Discrete Approximation by First-Degree Splines with Free Knots

Authors

  • Jens Kunath Numerische und Angewandte Mathematik, BTU Cottbus-Senftenberg, Platz der Deutschen Einheit 1, D-03046 Cottbus, Germany
  • Ludwig J. Cromme Numerische und Angewandte Mathematik, BTU Cottbus-Senftenberg, Platz der Deutschen Einheit 1, D-03046 Cottbus, Germany

DOI:

https://doi.org/10.53555/ms.v4i5.621

Keywords:

splines of degree one, splines with free knots, best approximation, discrete approx- imation, Lp-approximation (1 ≤ p ≤ ∞), existence theorem, first-degree splines, broken lines

Abstract

This paper deals with the approximation of discrete real-valued functions by first-degree splines (broken lines) with free knots for arbitrary Lp-norms (1 ≤ p ≤ ∞). We prove the existence of best approximations und derive statements on the position of the (free) knots of a best approximation. Building on this, elsewhere we develop an algorithm to determine a (global) best approximation in the L2-norm.

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References

Cromme, L. J., Kunath, J., Krebs, A.: Computing best discrete least-squares approximations by first-degree splines with free knots. Preprint at https://arxiv.org (2017).

Koci´c, L. M., Milovanovi´c, G. V.: Shape Preserving Approximations by Polynomials and Splines. Computers Math. Applic., 33(11), 59-97 (1997)

Koutsoyiannis, D.: Broken line smoothing: a simple method for interpolating and smoothing data series. Environmental Modelling and Software, 14, 139-149 (2000)

Rice, J.R.: The Approximation of Functions Vol. II - Nonlinear and multivariate theory. Addisson-Wesley, Reading, Massachusetts (1969)

Richards, F. B.: A Gibbs Phenomenon for Spline Functions. J. of Approximation Theory, 66(3), 344-351 (1991)

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Published

2018-05-31

How to Cite

Kunath, J., & Cromme, L. J. (2018). Discrete Approximation by First-Degree Splines with Free Knots. International Journal For Research In Mathematics And Statistics, 4(5), 26–39. https://doi.org/10.53555/ms.v4i5.621

Issue

Vol. 4 No. 5 (2018): International Journal For Research In Mathematics And Statistics

Section

Articles

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Copyright (c) 2018 International Journal For Research In Mathematics And Statistics (ISSN: 2208-2662)

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