Monotone approximation of random functions with multivariate domains in respect of lattice semi-norms. (English) Zbl 0741.41018
Summary: Consider the problem of approximating a random function which is defined on a compact and convex subset of a topological vector space. For monotone approximation procedures global and local error bounds with respect to lattice semi-norms are established.
MSC:
| 41A30 | Approximation by other special function classes |
| 41A25 | Rate of convergence, degree of approximation |
References:
| [1] | Anastassiou, G.A.: (1985a) Miscellaneous sharp inequalities and Korovkin-type convergence theorems involving sequences of probability measures. J. Approx. Theory 44, 384–390. · Zbl 0573.42004 · doi:10.1016/0021-9045(85)90088-7 |
| [2] | Anastassiou, G.A.: (1985b) A study of positive linear operators by the method of moments, one-dimensional case. J. Approx. Theory 45, 247–270. · Zbl 0571.41019 · doi:10.1016/0021-9045(85)90049-8 |
| [3] | Anastassiou, G.A.: (1987) On the degree of weak convergence of a sequence of finite measures to the unit measure under convexity. J. Approx. Theory 51, 333–349. · Zbl 0632.41014 · doi:10.1016/0021-9045(87)90042-6 |
| [4] | Anastassiou, G.A.: (1989a) Smooth rate of weak convergence of convex type positive finite measures. J. Math. Anal. Appl. (to appear). |
| [5] | Anastassiou, G.A.: (1989b) Rate of convergence of non-positive generalized convolution type operators. J. Math. Anal. Appl. (to appear). · Zbl 0678.41025 |
| [6] | Berens, H., Lorentz, G.G.: (1973) Theorems of Korovkin type for positive linear operators on Banach lattices. In: Approximation theory (edited by G.G. Lorentz), 1–30. New York: Academic Press. · Zbl 0322.41017 |
| [7] | Cheney, E.W.: (1966) Introduction to approximation theory. New York: McGraw-Hill. · Zbl 0161.25202 |
| [8] | Censor, E.: (1971) Quantitative results for positive linear approximation operators. J. Approx. Theory 4, 442–450. · Zbl 0233.41004 · doi:10.1016/0021-9045(71)90009-8 |
| [9] | De Vore, R.A.: (1972) The approximation of continuous functions by positive linear operators. Lecture notes in mathematics, Vol. 293. Berlin, New York: Springer. · Zbl 0252.41017 |
| [10] | Ditzian, Z.: (1986)Inverse theorems for multidimensional Bernstein operators. Pacific J. Math. 121, 293–319. · Zbl 0581.41023 · doi:10.2140/pjm.1986.121.293 |
| [11] | Donner, K.: (1982)Extensions of positive operators and Korovkin theorems. Lecture notes in mathematics, Vol. 904. Berlin, New York: Springer. · Zbl 0481.47024 |
| [12] | Gonska, H.H.: (1983) On approximation of continuously differentiable functions by positive linear operators. Bull. Austral. Math. Soc. 27, 73–81. · Zbl 0494.41014 · doi:10.1017/S0004972700011497 |
| [13] | Gonska, H.H., Meier-Gonska, J.: (1986) A bibliography on approximation of functions by Bernatein-type operators. In: Approximation Theory V (edited by C.K. Chui, L.L. Schumaker and J.D. Ward), 621–654. Orlando: Academic Press. · Zbl 0615.41019 |
| [14] | Hahn, L.: (1982) Stochastic methods in connections with approximation theorems for positive linear operators. Pacific J. Math. 101, 307–319. · Zbl 0458.41016 · doi:10.2140/pjm.1982.101.307 |
| [15] | Keimel, K., Roth, W.: (1988) A Korovkin type approximation theorem for set-valued functions. Proc. Amer. Math. Soc. 104, 819–823. · Zbl 0693.47032 · doi:10.1090/S0002-9939-1988-0964863-8 |
| [16] | Keimel, K., Roth, W.: (1990) Ordered cones and approximation. Preprint, Department of Mathematics, University of Bahrain. · Zbl 0752.41033 |
| [17] | Luxemburg, W.A.J., Zaanen, A.C.: (1971) Riesz spaces I. Amsterdam: North Holland. · Zbl 0231.46014 |
| [18] | Mond, B., Vasudevan, R.: (1980) On approximation by linear positive operators. J. Approx. Theory 30, 334–336. · Zbl 0521.41018 · doi:10.1016/0021-9045(80)90035-0 |
| [19] | Nishishiraho, T.: (1982) Quantitative theorems on approximation processes of positive linear operators. In: Multivariate approximation theory II (edited by W. Schempp and K. Zeller), ISNM 61, 297-311. Basel, Boston, Stuttgart: Birkhäuser. · Zbl 0498.41014 |
| [20] | Roth, W.: (1989) Families of convex subsets and Korovkin type theorems in locally convex spaces. Rev. Roumaine Math. Pures et Appl. 34, 329–346. · Zbl 0684.46014 |
| [21] | Sablonnière, P.: (1988) Positive spline operators and orthogonal splines. J. Approx. Theory 52, 28–42. · Zbl 0672.41008 · doi:10.1016/0021-9045(88)90035-4 |
| [22] | Schaefer, H.H.: (1974) Banach lattices and positive operators. Berlin: Springer. · Zbl 0296.47023 |
| [23] | Scheffold, E.: (1973) Ein allgemeiner Korovkin-Satz für lokalkonvexe Vektorverbände. Math. Z. 132, 209–214. · Zbl 0255.46007 · doi:10.1007/BF01213865 |
| [24] | Schempp, W.: (1972) A note on Korovkin test families. Arch. Math. 23, 521–524. · Zbl 0229.41013 · doi:10.1007/BF01304926 |
| [25] | Weba, M.: (1986) Korovkin systems of stochastic processes. Math. Z. 192, 73–80. · Zbl 0598.60058 · doi:10.1007/BF01162021 |
| [26] | Weba, M.: (1989) Quantitative results on monotone approximation of stochastic processes. Probab. Math. Statist. (to appear). · Zbl 0756.60049 |
| [27] | Weba, M.: (1990) A quantitative Korovkin theorem for random functions with multivariate domains. J. Approx. Theory 61, 74–87. · Zbl 0698.41016 · doi:10.1016/0021-9045(90)90024-K |
| [28] | Wolff, M.: (1978) On the universal Korovkin closure of subsets in vector lattices. J. Approx. Theory 22, 243–253. · Zbl 0392.41017 · doi:10.1016/0021-9045(78)90056-4 |
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