Fractional Weyl-Riesz Integrodifferentiation of Periodic Functions of Two Variables via the Periodization of the Riesz Kernel

Author: Samko S.G.

Source: Applicable Analysis, Volume 82, Number 3, 2003 , pp. 269-299(31)

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Abstract:

We consider the periodization of the Riesz fractional integrals (Riesz potentials) of two variables and show that already in this case we come across different effects, depending on whether we use the repeated periodization, first in one variable, and afterwards in another one, or the so called double periodization. We show that the naturally introduced doubly-periodic Weyl-Riesz kernel of order 0< <2 in general coincides with the periodization of the Riesz kernel, the repeated periodization being possible for all 0< <2, while the double one is applicable only for 0< <1. This is obtained as a realization of a certain general scheme of periodization, both repeated and double versions. We prove statements on coincidence of the corresponding periodic and nonperiodic convolutions and give an application to the case of the Riesz kernel.

Keywords: Fractional integration; Weyl integration; Periodic convolutions; Periodization

Document Type: Research article

DOI: http://dx.doi.org/10.1080/000368103763875715

Affiliations: 1: Faculdade de Ciencias e Tecnologia, Universidade do Algarve, Campus de Gambelas, Faro 8000, Portugal

Publication date: 2003-01-01

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