An extension of the dyadic calculus with fractional order derivatives. Further theory and applications.

*(English)*Zbl 0661.42018In their second paper dealing with the extended dyadic derivative the authors complete the theory by adding operational rules for their so- called ED-derivative, e.g. a Leibnizian rule valid for Walsh functions.

They also calculate the ED-derivatives for a couple of examples: piecewise constant functions, polynomials, piecewise polynomials given as a product of a monomial and a Walsh function, Walsh and Rademacher series.

Furthermore they investigate applications in Walsh-Fourier analysis and the theory of best approximation. Finally, they deal with differential equations, especially solve the dyadic wave equation. Although not all results are in strict analogy to classical calculus, the present investigations might help to answer the questions, in which of the modern fields - like digital communication or pattern recognition - dyadic derivatives will find their interpretation.

They also calculate the ED-derivatives for a couple of examples: piecewise constant functions, polynomials, piecewise polynomials given as a product of a monomial and a Walsh function, Walsh and Rademacher series.

Furthermore they investigate applications in Walsh-Fourier analysis and the theory of best approximation. Finally, they deal with differential equations, especially solve the dyadic wave equation. Although not all results are in strict analogy to classical calculus, the present investigations might help to answer the questions, in which of the modern fields - like digital communication or pattern recognition - dyadic derivatives will find their interpretation.

Reviewer: W.Splettstößer

##### MSC:

42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |

26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |

26A36 | Antidifferentiation |

40G05 | Cesàro, Euler, Nörlund and Hausdorff methods |

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\textit{P. L. Butzer} et al., Comput. Math. Appl., Part A 12, 921--943 (1986; Zbl 0661.42018)

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##### References:

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