2 September 2015 Relationship between fractional calculus and fractional Fourier transform
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Abstract
The fractional calculus (FC) deals with integrals and derivatives of arbitrary (i.e., non-integer) order, and shares its origins with classical integral and differential calculus. The fractional Fourier transform (FRFT), which has been found having many applications in optics and other areas, is a generalization of the usual Fourier transform. The FC and the FRFT are two of the most interesting and useful fractional areas. In recent years, it appears many papers on the FC and FRFT, however, few of them discuss the connection of the two fractional areas. We study their relationship. The relational expression between them is deduced. The expectation of interdisciplinary cross fertilization is our motivation. For example, we can use the properties of the FC (non-locality, etc.) to solve the problem which is difficult to be solved by the FRFT in optical engineering; we can also through the physical meaning of the FRFT optical implementation to explain the physical meaning of the FC. The FC and FRFT approaches can be transposed each other in the two fractional areas. It makes that the success of the fractional methodology is unquestionable with a lot of applications, namely in nonlinear and complex system dynamics and image processing.
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Yanshan Zhang, Yanshan Zhang, Feng Zhang, Feng Zhang, Mingfeng Lu, Mingfeng Lu, } "Relationship between fractional calculus and fractional Fourier transform", Proc. SPIE 9596, Signal and Data Processing of Small Targets 2015, 95960A (2 September 2015); doi: 10.1117/12.2187649; https://doi.org/10.1117/12.2187649
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