×

Note on fuzzy monotonic interpolating splines of odd degree. (English) Zbl 1368.65022

Summary: In this paper we propose a Hermite type monotonic fuzzy quintic spline for interpolating fuzzy data. Its monotonicity is proved and the corresponding interpolation error estimate is provided. This fuzzy quintic spline is generalized to odd degree monotonic fuzzy splines. The uniform approximation property of the corresponding fuzzy linear and positive operator is obtained. The monotonicity is tested on a numerical experiment.

MSC:

65D07 Numerical computation using splines
65D05 Numerical interpolation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abbasbandy, S.: Interpolation of fuzzy data by complete splines. Korean J. Comput. appl. Math. 8, 587-594 (2001) · Zbl 1009.65006
[2] Abbasbandy, S.; Babolian, E.: Interpolation of fuzzy data by natural splines. Korean J. Comput. appl. Math. 5, 457-463 (1998) · Zbl 0914.65001
[3] Abbasbandy, S.; Amirfakhrian, M.: Numerical approximation of fuzzy functions by fuzzy polynomials. Appl. math. Comput. 174, 1001-1006 (2006) · Zbl 1094.65015 · doi:10.1016/j.amc.2005.05.028
[4] Abbasbandy, S.; Ezzati, R.; Behforooz, H.: Interpolation of fuzzy data by using fuzzy splines. Int. J. Uncertain. fuzziness knowl.-based syst. 16, No. 1, 107-115 (2008) · Zbl 1154.65011 · doi:10.1142/S0218488508005078
[5] Abel, U.; Ivan, M.: New representation of the remainder in the Bernstein approximation. J. math. Anal. appl. 381, 952-956 (2011) · Zbl 1220.41002 · doi:10.1016/j.jmaa.2011.02.005
[6] Anastassiou, G. A.: On basic fuzzy Korovkin theory, studia univ. “Babes-bolyai”. Mathematica 50, No. 4, 3-10 (2005) · Zbl 1138.26315
[7] Anastassiou, G. A.: Basic fuzzy Korovkin theory. Studies in fuzziness and soft computing 251, 99-104 (2010)
[8] Anile, A. M.; Spinella, S.: Modeling uncertain sparse data with fuzzy B-splines. Reliab. comput. 10, 335-355 (2004) · Zbl 1048.65015 · doi:10.1023/B:REOM.0000032117.04378.9a
[9] Anile, A. M.; Falcidieno, B.; Gallo, G.; Spagnuolo, M.; Spinella, S.: Modeling uncertain data with fuzzy B-splines. Fuzzy sets syst. 113, 397-410 (2000) · Zbl 1147.65301 · doi:10.1016/S0165-0114(98)00146-8
[10] Blaga, P.; Bede, B.: Approximation by fuzzy B-spline series. J. appl. Math. comput. 20, 157-169 (2006) · Zbl 1124.26019 · doi:10.1007/BF02831930
[11] Bede, B.: Mathematics of fuzzy sets and fuzzy logic. (2013) · Zbl 1271.03001 · doi:10.1007/978-3-642-35221-8
[12] Behforooz, H.; Ezzati, R.; Abbasbandy, S.: Interpolation of fuzzy data by using \(E(3)\) cubic splines. Int. J. Pure appl. Math. sci. 60, No. 4, 383-392 (2010) · Zbl 1197.65020
[13] Behforooz, H.; Ezzati, R.; Abbasbandy, S.: Interpolation of fuzzy data by using quartic piecewise polynomials induced from \(E(3)\) cubic splines. Math. sci. 6, 40 (2012) · Zbl 1271.65026 · doi:10.1186/2251-7456-6-40
[14] Bica, A. M.; Popescu, C.: Fuzzy spline interpolation with optimal property in parametric form. Inf. sci. 236, 138-155 (2013) · Zbl 1284.65023 · doi:10.1016/j.ins.2013.02.047
[15] Bica, A. M.: Algebraic structures for fuzzy numbers from categorial point of view. Soft comput. 11, 1099-1105 (2007) · Zbl 1125.03039 · doi:10.1007/s00500-007-0167-x
[16] Bica, A. M.: The middle-parametric representation of fuzzy numbers and applications to fuzzy interpolation. Int. J. Approx. reason. 68, 27-44 (2016) · Zbl 1346.03048 · doi:10.1016/j.ijar.2015.10.001
[17] Chen, Q.; Kawase, S.: On operations and order relations between fuzzy values. Fuzzy sets syst. 108, 313-324 (1999) · Zbl 0988.03081 · doi:10.1016/S0165-0114(97)00327-8
[18] Ezzati, R.; Rohani-Rahab, N.; Mokhtarnejad, F.; Aghamohammadi, M.; Hassasi, N.: Fuzzy splines and their applications to interpolate fuzzy data. Int. J. Fuzzy syst. 15, No. 2, 127-132 (2013)
[19] Gal, S. G.: Approximation theory in fuzzy setting. Handbook of analytic-computational methods in applied mathematics, 617-666 (2000) · Zbl 0968.41018
[20] Gal, S. G.: A fuzzy variant of the Weierstrass’s approximation theorem. J. fuzzy math. 1, 865-872 (1993) · Zbl 0815.26015
[21] Gal, S. G.: Interpolation of fuzzy mappings. Mathematica (Cluj) 38, No. 61, 61-65 (1996) · Zbl 0880.41032
[22] Goghary, H. S.; Abbasbandy, S.: Interpolation of fuzzy data by Hermite polynomial. Int. J. Comput. math. 82, No. 12, 1541-1545 (2005) · Zbl 1083.65007 · doi:10.1080/00207160410001714592
[23] Jenei, S.: Interpolation and extrapolation of fuzzy quantities revisited - an axiomatic approach. Soft comput. 5, 179-193 (2001) · Zbl 0993.68123 · doi:10.1007/s005000100080
[24] Kaleva, O.: Interpolation of fuzzy data. Fuzzy sets syst. 61, 63-70 (1994) · Zbl 0827.65007 · doi:10.1016/0165-0114(94)90285-2
[25] Lowen, R.: A fuzzy Lagrange interpolation theorem. Fuzzy sets syst. 34, 33-38 (1990) · Zbl 0685.41005 · doi:10.1016/0165-0114(90)90124-O
[26] Nobuhara, H.; Bede, B.; Hirota, K.: On various eigen fuzzy sets and their application to image reconstruction. Inf. sci. 176, No. 20, 2988-3010 (2006) · Zbl 1102.68697 · doi:10.1016/j.ins.2005.11.008
[27] Perfilieva, I.; Dubois, D.; Prade, H.; Esteva, F.; Godo, L.; Hodakova, P.: Interpolation of fuzzy data: analytical approach and overview. Fuzzy sets syst. 192, 134-158 (2012) · Zbl 1238.68161 · doi:10.1016/j.fss.2010.08.005
[28] Valenzuela, O.; Pasadas, M.: Fuzzy data approximation using smoothing cubic splines: similarity and error analysis. Appl. math. Model. 35, 2122-2144 (2011) · Zbl 1217.41013 · doi:10.1016/j.apm.2010.11.046
[29] Congxin, Wu; Cong, Wu: The supremum and infimum of the set of fuzzy numbers and its applications. J. math. Anal. appl. 210, 499-511 (1997) · Zbl 0883.04008 · doi:10.1006/jmaa.1997.5406
[30] Zeinali, M.; Shahmorad, S.; Mirnia, K.: Hermite and piecewise cubic Hermite interpolation of fuzzy data. J. intell. Fuzzy syst. 26, No. 6, 2889-2898 (2014) · Zbl 1305.65091 · doi:10.3233/IFS-130955
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.