Quantum symmetric derivatives |
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Authors: | J. Marshall Ash Stefan Catoiu |
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Affiliation: | Department of Mathematics, DePaul University, Chicago, Illinois 60614 ; Department of Mathematics, DePaul University, Chicago, Illinois 60614 |
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Abstract: | For , a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For , symmetrization holds, that is, whenever the th Peano derivative exists at a point, all of these derivatives of order also exist at that point. The main result, desymmetrization, is that conversely, for , each symmetric quantum derivative is a.e. equivalent to the Peano derivative of the same order. For and , each th symmetric quantum derivative coincides with both corresponding th Riemann symmetric quantum derivatives, so, in particular, for and , both th Riemann symmetric quantum derivatives are a.e. equivalent to the Peano derivative. |
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Keywords: | Generalized derivatives quantum derivatives $L^{p}$ derivatives |
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