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Tchebycheffian B-Splines Revisited: An Introductory Exposition

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Advanced Methods for Geometric Modeling and Numerical Simulation

Part of the book series: Springer INdAM Series ((SINDAMS,volume 35))

Abstract

Tchebycheffian splines are smooth piecewise functions where the different pieces are drawn from extended Tchebycheff spaces. They are a natural generalization of polynomial splines and can be represented in terms of an interesting set of basis functions, the so-called Tchebycheffian B-splines, which generalize the standard polynomial B-splines. We provide an accessible and self-contained exposition of Tchebycheffian B-splines and their main properties. Our construction is based on an integral recurrence relation and allows for the use of different extended Tchebycheff spaces on different intervals. The special class of generalized B-splines is also discussed in detail.

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Notes

  1. 1.

    The space \({\mathbb {T}}_{p}(I)\) is called a Tchebycheff (T-) space if any solution of (1) with \(m_0=\cdots =m_p=1\) is unique in \({\mathbb {T}}_{p}(I)\). In such a case, (1) is a Lagrange interpolation problem.

  2. 2.

    Our Tchebycheffian B-spline construction follows the approach of [3, 4], while it differs from [26] in two ways: the indexing of the weight functions and the positioning of the weight functions with respect to the integration. This provides a more intuitive notation.

  3. 3.

    The term “generalized splines” has several different meanings in the literature. For example, the splines considered here are much less general than those described in [30, Chap. 11]. We follow the definition given in [17]. This definition was already used before for special choices of U and V; see, for example, [15, 16].

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Acknowledgements

C. Manni and H. Speleers are members of the Gruppo Nazionale Calcolo Scientifico—Istituto Nazionale di Alta Matematica (GNCS-INdAM), and acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C18000100006). The authors are grateful to the Centre International de Rencontres Mathématiques (CIRM, Luminy) and the Centro Internazionale per la Ricerca Matematica (CIRM, Trento) for the Research-in-Pairs support.

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Authors and Affiliations

  1. Department of Mathematics, University of Oslo, Oslo, Norway

    Tom Lyche

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Carla Manni & Hendrik Speleers

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  1. Tom Lyche
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  2. Carla Manni
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Correspondence to Hendrik Speleers .

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Editors and Affiliations

  1. Department of Mathematics and Computer Science "U. Dini", University of Florence, Florence, Italy

    Carlotta Giannelli

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Hendrik Speleers

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Lyche, T., Manni, C., Speleers, H. (2019). Tchebycheffian B-Splines Revisited: An Introductory Exposition. In: Giannelli, C., Speleers, H. (eds) Advanced Methods for Geometric Modeling and Numerical Simulation. Springer INdAM Series, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-030-27331-6_8

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