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On nonlinear simultaneous Chebyshev approximation problems. (English) Zbl 1039.41013

The authors consider the problem of nonlinear simultaneous Chebyshev approximation in a real continuous function space. The results deal with existence of solutions and characterization conditions of Kolmogorov type and also of alternation type. As special cases, previous results from the literature are obtained under less restrictive hypotheses. Applications are given to nonlinear approximation by rational functions, by exponential sums and by Chebyshev splines with free knots.

MSC:

41A28 Simultaneous approximation
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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[1] Braess, D., Nonlinear Approximation Theory (1986), Springer-Verlag · Zbl 0656.41001
[2] Cudia, D. F., The geometry of Banach spaces. Smoothness, Trans. Amer. Math. Soc., 110, 284-314 (1964) · Zbl 0123.30701
[3] Deutsch, F., Existence of best approximations, J. Approx. Theory, 28, 132-154 (1980) · Zbl 0464.41016
[4] Diestel, J.; Uhl, J. J., Vector Measures. Vector Measures, Math. Surveys, 15 (1977), American Mathematical Society · Zbl 0369.46039
[5] Dunham, C. B., Simultaneous Chebyshev approximation of functions on an interval, Proc. Amer. Math. Soc., 18, 472-477 (1967) · Zbl 0154.05904
[6] Li, C., Simultaneous Chebyshev approximation, J. Math. (PRC), 5, 231-240 (1985), (in Chinese) · Zbl 0592.41028
[7] Li, C., Nonlinear simultaneous Chebyshev approximation, J. Math. (PRC), 7, 335-337 (1987), (in Chinese)
[8] Li, C.; Watson, G. A., On best simultaneous approximation, J. Approx. Theory, 91, 332-348 (1997) · Zbl 0892.41012
[9] Li, C.; Watson, G. A., Best simultaneous approximation of an infinite set of functions, Comput. Math. Appl., 37, 1-9 (1999) · Zbl 0974.41025
[10] Li, C.; Watson, G. A., A class of best simultaneous approximation problems, Comput. Math. Appl., 31, 45-53 (1996) · Zbl 0863.41012
[11] Rice, J. R., The Approximation of Functions, Vol. II, Nonlinear and Multivariate Theory (1969), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0185.30601
[12] Pinkus, A., Uniqueness in vector-valued approximation, J. Approx. Theory, 73, 17-92 (1993) · Zbl 0780.41018
[13] Shi, Y. G., Weighted simultaneous Chebyshev approximation, J. Approx. Theory, 32, 305-315 (1981) · Zbl 0475.41028
[14] Shi, Y. G., Simultaneous best rational approximation, Chinese Ann. Math., 1, 477-484 (1980), (in Chinese) · Zbl 0466.41006
[15] Shi, J.; Huotari, R., Simultaneous approximation from convex sets, Comput. Math. Appl., 30, 197-206 (1995) · Zbl 0842.46011
[16] Smarzewski, R., On characterization of Chebyshev optimal starting and transformed approximation by families having a degree, Ann. Univ. Mariae Curie-Skłodowska, 14, 111-118 (1977) · Zbl 0473.41021
[17] Smarzewski, R., Chebyshev additive weight approximation by maximal families, J. Approx. Theory, 35, 195-220 (1982) · Zbl 0488.41026
[18] Tanimoto, S., A characterization of best simultaneous approximations, J. Approx. Theory, 59, 359-361 (1989) · Zbl 0697.41014
[19] Tapia, R. A., A characterization of inner product spaces, Proc. Amer. Math. Soc., 41, 569-574 (1973) · Zbl 0286.46025
[20] Watson, G. A., A characterization of best simultaneous approximations, J. Approx. Theory, 75, 175-182 (1993) · Zbl 0794.41019
[21] Xu, S. Y.; Li, C., Characterization of the best simultaneous approximation, Acta Math. Sinica, 30, 528-534 (1987), (in Chinese)
[22] Xu, S. Y.; Li, C., Characterization of best simultaneous approximation, Approx. Theory Appl., 3, 190-198 (1987) · Zbl 0676.41023
[23] Xu, S. Y.; Li, C.; Yang, W. S., Nonlinear Approximation Theory in Banach Spaces (1997), Science Press: Science Press Beijing, (in Chinese)
[24] Yang, W. S.; Li, C.; Watson, G. A., Characterization and uniqueness of nonlinear uniform approximation, Proc. Edinburgh Math. Soc., 40, 473-482 (1997) · Zbl 0891.41018
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