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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Fourier-Bessel series representation of the pseudo-differential operator $ (-x\sp {-1}D)\sp \nu$


Authors: O. P. Singh and J. N. Pandey
Journal: Proc. Amer. Math. Soc. 115 (1992), 969-976
MSC: Primary 46F12; Secondary 26A33, 47G30
MathSciNet review: 1107924
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Abstract: For a certain Fréchet space $ F$ consisting of complex-valued $ {C^\infty }$ functions defined on $ I = (0,\infty )$ and characterized by their asymptotic behaviour near the boundaries, we show that:

(I) The pseudo-differential operator $ {( - {x^{ - 1}}D)^\nu },\nu \in \mathbb{R},D = d/dx$, is an automorphism (in the topological sense) on $ F$;

(II) $ {( - {x^{ - 1}}D)^\nu }$ is almost an inverse of the Hankel transform $ {h_\nu }$ in the sense that

$\displaystyle {h_\nu } \circ {({x^{ - 1}}D)^\nu }(\varphi ) = {h_0}(\varphi ),\quad \forall \varphi \in F,\quad \forall \nu \in \mathbb{R};$

(III) $ {( - {x^{ - 1}}D)^\nu }$ has a Fourier-Bessel series representation on a subspace $ {F_b} \subset F$ and also on its dual $ {F'_b}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1107924-6
PII: S 0002-9939(1992)1107924-6
Keywords: Pseudo-differential operator, Hankel transform of distributions, open mapping theorem, almost inverse
Article copyright: © Copyright 1992 American Mathematical Society