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Extending fundamental formulas from classical B-splines to quantum B-splines. (English) Zbl 1309.65018
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.

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
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