Budakçı, Gülter; Dişibüyük, Çetin; Goldman, Ron; Oruç, Halil Extending fundamental formulas from classical B-splines to quantum B-splines. (English) Zbl 1309.65018 J. Comput. Appl. Math. 282, 17-33 (2015). Summary: We derive a collection of fundamental formulas for quantum B-splines analogous to known fundamental formulas for classical B-splines. Starting from known recursive formulas for evaluation and quantum differentiation along with quantum analogues of the Marsden identity, we derive quantum analogues of the de Boor-Fix formula for the dual functionals, explicit formulas for the quantum B-splines in terms of divided differences of truncated power functions, formulas for computing divided differences of arbitrary functions by quantum integrating certain quantum derivatives of these functions with respect to the quantum B-splines, closed formulas for the quantum integral of the quantum B-splines over their support, and finally a \(1 / q\)-convolution formula for uniform \(q\)-B-splines. Cited in 5 Documents MSC: 65D07 Numerical computation using splines 81P68 Quantum computation 68Q12 Quantum algorithms and complexity in the theory of computing 41A15 Spline approximation 65D25 Numerical differentiation 65D32 Numerical quadrature and cubature formulas Keywords:quantum splines; \(q\)-B-splines; \(h\)-B-splines; divided differences; quantum derivatives; quantum integrals; de Boor-Fix formula PDF BibTeX XML Cite \textit{G. Budakçı} et al., J. Comput. Appl. Math. 282, 17--33 (2015; Zbl 1309.65018) Full Text: DOI References: [1] Mangasarian, O.; Schumaker, L., Discrete splines via mathematical programming, SIAM J. Control, 9, 174-183, (1971) · Zbl 0223.65004 [2] Mangasarian, O.; Schumaker, L., Best summation formula and discrete splines, SIAM J. Numer. Anal., 10, 448-459, (1973) · Zbl 0256.65003 [3] Lyche, T., Discrete polynomial spline approximation methods, (1975), Department of Mathematics, University of Texas Austin, (Ph.D. thesis) [4] Simeonov, P.; Goldman, R., Quantum B-splines, BIT, 53, 193-223, (2013) · Zbl 1275.65011 [5] Kac, V.; Cheung, P., (Quantum Calculus, Universitext Series, vol. IX, (2002), Springer Verlag) · Zbl 0986.05001 [6] Goldman, R., Pyramid algorithms: A dynamic programming approach to curves and surfaces for geometric modeling, (2002), Morgan Kaufmann Publishers, Academic Press San Diego [7] Ernst, T., A comprehensive treatment of \(q\)-calculus, (2012), Springer, Birkhäuser · Zbl 1256.33001 [8] de Boor, C., B-spline basics, (Piegl, Les, Fundamental Developments of Computer-Aided Geometric Modeling, (1993), Academic Press London), 27-49 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.