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Algebraic Proof of the B-Spline Derivative Formula

In: Proceedings of the Conference on Applied Mathematics and Scientific Computing

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  • Mladen Rogina

    (University of Zagreb, Department of Mathematics)

Abstract

We prove a well known formula for the generalized derivatives of Chebyshev B-splines: $$L_1 B_i^k (x) = \frac{{B_i^{k - 1} (x)}} {{C_{k - 1} (i)}} - \frac{{B_{i + 1}^{k - 1} (x)}} {{C_{k - 1} (i + 1)'}}$$ where $$C_{k - 1} (i) = \int \begin{gathered} t_{i + k - 1} \hfill \\ t_{} \hfill \\ \end{gathered} B_i^{k - 1} (x)d\sigma $$ in a purely algebraic fashion, and thus show that it holds for the most general spaces of splines. The integration is performed with respect to a certain measure associated in a natural way to the underlying Chebyshev system of functions. Next, we discuss the implications of the formula for some special spline spaces, with an emphasis on those that are not associated with ECC-systems.

Suggested Citation

  • Mladen Rogina, 2005. "Algebraic Proof of the B-Spline Derivative Formula," Springer Books, in: Zlatko Drmač & Miljenko Marušić & Zvonimir Tutek (ed.), Proceedings of the Conference on Applied Mathematics and Scientific Computing, pages 273-282, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4020-3197-7_20
    DOI: 10.1007/1-4020-3197-1_20
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