On monotone and convex spline interpolation

Author:
Paolo Costantini

Journal:
Math. Comp. **46** (1986), 203-214

MSC:
Primary 65D05; Secondary 41A05, 41A15

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815841-7

MathSciNet review:
815841

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Abstract: This paper is concerned with the problem of existence of monotone and/or convex splines, having degree *n* and order of continuity *k*, which interpolate to a set of data at the knots. The interpolating splines are obtained by using Bernstein polynomials of suitable continuous piecewise linear functions; they satisfy the inequality . The theorems presented here are useful in developing algorithms for the construction of shape-preserving splines interpolating arbitrary sets of data points. Earlier results of McAllister, Passow and Roulier can be deduced from those given in this paper.

**[1]**P. Costantini, "An algorithm for computing shape-preserving interpolating splines of arbitrary degree,"*J. Comput. Appl. Math.*(Submitted.) MR**948888 (89e:65165)****[2]**P. Costantini & R. Morandi, "Monotone and convex cubic spline interpolation,"*Calcolo*, v. 21, 1984, pp. 281-294. MR**799625 (86m:65014)****[3]**P. Costantini & R. Morandi, "An algorithm for computing shape-preserving cubic spline interpolation to data,"*Calcolo*, v. 21, 1984, pp. 295-305. MR**799994 (86j:65016)****[4]**P. Costantini & R. Morandi, "Piecewise monotone quadratic histosplines." (Preprint.) MR**982231 (90a:65026)****[5]**C. de Boor & B. Swartz, "Piecewise monotone interpolation,"*J. Approx. Theory*, v. 21, 1977, pp. 411-416. MR**0481727 (58:1826)****[6]**G. G. Lorentz,*Bernstein Polynomials*, University of Toronto Press, Toronto, 1953. MR**0057370 (15:217a)****[7]**D. F. McAllister, E. Passow & J. A. Roulier, "Algorithms for computing shape-preserving spline interpolation to data,"*Math. Comp.*, v. 31, 1977, pp. 717-725. MR**0448805 (56:7110)****[8]**E. Neuman,*Shape-Preserving Interpolation by Polynomial Splines*, Report no. N112, Institute of Computer Science, Wrocław University, 1982.**[9]**E. Passow & J. A. Roulier, "Monotone and convex spline interpolation,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 904-909. MR**0470566 (57:10316)**

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0815841-7

Article copyright:
© Copyright 1986
American Mathematical Society