The Fourier-Bessel series representation of the pseudo-differential operator (-𝑥⁻¹𝐷)^{𝜈}

Singh, O. P.; Pandey, J. N.

doi:10.1090/S0002-9939-1992-1107924-6

The Fourier-Bessel series representation of the pseudo-differential operator $(-x^ {-1}D)^ \nu$
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by O. P. Singh and J. N. Pandey PDF

Proc. Amer. Math. Soc. 115 (1992), 969-976 Request permission

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 \[ {h_\nu } \circ {({x^{ - 1}}D)^\nu }(\varphi ) = {h_0}(\varphi ),\quad \forall \varphi \in F,\quad \forall \nu \in \mathbb {R};\] (III) ${( - {x^{ - 1}}D)^\nu }$ has a Fourier-Bessel series representation on a subspace ${F_b} \subset F$ and also on its dual ${F’_b}$.

References

L. S. Dube and J. N. Pandey, On the Hankel transform of distributions, Tohoku Math. J. (2) 27 (1975), no. 3, 337–354. MR 450964, DOI 10.2748/tmj/1203529246
James L. Griffith, Hankel transforms of functions zero outside a finite interval, J. Proc. Roy. Soc. New South Wales 89 (1955), 109–115 (1956). MR 77616
Will Y. Lee, On Schwartz’s Hankel transformation of certain spaces of distributions, SIAM J. Math. Anal. 6 (1975), 427–432. MR 361766, DOI 10.1137/0506037
Alan L. Schwartz, An inversion theorem for Hankel transforms, Proc. Amer. Math. Soc. 22 (1969), 713–717. MR 243294, DOI 10.1090/S0002-9939-1969-0243294-0
Om P. Singh, On distributional finite Hankel transform, Appl. Anal. 21 (1986), no. 3, 245–260. MR 840316, DOI 10.1080/00036818608839595
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
A. H. Zemanian, Generalized integral transformations, Pure and Applied Mathematics, Vol. XVIII, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. MR 0423007
A. H. Zemanian, A distributional Hankel transformation, SIAM J. Appl. Math. 14 (1966), 561–576. MR 201930, DOI 10.1137/0114049
A. H. Zemanian, The Hankel transformation of certain distributions of rapid growth, SIAM J. Appl. Math. 14 (1966), 678–690. MR 211211, DOI 10.1137/0114056