Fuzzy Mathematics: Approximation Theory

Studies in Fuzziness and Soft Computing

Volumn 251, Issue , 2010, Pages 1-451

  Anastassiou, George A ^a  

^a University of Memphis   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77950158382     PISSN: 14349922     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-11220-1_1     Document Type: Article

References (103)

1. C. D. Aliprantis, O. Burkinshaw, Positive Operators, Academic Press, New York and London, 1985.
2. C. D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, Third edition, Academic Press, Inc., San Diego, CA, 1998.
3. F. Altomare , M. Campiti, Korovkin type approximation theory and its applications, Walter de Gruyter Publ. Berlin, 1994.
4. G.A. Anastassiou, Moments in Probability and Approximation The- ory, Pitman/Longman, #287, UK, 1993.
5. G.A. Anastassiou, Ostrowski type inequalities, Proc. Amer. Math. Soc. 123 (1995), No.12, 3775-3781.
6. G.A. Anastassiou, G.A., Rate of convergence of some neural net- work operators to the unit-univariate case, J. Math. Anal. Appl. 212, (1997), 237-262.
7. G.A. Anastassiou, Shape and probability preserving univariate wavelet type operators, Commun. Appl. and Anal., 1, No.3 (1997), 303-314.
8. G.A. Anastassiou, Higher order univariate wavelet type approxi- mation, Approximation Theory, in Memory of A.K. Varma, Marcel Dekker, New York, 1998, pp. 43-60.
9. G.A. Anastassiou, Quantitative Approximations, Chapman & Hall/CRC, 2001, Boca Raton, New York.
10. G.A. Anastassiou, On H-fuzzy differentiation, Mathematica Balkan- ica, New Series, Vol.16, Fasc. 1-4 (2002), 153-193.
11. G.A. Anastassiou, Rate of convergence of fuzzy neural network op- erators, univariate case, Journal of Fuzzy Mathematics, 10, No.3 (2002), 755-780.
12. G.A. Anastassiou, High order approximation by univariate shift- invariant integral operators, in Nonlinear Analysis and Applications, edited by Ravi Agarwal and Donal O.Regan, Vol.I, 141-164, Kluwer, 2003
13. G.A. Anastassiou, Fuzzy Ostrowski type inequalities, Comput. Appl. Math. 22 (2003), No.2, 279-292.
14. G.A. Anastassiou, Fuzzy wavelet type operators, Nonlinear Func- tional Analysis and Applications, 9, No.2 (2004), 251-269.
15. G.A. Anastassiou, Fuzzy approximation by fuzzy Convolution type operators, Computers & Mathematics with Applications, 48 (2004), 1369-1386.
16. G.A. Anastassiou, High Order Fuzzy Approximation by Fuzzy Wavelet type and Neural Network Operators, Computers and Math. with Appl., special issue edited by G. Anastassiou, Vol.48, 2004, 1387-1401.
17. G.A. Anastassiou, Univariate fuzzy-random neural network approx- imation operators, Computers and Mathematics with Applications, Special issue/Proceedings edited by G.Anastassiou of special ses- sion "Comutational Methods in Analysis", AMS meeting in Or- lando,Florida, November 2002, Vol.48 (2004), 1263-1283
18. G.A. Anastassiou, On basic Fuzzy Korovkin Theory, Studia Univ. Babes - Bolyai, Mathematica, Vol.L, No.4, 2005, 3-10.
19. G.A. Anastassiou, Fuzzy Taylor formulae, Cubo, Vol.7, No.3, 2005, 1-13-
20. G.A. Anastassiou, Higher order Fuzzy Korovkin Theory via Inequali- ties, Communications in Applied Analysis, 10 (2006), no.2, 359-392
21. G.A. Anastassiou, On Fuzzy Global Smoothness Preservation, J. Fuzzy Math., vol 15, no.1, 2007, 219-231
22. G.A. Anastassiou, Fuzzy Random Korovkin Theory and Inequalities, Mathematical Inequalities & Applications, Vol.10, No.1, 2007, 63-94
23. G.A. Anastassiou, Quantitative estimates for Distance between Fuzzy Wavelet type operators, Journal of Concrete and Applicable Mathe- matics, Vol 5, No.1 (2007), 25-52
24. G.A. Anastassiou, Fuzzy Korovkin Theorems and Inequalities, J. Fuzzy Mathematics, Vol.15, No.1 (2007), 169-205
25. G.A. Anastassiou, C. Cottin, and H.H. Gonska, Global smoothness preservation by multivariate approximation operators, in Israel Math- ematical Conference Proceedings (S. Baron and D. Leviatan, eds.), Weizmann Science Press, Vol.IV, 1991, pp. 31-44-
26. G.A. Anastassiou, C. Cottin, and H.H. Gonska, Global smoothness of approximating functions, Analysis, 11 (1991), 43-57
27. G.A. Anastassiou, K. Demirci, S. Karakus, A-summability and fuzzy Korovkin-type approximation, submitted 2009
28. G.A. Anastassiou, K. Demirci, S. Karakus, A-summability and fuzzy trigonometric Korovkin-type approximation, submitted 2009-
29. G.A. Anastassiou and O. Duman, Statistical fuzzy approximation by fuzzy positive linear operators, Comput. Math. Appl. 55 (2008), 573-580.
30. G.A. Anastassiou and S. G. Gal, Approximation Theory. Moduli of Continuity and Global Smoothness Preservation, Birkhäuser, Boston, Basel, Berlin, 2000.
31. G.A. Anastassiou and S. Gal, On a fuzzy trigonometric approxima- tion theorem of Weierstrass-type, Journal of Fuzzy Mathematics, 9, No.3 (2001), Los Angeles, 701-708
32. G.A. Anastassiou, S.G. Gal, On fuzzy trigonometric Korovkin theory, Nonlinear Funct. Anal. Appl. 11 (2006), 385-395
33. G.A. Anastassiou and H.H. Gonska, On stochastic global smoothness, Rev. Acad. Cienc. Zaragoza, 49 (1994), 119-136
34. G.A. Anastassiou and H. H. Gonska, On some shift-invariant inte- gral operators, multivariate case, in Proc. Int. Conf. Approx. Prob. and Related Fields, U.C.S.B., Santa Barbara, CA, May 20.22, 1993 (Anastassiou, G.A. and Rachev, S.T., eds.), Plenum Press, New York, 1994, 41-64
35. G.A. Anastassiou and H. H. Gonska, On some shift invariant integral operators, univariate case, Ann. Polon. Math., LXI, 3 (1995), 225- 243
36. G.A. Anastassiou and X.M. Yu, Monotone and probabilistic wavelet approximation, Stochastic Anal. Appl., 10(3) (1992), 251-264
37. Tom M. Apostol, Mathematical Analysis, Addison-Wesley Publ. Company, Reading, Massachusetts, 1957
38. Ö.G. Atlihan, C. Orhan, Matrix summability and positive linear op- erators, Positivity 11 (2007), 387-389
39. Ö.G. Atlihan, C. Orhan, Summation process of positive linear oper- ators, Computers & Mathematics with Applications, 56 (2008) 1188-1195
40. B. Bede and S. G. Gal, Quadrature rules for integrals of fuzzy- number-valued functions, Fuzzy Sets and Systems, 145 (2004), no.3, 359-380.
41. B. Bede and S. G. Gal, Best Approximation and Jackson-Type esti- mates by generalized fuzzy polynomials, J. of Concrete and Applica- ble Mathematics, Vol.2 (2004), No.3, 213-232
42. H.T. Bell, Order summability and almost convergence, Proc. Am. Math. Soc. 38 (1973) 548-553-
43. E.K. Blum, Numerical Analysis and Computation Theory and Prac- tice, Addison-Wesley Series in Mathematics, Addison-Wesley Pub- lishing Company, XII, 1972
44. R. Bojanic and F. Cheng. Estimates for the rate of approximation of functions of bounded variation by Hermite-Fejér polynomials, Pro- ceedings of the conference of Canadian Math. Soc. 3 (1983), 5-17-
45. R. Bojanic and M. K. Khan. Summability of Hermite-Fejér interpo- lation for functions of bounded variation, J. Nat. Sci. Math. 32 No.1 (1992), 5-10.
46. D. Butnariu, measurability concepts of fuzzy mappings, Fuzzy Sets and Systems, 31 (1989), 77-82
47. P. Cardaliaguet and G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Networks, 5 (1992), 207- 220.
48. A. Colubi, J.S. Dominguez-Menchero, M. Lopez-Diaz, and D. Ralescu, A DE[0; 1] representation of random upper semicontinuous functions, Proc. A.M.S., 130, No.11 (2002), 3237-3242
49. Wu Congxin and Liu Danhong, A fuzzy Weierstrass approximation theorem, J. Fuzzy Mathematics 7, No.1 (1999), 101-104
50. Congxin Wu and Ming Ma. On embedding problem of fuzzy number space: Part 1, Fuzzy Sets and Systems, 44 (1991), 33- 38
51. Congxin Wu and Ming Ma. On embedding problem of fuzzy number space: Part 2, Fuzzy Sets and Systems, 45, 1992, 189-202
52. Wu Congxin and Gong Zengtai, On Henstock integral of fuzzy- number-valued functions, I, Fuzzy Sets and Systems, 115, No.3 (2000), 377-391
53. Congxin Wu and Zengtai Go n g. On Henstock integral of fuzzy num- ber valued functions (I), Fuzzy Sets and Systems, 120, No.3, 2001, 523-532-
54. R. Cristescu, Ordered structures in ordered vector spaces, Ed. St. Enciclop., Bucharest, 1983 (in Romanian).
55. R. A. DeVore, the Approximation of Continuous Functions by Pos- itive Linear Operators, Lecture Notes in Mathematics, Springer- Verlag, 293 (1972), Berlin.
56. Ronald A. DeVore and George G. Lorentz, Constructive Approxima- tion, Vol.303, Springer-Verlag, Berlin, New York, 1993
57. P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World Scien- ti.c, New Jersey, 1994
58. P. Diamond and P. Kloeden, Metric topology of fuzzy numbers and fuzzy analysis, Chapter 11 in Fundamentals of Fuzzy Sets, Handbook of Fuzzy Sets Series, 7, Kluwer, Boston, 2000.
59. P. Diamond, P. Kloeden, A. Vladimirov, Spikes, broken planes and the approximation of convex sets, Fuzzy Sets and Systems 99(1999), 225-232
60. P. Diamond and A. Ramer, Approximation of knowledge and fuzzy sets, Cybernetics and Systems: An International Journal, 24 (1993), 407-417
61. M. Lopez-Diaz and M. Angeles Gil, Approximating integrably bounded fuzzy random variables in terms of the .Generalized.Haus- dorff metric, Information Sciences, 104 (1998), 279-291
62. R. Dudley, Real Analysis and Probability, Wadsworth & Brooks/Cole Mathematics Series, Paci.c Grove, CA, 1989
63. O. Duman and G.A. Anastassiou, On statistical fuzzy trigonometric Korovkin theory, J. Comput. Anal. Appl. 10 (2008), 333-344
64. A.M. Fink, Bounds on the deviation of a function from its averages, Czechoslavak Math. J., 42 (117) (1992), 289-310.
65. A.R. Freedman, J.J. Sember, Densities and summability, Paci.c J. Math. 95 (1981) 293-305.
66. S. Gal, Approximation Theory in Fuzzy Setting, Chapter 13 in Hand- book of Analytic-Computational Methods in Applied Mathematics, pp. 617-666, edited by G. Anastassiou, Chapman & Hall/CRC, 2000, Boca Raton, New York.
67. R. Goetschel, Jr. and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43.
68. W.J. Gordon and J.A. Wixom, Shepard.s method of metric interpo- lation to bivariate and multivariate interpolation, Math. Comp., 32 (1978), 253-264.
69. C.W. Groetsch, Elements of Applicable Functional Analysis, Marcel Dekker, Inc., New York and Basel, 1980.
70. G. H. Hardy, Divergent series, Oxford Uni. Press, London, 1949-
71. O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
72. O. Kaleva, Interpolation of fuzzy data, Fuzzy Sets and Systems 61(1994), 63-70.
73. L.V. Kantorovich, On the moment problem for a .nite interval, Dokl. Akad. Nauk SSSR, 14 (1937), 531-537 (in Russian).
74. J.H.B. Kemperman, The general moment problem, a geometric ap- proach, The Annals of Mathematical Statistics, 39, No.1 (1968), 93-122.
75. Yun Kyong Kim and Byung Moon Ghil, Integrals of fuzzy-number- valued functions, Fuzzy Sets and Systems, 86 (1997), 213-222.
76. J.P. King, J.J. Swetits, Positive linear operators and summability, J. Aust. Math. Soc. 11 (1970) 281-290.
77. E. Kolk, Matrix summability of statistically convergent sequences, Analysis 13 (1993), 77-83.
78. P.P. Korovkin, Linear Operators and Approximation Theory, Hindus- tan Publ. Corp., Delhi, India, 1960.
79. B. Kuttner. On the Gibbs phenomenon for Riesz means, J. London Math. Soc. 19 (1944), 153-161-
80. Puyin Liu, Analysis of approximation of continuous fuzzy func- tions by multivariate fuzzy polynomials, Fuzzy Sets and Systems 127(2002), 299-313.
81. W. Lodwick, J. Santos, Constructing consistent fuzzy surfaces from fuzzy data, Fuzzy Sets and Systems, 135(2003), 259-277.
82. G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948) 167-190.
83. G.G. Lorentz, Appproximation of Functions, Chelsea Publishing Company, New York, 1986 (second edition).
84. R. Lowen, A fuzzy Lagrange interpolation theorem, Fuzzy Sets and Systems 34(1990) 33-38.
85. M. Matloka, Sequences of fuzzy numbers, BUSEFAL 28 (1986) 28-37.
86. H.I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995) 1811- 1819.
87. H.I. Miller, C. Orhan, On almost convergent and statistically conver- gent subsequences. Acta Math. Hungar. 93 (2001), 135-151.
88. R.N. Mohapatra, Quantitative results on almost convergence of a sequence of positive linear operators, J. Approx. Theory 20 (1977) 239-250.
89. T. Nishishiraho, Quantitative theorems on linear approximation processes of convolution operators in Banach spaces, Tôhoku Math. J. 33(1981) 109-126.
90. T. Nishishiraho, Convergence of positive linear approximation processes, Tôhoku Math. J. 35 (1983) 441-458.
91. F. Nuray, E. Sava̧s, Statistical convergence of sequences of fuzzy num- bers, Math. Slovaca 45 (1995), 269-273.
92. A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1938), 226-227.
93. M. L. Puri and D. A. Ralescu, Differentials of fuzzy functions, J. of Math. Analysis and Appl., 91, 1983, 552-558.
94. C. Radu, A-summability and approximation of continuous periodic functions, Studia Univ. Babȩs-Bolyai Math. 52 (2007) 155-161.
95. D. Shepard, A two-dimensional interpolation function for irregularly- spaced data, in Proc. 1968 ACM National Conference, 1968, 517-524.
96. O. Shisha and B. Mond, The degree of convergence of sequences of linear positive operators, Nat. Acad. of Sci. U.S., 60 (1968), 1196- 1200.
97. O. Shisha and B. Mond, The degree of approximation to periodic functions by linear positive operators, J. Approx. Th., 1 (1968), 335- 339.
98. Murray R. Spiegel, Advanced Calculus, Schaum.s Outline Series, McGraw-Hill Book Company, New York. 1963.
99. M. Steiglitz, Eine verallgemeinerung des begriffs der fastkonvergenz, Math. Japon. 18 (1973) 53-70.
100. J.J. Swetits, On summability and positive linear operators, J. Ap- prox. Theory 25 (1979) 186-188.
101. J. Szabados, On a problem of R. DeVore, Acta Math. Hungar., 27 (1-2)(1976) 219-223.
102. P. Teran and M. Lopez-Diaz, Remarks on Korovkin-type approxi- mation of fuzzy random variables, in Statistical Modeling, Analysis and Management of Fuzzy Data, edited by C. Bertoluzza, M. Gil, D. Ralescu, Physica-Verlag, 2002, pp. 90-103.
103. L. A. Zadeh, Fuzzy sets, Information and Control, 8, 1965, 338-353.