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arXiv · 2101.03629

A series representation of the discrete fractional Laplace operator of arbitrary order

Tiffany Frugé Jones·Evdokiya Georgieva Kostadinova·Joshua Lee Padgett·Qin Sheng
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Abstract

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.

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Tiffany Frugé Jones, Evdokiya Georgieva Kostadinova, Joshua Lee Padgett, Qin Sheng. 2021-01-10. A series representation of the discrete fractional Laplace operator of arbitrary order. https://arxiv.org/abs/2101.03629

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