Czechoslovak Mathematical Journal, online first, 23 pp.

# A new algorithm for approximating the least concave majorant

## Martin Franců, Ron Kerman, Gord Sinnamon

#### Received August 1, 2016.   First published August 16, 2017.

Martin Franců, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: martinfrancu@gmail.com; Ron Kerman, Department of Mathematics and Statistics, Brock University, 1812 Sir Isaac Brock Way, St. Catharines, Ontario L2S 3A1, Canada, e-mail: rkerman@brocku.ca; Gord Sinnamon, Department of Mathematics, University of Western Ontario, 2004 Perth Dr, London, Ontario N6G, Canada, e-mail: sinnamon@uwo.ca

Abstract: The least concave majorant, $\hat F$, of a continuous function $F$ on a closed interval, $I$, is defined by $\hat F (x) = \inf\{ G(x) G \geq F, G \text{ concave}\},\quad x \in I.$ 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 $\hat F$. We give two examples, one to illustrate, the other to apply our algorithm.

Keywords: least concave majorant; level function; spline approximation

Classification (MSC 2010): 26A51, 52A41, 46N10

DOI: 10.21136/CMJ.2017.0408-16

Full text available as PDF.

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