Vol. 37, No. 2, 1971

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ISSN: 0030-8730
Inversion of the Hankel potential transform

Frank Michael Cholewinski and Deborah Tepper Haimo

Vol. 37 (1971), No. 2, 319–330
Abstract

We consider the Hankel potential transform

            ∫
∞-----t-----
(1.1) f (x) =  0 (x2 +t2)ν+1ϕ(t)dμ (t),ν > 0,

where

                    1
(1.2)  dμ(x) =-ν−1∕2---------x2ν dx.
2     Γ (ν + 1∕2)

This transform is intimately related to the Hankel transform. Indeed, its kernel is the Hankel transform

          t             √π-     ∫ ∞      − tx
(1.3)  (x2-+-t2)ν+1-= 2ν+1∕2Γ (ν-+-1)  𝒥 (xy)e  dμ (y),
0

where

𝒥 (z) = 2v−1∕2Γ (ν + 12)z−ν+1∕2Jν−1∕2(z),

Jγ(z) being the ordinary Bessel function of order γ. Our object is to develop an inversion theory for (1.1) and to exploit the relationship of (1.1) to the Hankel transform.

Mathematical Subject Classification 2000
Primary: 44A15
Milestones
Received: 2 November 1970
Revised: 13 January 1971
Published: 1 May 1971
Authors
Frank Michael Cholewinski
Deborah Tepper Haimo