Abstract
Differential operators are usually used to determine the rate of change and the direction of change of a signal modeled by a function in some appropriately selected function space. Gibbs derivatives are introduced as operators permitting differentiation of piecewise constant functions. Being initially intended for applications in Walsh dyadic analysis, they are defined as operators having Walsh functions as eigenfunctions. This feature was used in different generalizations and extensions of the concept firstly defined for functions on finite dyadic groups. In this paper, we provide a brief overview of the evolution of this concept into a particlar class of differential operators for functions on various groups.
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Stanković, R.S., Butzer, P.L., Schipp, F., Wade, W.R., Su, W., Endow, Y., Fridli, S., Golubov, B.I., Pichler, F., Onneweer, K.C.W.: Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science Volume 1, Foundations. A Monograph Based on Articles of the Founding Authors, Reproduced in Full. Atlantis Studies in Mathematics for Engineering and Science, Vol. 12. Atlantis Press, Springer (2015)
Stanković, R.S., Butzer, P.L., Schipp, F., Wade, W.R., Su, W., Endow, Y., Fridli, S., Golubov, B.I., Pichler, F., Onneweer, K.C.W.: Dyadic Walsh Analysis from 1924 Onwards, Walsh-Gibbs-Butzer Dyadic Differentiation in Science, Volume 2, Extensions and Generalizations. A Monograph Based on Articles of the Founding Authors, Reproduced in Full, Atlantis Studies in Mathematics for Engineering and Science. Atlantis Press (2015)
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Stanković, R.S., Astola, J., Moraga, C. (2018). Gibbs Dyadic Differentiation on Groups - Evolution of the Concept. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2017. EUROCAST 2017. Lecture Notes in Computer Science(), vol 10672. Springer, Cham. https://doi.org/10.1007/978-3-319-74727-9_27
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DOI: https://doi.org/10.1007/978-3-319-74727-9_27
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