Abstract
We derive a closed formula for the generating functions of the uniform B-splines. We begin by constructing a PDE for these generating functions starting from the de Boor recurrence. By solving this PDE, we find that we can express these generating functions explicitly as sums of polynomials times exponentials. Using these generating functions, we derive some known identities, including the Schoenberg identity, the two term formula for the derivatives in terms of B-splines of lower degree, and the partition of unity property. We also derive several new identities for uniform B-splines not previously available from classical methods, including formulas for sums and alternating sums, for moments and reciprocal moments, and for Laplace transforms and convolutions with monomials.
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References
Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison Wesley, Reading (1992)
Phillips, G.: Interpolation and Approximation by Polynomials. Canadian Mathematical Society, Springer, New York (2003)
Schoenberg, I.J.: Contributions to the problem of approximation of equidistant data by analytic functions: Part A—On the problem of smoothing or graduation. A first class of analytic approximation formulae. Quarterly of Applied Mathematics 4, 45–99 (1964)
Simsek, Y.: Interpolation function of generalized q-bernstein-type basis polynomials and applications. In: Conference on Mathematical Methods for Curves and Surfaces, Avignon, France (2010)
Simsek, Y.: Generating functions for the Bernstein polynomials: A unified approach to deriving identities for the Bernstein basis functions, arXiv:1012.5538v1 (math.CA)
Simsek, Y.: Functional equations from generating functions: A novel approach to deriving identities for the Bernstein basis functions (2011) (preprint)
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Goldman, R. (2014). Generating Functions for Uniform B-Splines. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2012. Lecture Notes in Computer Science, vol 8177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54382-1_10
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DOI: https://doi.org/10.1007/978-3-642-54382-1_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54381-4
Online ISBN: 978-3-642-54382-1
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