| Title:
|
A new algorithm for approximating the least concave majorant (English) |
| Author:
|
Franců, Martin |
| Author:
|
Kerman, Ron |
| Author:
|
Sinnamon, Gord |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
4 |
| Year:
|
2017 |
| Pages:
|
1071-1093 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
least concave majorant |
| Keyword:
|
level function |
| Keyword:
|
spline approximation |
| MSC:
|
26A51 |
| MSC:
|
46N10 |
| MSC:
|
52A41 |
| idZBL:
|
Zbl 06819574 |
| idMR:
|
MR3736020 |
| DOI:
|
10.21136/CMJ.2017.0408-16 |
| . |
| Date available:
|
2017-11-20T14:57:18Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146968 |
| . |
| Reference:
|
[1] Brudnyĭ, Y. A., Krugljak, N. Y.: Interpolation Functors and Interpolation Spaces. Vol. 1.North-Holland Mathematical Library 47, Amsterdam (1991). Zbl 0743.46082, MR 1107298 |
| Reference:
|
[2] Carolan, C. A.: The least concave majorant of the empirical distribution function.Can. J. Stat. 30 (2002), 317-328. Zbl 1012.62052, MR 1926068, 10.2307/3315954 |
| Reference:
|
[3] Debreu, G.: Least concave utility functions.J. Math. Econ. 3 (1976), 121-129. Zbl 0361.90007, MR 0411563, 10.1016/0304-4068(76)90020-3 |
| Reference:
|
[4] Hall, C. A., Meyer, W. W.: Optimal error bounds for cubic spline interpolation.J. Approximation Theory 16 (1976), 105-122. Zbl 0316.41007, MR 0397247, 10.1016/0021-9045(76)90040-x |
| Reference:
|
[5] Halperin, I.: Function spaces.Can. J. Math. 5 (1953), 273-288. Zbl 0052.11303, MR 0056195, 10.4153/CJM-1953-031-3 |
| Reference:
|
[6] Härdle, W., Kerkyacharian, G., Picard, D., Tsybakov, A.: Wavelets, Approximation, and Statistical Applications.Lecture Notes in Statistics 129, Springer, Berlin (1998). Zbl 0899.62002, MR 1618204, 10.1007/978-1-4612-2222-4 |
| Reference:
|
[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. Zbl 0256.68041, MR 0381227, 10.1016/0020-0190(73)90020-3 |
| Reference:
|
[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. Zbl 1144.46058, MR 2351114, 10.5209/rev_rema.2007.v20.n2.16492 |
| Reference:
|
[9] Lorentz, G. G.: Bernstein Polynomials.Mathematical Expositions, no. 8. University of Toronto Press X, Toronto (1953). Zbl 0051.05001, MR 0057370 |
| Reference:
|
[10] Mastyło, M., Sinnamon, G.: A Calderón couple of down spaces.J. Funct. Anal. 240 (2006), 192-225. Zbl 1116.46015, MR 2259895, 10.1016/j.jfa.2006.05.007 |
| Reference:
|
[11] Peetre, J.: Concave majorants of positive functions.Acta Math. Acad. Sci. Hung. 21 (1970), 327-333. Zbl 0204.38002, MR 0272960, 10.1007/BF01894779 |
| . |
