On the Generalized Riesz Derivative

Chenkuan Li and Joshua Beaudin
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Chenkuan Li: Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
Joshua Beaudin: Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada

Mathematics, 2020, vol. 8, issue 7, 1-22

Abstract: The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ? , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n -tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.

Keywords: Riesz derivative; fractional Laplacian; normalization; distribution; Gamma function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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