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Dual polynomial bases. (English) Zbl 0824.41007
The author has presented techniques for extending M. J. Marsden’s identity [J. Approximation Theory 3, 7-49 (1970; Zbl 0192.421)] the blossoming and de Boor-Fix forms of the dual functionals, the Oslo algorithm, and recursive procedures for evaluation, differentiation and blossoming from $$B$$-spline and progressive polynomial bases to arbitrary polynomial and locally linearly independent spline bases. He first generalizes Marsden’s identity and then derives the other formulas and procedures as consequence of this basic identity for working with the dual bases. He also gives some concrete examples to flesh out the details of the theory, and extends his results from polynomial bases to locally linearly independent spline bases. He further obtains necessary and sufficient conditions for extending the blossoming form of the dual functionals to multivariate polynomial bases. Finally he gives a brief summary of his work and raises a few open questions.

MSC:
 41A15 Spline approximation 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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