Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
least concave majorant; level function; spline approximation
Summary:
The least concave majorant, $\hat F$, of a continuous function $F$ on a closed interval, $I$, is defined by \[ \hat F (x) = \inf \{ G(x)\colon G \geq F,\ G \text { concave}\},\quad x \in I. \] We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$. Given any function $F \in \mathcal {C}^4(I)$, it can be well-approximated on $I$ by a clamped cubic spline $S$. We show that $\hat S$ is then a good approximation to $\hat F$. \endgraf We give two examples, one to illustrate, the other to apply our algorithm.
References:
[1] Brudnyĭ, Y. A., Krugljak, N. Y.: Interpolation Functors and Interpolation Spaces. Vol. 1. North-Holland Mathematical Library 47, Amsterdam (1991). MR 1107298 | Zbl 0743.46082
[2] Carolan, C. A.: The least concave majorant of the empirical distribution function. Can. J. Stat. 30 (2002), 317-328. DOI 10.2307/3315954 | MR 1926068 | Zbl 1012.62052
[3] Debreu, G.: Least concave utility functions. J. Math. Econ. 3 (1976), 121-129. DOI 10.1016/0304-4068(76)90020-3 | MR 0411563 | Zbl 0361.90007
[4] Hall, C. A., Meyer, W. W.: Optimal error bounds for cubic spline interpolation. J. Approximation Theory 16 (1976), 105-122. DOI 10.1016/0021-9045(76)90040-x | MR 0397247 | Zbl 0316.41007
[5] Halperin, I.: Function spaces. Can. J. Math. 5 (1953), 273-288. DOI 10.4153/CJM-1953-031-3 | MR 0056195 | Zbl 0052.11303
[6] Härdle, W., Kerkyacharian, G., Picard, D., Tsybakov, A.: Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics 129, Springer, Berlin (1998). DOI 10.1007/978-1-4612-2222-4 | MR 1618204 | Zbl 0899.62002
[7] Jarvis, R. A.: On the identification of the convex hull of a finite set of points in the plane. Inf. Process. Lett. 2 (1973), 18-21. DOI 10.1016/0020-0190(73)90020-3 | MR 0381227 | Zbl 0256.68041
[8] Kerman, R., Milman, M., Sinnamon, G.: On the Brudnyĭ-Krugljak duality theory of spaces formed by the $\mathcal{K}$-method of interpolation. Rev. Mat. Complut. 20 (2007), 367-389. DOI 10.5209/rev_rema.2007.v20.n2.16492 | MR 2351114 | Zbl 1144.46058
[9] Lorentz, G. G.: Bernstein Polynomials. Mathematical Expositions, no. 8. University of Toronto Press X, Toronto (1953). MR 0057370 | Zbl 0051.05001
[10] Mastyło, M., Sinnamon, G.: A Calderón couple of down spaces. J. Funct. Anal. 240 (2006), 192-225. DOI 10.1016/j.jfa.2006.05.007 | MR 2259895 | Zbl 1116.46015
[11] Peetre, J.: Concave majorants of positive functions. Acta Math. Acad. Sci. Hung. 21 (1970), 327-333. DOI 10.1007/BF01894779 | MR 0272960 | Zbl 0204.38002
Partner of
EuDML logo