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Splines and PDEs: From Approximation Theory to Numerical Linear Algebra pp 1–76Cite as

Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement

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Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2219))

Abstract

This chapter presents an overview of polynomial spline theory, with special emphasis on the B-spline representation, spline approximation properties, and hierarchical spline refinement. We start with the definition of B-splines by means of a recurrence relation, and derive several of their most important properties. In particular, we analyze the piecewise polynomial space they span. Then, we present the construction of a suitable spline quasi-interpolant based on local integrals, in order to show how well any function and its derivatives can be approximated in a given spline space. Finally, we provide a unified treatment of recent results on hierarchical splines. We especially focus on the so-called truncated hierarchical B-splines and their main properties. Our presentation is mainly confined to the univariate spline setting, but we also briefly address the multivariate setting via the tensor-product construction and the multivariate extension of the hierarchical approach.

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Notes

  1. 1.

    The original meaning of the word “spline” is a flexible ruler used to draw curves, mainly in the aircraft and shipbuilding industries. The “B” in B-splines stands for basis or basic.

  2. 2.

    The recurrence relation is due to de Boor, Cox and Mansfield [4, 14]. However, it appears already in works by Popoviciu and Chakalov in the 1930s; see [8] for an account of the early history of splines. For the modern theory of splines we refer the reader to the seminal papers by Schoenberg [41,42,43] and Curry/Schoenberg [15, 16]. In their works, B-splines were defined by divided differences of truncated power functions.

  3. 3.

    An explicit expression of (1.51) was given by Greville in [24]. According to Schoenberg [43], Greville reviewed the paper [43] introducing some elegant simplifications.

  4. 4.

    The value Λ j,p,ξ(s) is known as the de Boor–Fix functional [7] applied to s.

  5. 5.

    The inner product formula for cardinal B-splines traces back to [44]. The formula for derivatives of cardinal B-splines can be found in [21] and a generalization for multivariate box splines in [48].

  6. 6.

    This notation means that if λ j(f) uses the value of f or one of its derivatives at \(\xi _{m_j}\) (or \(\xi _{m_j+1}\)) then this value is obtained by taking the one sided limit from the right (or the left).

  7. 7.

    The HB-splines in Definition 7 were introduced by Kraft [28, 29] and further elaborated in [53]. However, the concept of hierarchical splines has a long history; for example, it was used in preconditioning [18, 54], adaptive modeling [19, 20] and adaptive finite elements [25, 30].

  8. 8.

    The truncation approach was introduced in [22] for hierarchical tensor-product splines, but was already developed before in the context of hierarchical Powell–Sabin splines [50]. A generalization towards a broad class of hierarchical spaces can be found in [23].

  9. 9.

    The general telescopic expression for the hierarchical quasi-interpolant was presented in [51]. A special telescopic approximation in the hierarchical setting was already considered in [29].

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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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Editors and Affiliations

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

    Tom Lyche

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

    Carla Manni

  3. Department of Mathematics, University of Rome Tor Vergata, ROMA, Roma, Italy

    Hendrik Speleers

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Lyche, T., Manni, C., Speleers, H. (2018). Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement. In: Lyche, T., Manni, C., Speleers, H. (eds) Splines and PDEs: From Approximation Theory to Numerical Linear Algebra. Lecture Notes in Mathematics(), vol 2219. Springer, Cham. https://doi.org/10.1007/978-3-319-94911-6_1

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