# American Institute of Mathematical Sciences

March  2014, 34(3): 1041-1060. doi: 10.3934/dcds.2014.34.1041

## Bernstein-type approximation of set-valued functions in the symmetric difference metric

 1 School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel, Israel

Received  November 2012 Revised  February 2013 Published  August 2013

We study the approximation of univariate and multivariate set-valued functions (SVFs) by the adaptation to SVFs of positive sample-based approximation operators for real-valued functions. To this end, we introduce a new weighted average of several sets and study its properties. The approximation results are obtained in the space of Lebesgue measurable sets with the symmetric difference metric.
In particular, we apply the new average of sets to adapt to SVFs the classical Bernstein approximation operators, and show that these operators approximate continuous SVFs. The rate of approximation of Hölder continuous SVFs by the adapted Bernstein operators is studied and shown to be asymptotically equal to the one for real-valued functions. Finally, the results obtained in the metric space of sets are generalized to metric spaces endowed with an average satisfying certain properties.
Citation: Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041
##### References:
 [1] Z. Artstein, Piecewise linear approximations of set-valued maps, Journal of Approximation Theory, 56 (1989), 41-47. doi: 10.1016/0021-9045(89)90131-7. [2] R. Baier and E. Farkhi, Differences of convex compact sets in the space of directed sets. Part I: The space of directed sets, Set-Valued Analysis, 9 (2001), 217-245. doi: 10.1023/A:1012046027626. [3] R. Baier and G. Perria, Set-valued hermite interpolation, Journal of Approximation Theory, 163 (2011), 1349-1372. doi: 10.1016/j.jat.2010.11.004. [4] S. Bernstein, Démonstration du théoreme de weierstrass fondée sur le calcul des probabilités, Commun. Soc. Math. Kharkow, 13 (1912), 1-2. [5] D. Burago, Y. Burago, S. Ivanov and A. M. Society, "A Course in Metric Geometry," American Mathematical Society, 2001. [6] C. De Boor, "A Practical Guide to Splines," Springer Verlag, 2001. [7] R. DeVore and G. Lorentz, "Constructive Approximation," Springer, 1993. [8] N. Dyn and E. Farkhi, Spline subdivision schemes for convex compact sets, Journal of Computational and Applied Mathematics, 119 (2000), 133-144. doi: 10.1016/S0377-0427(00)00375-7. [9] N. Dyn and E. Farkhi, Spline subdivision schemes for compact sets with metric averages, Trends in Approximation Theory, (2001), 93-102. [10] N. Dyn and E. Farkhi, Set-valued approximations with Minkowski averages-convergence and convexification rates, Numerical Functional Analysis and Optimization, 25 (2004), 363-377. doi: 10.1081/NFA-120039682. [11] N. Dyn, E. Farkhi and A. Mokhov, Approximation of univariate set-valued functions-an overview, Serdica Math. J., 33 (2007), 495-514. [12] N. Dyn, E. Farkhi and A. Mokhov, Approximations of set-valued functions by metric linear operators, Constructive Approximation, 25 (2007), 193-209. doi: 10.1007/s00365-006-0632-9. [13] N. Dyn and A. Mokhov, Approximations of set-valued functions based on the metric average, Rendiconti di Matematica, 26 (2006), 249-266. [14] G. Farin, "Curves and Surfaces for CAGD: A Practical Guide," Morgan Kaufmann Pub, 2002. [15] W. Feller, "An Introduction to Probability Theory and Its Applications," I, John Wiley & Sons, 1968. [16] P. Halmos, "Naive Set Theory," Springer-Verlag, New York-Heidelberg, 1974. [17] M. Kac, Une remarque sur les polynomes de m.s. bernstein, Studia Math, 7 (1938), 49-51. [18] M. Kac, Reconnaissance de priorité relative a ma note, une remarque sur les polynomes de m.s. bernstein, Studia Math, 8 (1939), 170. [19] S. Kels and N. Dyn, Subdivision schemes of sets and the approximation of set-valued functions in the symmetric difference metric, Arxiv Preprint arXiv:1111.6844, (2011). doi: 10.1007/s10208-013-9146-z. [20] K. Levasseur, A probabilistic proof of the Weierstrass approximation theorem, Amer. Math. Monthly, 91 (1984), 249-250. doi: 10.2307/2322960. [21] P. Mathé, Approximation of holder continuous functions by Bernstein polynomials, The American Mathematical Monthly, 106 (1999), 568-574. doi: 10.2307/2589469. [22] P. Mathé, Asymptotic constants for multivariate Bernstein polynomials, Studia Scientiarum Mathematicarum Hungarica, 40 (2003), 59-69. doi: 10.1556/SScMath.40.2003.1-2.5. [23] I. Molchanov, "Theory of Random Sets," Springer Verlag, 2005. [24] M. Muresan, Set-valued approximation of multifunctions, Studia Univ. Babes-Bolyai, Mathematica, 55 (2010), 107-148. [25] A. Papadopoulos, "Metric Spaces, Convexity and Nonpositive Curvature," 6 European Mathematical Society, 2005. [26] C. Rabut, An introduction to Schoenberg's approximation, Computers & Mathematics with Applications, 24 (1992), 149-175. doi: 10.1016/0898-1221(92)90177-J. [27] R. Vitale, Approximation of convex set-valued functions, Journal of Approximation Theory, 26 (1979), 301-316. doi: 10.1016/0021-9045(79)90067-4. [28] M. Zelen and N. Severo, "Probability Functions," in "Handbook of Mathematical Functions" (editors, M. Abramowitz and Stegun), Dover, New York, 5 (1964), 925-995.

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##### References:
 [1] Z. Artstein, Piecewise linear approximations of set-valued maps, Journal of Approximation Theory, 56 (1989), 41-47. doi: 10.1016/0021-9045(89)90131-7. [2] R. Baier and E. Farkhi, Differences of convex compact sets in the space of directed sets. Part I: The space of directed sets, Set-Valued Analysis, 9 (2001), 217-245. doi: 10.1023/A:1012046027626. [3] R. Baier and G. Perria, Set-valued hermite interpolation, Journal of Approximation Theory, 163 (2011), 1349-1372. doi: 10.1016/j.jat.2010.11.004. [4] S. Bernstein, Démonstration du théoreme de weierstrass fondée sur le calcul des probabilités, Commun. Soc. Math. Kharkow, 13 (1912), 1-2. [5] D. Burago, Y. Burago, S. Ivanov and A. M. Society, "A Course in Metric Geometry," American Mathematical Society, 2001. [6] C. De Boor, "A Practical Guide to Splines," Springer Verlag, 2001. [7] R. DeVore and G. Lorentz, "Constructive Approximation," Springer, 1993. [8] N. Dyn and E. Farkhi, Spline subdivision schemes for convex compact sets, Journal of Computational and Applied Mathematics, 119 (2000), 133-144. doi: 10.1016/S0377-0427(00)00375-7. [9] N. Dyn and E. Farkhi, Spline subdivision schemes for compact sets with metric averages, Trends in Approximation Theory, (2001), 93-102. [10] N. Dyn and E. Farkhi, Set-valued approximations with Minkowski averages-convergence and convexification rates, Numerical Functional Analysis and Optimization, 25 (2004), 363-377. doi: 10.1081/NFA-120039682. [11] N. Dyn, E. Farkhi and A. Mokhov, Approximation of univariate set-valued functions-an overview, Serdica Math. J., 33 (2007), 495-514. [12] N. Dyn, E. Farkhi and A. Mokhov, Approximations of set-valued functions by metric linear operators, Constructive Approximation, 25 (2007), 193-209. doi: 10.1007/s00365-006-0632-9. [13] N. Dyn and A. Mokhov, Approximations of set-valued functions based on the metric average, Rendiconti di Matematica, 26 (2006), 249-266. [14] G. Farin, "Curves and Surfaces for CAGD: A Practical Guide," Morgan Kaufmann Pub, 2002. [15] W. Feller, "An Introduction to Probability Theory and Its Applications," I, John Wiley & Sons, 1968. [16] P. Halmos, "Naive Set Theory," Springer-Verlag, New York-Heidelberg, 1974. [17] M. Kac, Une remarque sur les polynomes de m.s. bernstein, Studia Math, 7 (1938), 49-51. [18] M. Kac, Reconnaissance de priorité relative a ma note, une remarque sur les polynomes de m.s. bernstein, Studia Math, 8 (1939), 170. [19] S. Kels and N. Dyn, Subdivision schemes of sets and the approximation of set-valued functions in the symmetric difference metric, Arxiv Preprint arXiv:1111.6844, (2011). doi: 10.1007/s10208-013-9146-z. [20] K. Levasseur, A probabilistic proof of the Weierstrass approximation theorem, Amer. Math. Monthly, 91 (1984), 249-250. doi: 10.2307/2322960. [21] P. Mathé, Approximation of holder continuous functions by Bernstein polynomials, The American Mathematical Monthly, 106 (1999), 568-574. doi: 10.2307/2589469. [22] P. Mathé, Asymptotic constants for multivariate Bernstein polynomials, Studia Scientiarum Mathematicarum Hungarica, 40 (2003), 59-69. doi: 10.1556/SScMath.40.2003.1-2.5. [23] I. Molchanov, "Theory of Random Sets," Springer Verlag, 2005. [24] M. Muresan, Set-valued approximation of multifunctions, Studia Univ. Babes-Bolyai, Mathematica, 55 (2010), 107-148. [25] A. Papadopoulos, "Metric Spaces, Convexity and Nonpositive Curvature," 6 European Mathematical Society, 2005. [26] C. Rabut, An introduction to Schoenberg's approximation, Computers & Mathematics with Applications, 24 (1992), 149-175. doi: 10.1016/0898-1221(92)90177-J. [27] R. Vitale, Approximation of convex set-valued functions, Journal of Approximation Theory, 26 (1979), 301-316. doi: 10.1016/0021-9045(79)90067-4. [28] M. Zelen and N. Severo, "Probability Functions," in "Handbook of Mathematical Functions" (editors, M. Abramowitz and Stegun), Dover, New York, 5 (1964), 925-995.
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