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Selected Topics in Approximation and Computation$
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Marek A. Kowalski, Krzystof A. Sikorski, and Frank Stenger

Print publication date: 1995

Print ISBN-13: 9780195080599

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195080599.001.0001

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(p.93) Chapter 2 Splines
Selected Topics in Approximation and Computation

Marek A. Kowalski

Krzysztof A. Sikorski

Frank Stenger

Oxford University Press

Spline functions are important approximation tools in numerous applications for which high degree polynomial methods perform poorly, such as in computer graphics and geometric modelling, as well as for various engineering problems—especially those involving graphing of numerical solutions and noisy data. Algorithms based on spline functions enjoy minimal approximation errors in wide classes of problems and minimal complexity bounds. In this Chapter we provide a brief introduction to basic classes of polynomial splines, B-Splines, and abstract splines. Further study of spline algorithms as applied to linear problems is outlined in Chapter 7. In this section we define polynomial spline functions, exhibit their interpolatory properties, and construct algorithms to compute them. It turns out that these splines provide interpolating curves that do not exhibit the large oscillations associated with high degree interpolatory polynomials. This is why they find applications in univariate curve matching in computer graphics.

Keywords:   Homer’s algorithm, Steffenson’s rule, curvature, general splines, interpolation problem, knot sequence, natural cubic spline, natural spline, spline function

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