Fractional Laplace Operator and Meijer G-function

We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of | x | 2 , or generalized hypergeometric functions of - | x | 2 , multiplied by a solid...

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Bibliographic Details
Published in:Constructive approximation 2017-06, Vol.45 (3), p.427-448
Main Authors: Dyda, Bartłomiej, Kuznetsov, Alexey, Kwaśnicki, Mateusz
Format: Article
Language:English
Subjects:
Analysis
Hypergeometric functions
Mathematics
Mathematics and Statistics
Numerical Analysis
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Summary:We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of | x | 2 , or generalized hypergeometric functions of - | x | 2 , multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator ( 1 - | x | 2 ) + α / 2 ( - Δ ) α / 2 with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper (Dyda et al., Eigenvalues of the fractional Laplace operator in the unit ball, 2015 , arXiv:1509.08533 ).
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-016-9336-4