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On the local fractional derivative |
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| Authors: | Yan Chen Kewei Zhang |
| Institution: | a College of Resource and Environment, China Agricultural University, Beijing 100094, China b School of Mathematical Sciences, Queen Mary College, University of London, London E1 4NS, UK c Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK |
| Abstract: | We present the necessary conditions for the existence of the Kolwankar-Gangal local fractional derivatives (KG-LFD) and introduce more general but weaker notions of LFDs by using limits of certain integral averages of the difference-quotient. By applying classical results due to Stein and Zygmund (1965) 16] we show that the KG-LFD is almost everywhere zero in any given intervals. We generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing. |
| Keywords: | Local fractional derivative Necessary condition Structure theorem Alternative definitions Numerical approximation |
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