points in convex metric spaces and linear metric spaces, Pub. Inst. Math. 95 (2014), no. 109, 229–238.
[16] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, New York-Berlin, 1970.
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Volume 14, Issue 4
April 2023Pages 313-322
- Receive Date: 08 August 2021
- Accept Date: 16 December 2021