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A generalization of the Peano kernel and its applications. (English) Zbl 1411.65027
Summary: Based on the \(q\)-Taylor Theorem, we introduce a more general form of the Peano kernel (\(q\)-Peano) which is also applicable to non-differentiable functions. Then we show that quantum B-splines are the \(q\)-Peano kernels of divided differences. We also give applications to polynomial interpolation and construct examples in which classical remainder theory fails whereas \(q\)-Peano kernel works.
MSC:
65D07 Numerical computation using splines
41A05 Interpolation in approximation theory
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